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\title{\textbf{The Riemann Hypothesis on Young's Lattice}}
\author{José Damián Espinosa}
\date{}
\raggedbottom
\begin{document}

\begin{abstract}
This work establishes a novel structural connection between the Riemann
Hypothesis and Young's Lattice through partition-indexed harmonic numbers and
the Generalized Assembly Theorem. We establish a new family of equivalences for the conjecture,
define the partition-based harmonic numbers $H_n^{\boldsymbol{\lambda}}$, and
reformulate the Riemann Hypothesis as the Young's Lattice equivalence:
$$\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k
 >  \sum_{d \mid n} d$$ for every $n \in \mathbb{N}$, where $\boldsymbol{\lambda} \vdash k$ denotes a partition of $k$,
$\boldsymbol{\lambda}!$ is the partition factorial, and
$a_k = \frac{1}{k} + \dots + \frac{1}{k^2}$ or $a_k=\log(k+1)$. We then prove the Generalized
Assembly Theorem, which shows that the assembly sum
$\mathfrak{S}_n^m = \sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in
\wp_k^{k+m-1}} H_n^{\boldsymbol{\lambda}}$ equals $n$ for every excess
parameter $m-1$, providing a systematic framework to analyze divisor bounds
through the branch decomposition of the partition-indexed series. The
resulting structure yields an exact, non-asymptotic realization of the divisors
of $n$. Finally, we propose and explore a hierarchical
expansion connecting the exponential of the Euler--Mascheroni constant,
$e^\gamma$, to Multiple Zeta Values under Abel summability. $$e^\gamma = 1 +
\sum_{a\ge2}
\frac1{a!}
\sum_{m\ge0}
(-1)^m\zeta^{\mathrm{A}}(a+m)
+
\sum_{a,b\ge2}
\frac1{a!\,b!}
\sum_{m,n\ge0}
(-1)^{m+n}\zeta^{\mathrm{A}}(a+m,b+n)
+
\cdots$$
\end{abstract}

\maketitle

\section{Introduction}
\label{sec:intro}

The Riemann Hypothesis, first proposed by Bernhard Riemann in his foundational
1859 memoir \cite{riemann1859}, asserts that all non-trivial zeros of the
Riemann zeta function $\zeta(s)$ lie on the critical line
$\operatorname{Re}(s) = \frac{1}{2}$. Its resolution would have profound
consequences for the distribution of prime numbers and, more broadly, for
analytic number theory.

Since a direct attack on the conjecture in its original form has proven
extremely difficult, a significant body of research has been dedicated to
establishing elementary equivalent formulations. These translate the problem
from the complex plane into inequalities involving classical arithmetic
functions, most notably the sum-of-divisors function
$\sum_{d \mid n} d$ and the harmonic numbers
$H_n = \sum_{i=1}^{n} \frac{1}{i}$.
A landmark achievement in this direction is due to Guy Robin \cite{robin},
who characterized the conjecture through the maximal growth rate of
$\sum_{d \mid n} d$: the Riemann Hypothesis holds if and only if
\begin{equation}\label{eq:robin-intro}
\sum_{d \mid n} d \ < \ e^{\gamma} \, n \log \log n
\qquad \text{for every } n \ge 5041,
\end{equation}
where $\gamma$ is the Euler--Mascheroni constant, and if the hypothesis fails,
then the reverse inequality holds for infinitely many $n$ with a quantitative
error term. Complementing this framework, Jeffrey C. Lagarias \cite{lagarias}
proved that the Riemann Hypothesis is equivalent to the inequality
\begin{equation}\label{eq:lagarias-intro}
\sum_{d \mid n} d \ \le \ H_n + e^{H_n} \log H_n
\qquad \text{for every } n \in \mathbb{N},
\end{equation}
and, equivalently, to
\begin{equation}\label{eq:lagarias-intro-small}
\sum_{d \mid n} d \ < \ e^{H_n} \log H_n
\qquad \text{for every } n > 60.
\end{equation}
Both criteria express the hypothesis as an elementary inequality involving
the sum-of-divisors function.
The present work compiles and offers a unified presentation of all the results previously established and rigorously detailed across a series of ten foundational papers, with \cite{espinosa-part1, espinosa-part2, espinosa-part3, espinosa-part4, espinosa-part5, espinosa-part6, espinosa-part7} published between December 2025 and January 2026, and \cite{espinosa-part8a,espinosa-part8b,espinosa-part8c} published between June 2026 and July 2026.
We begin with a new elementary equivalence of the Riemann
Hypothesis, expressed as an inequality involving the sum-of-divisors function
and a partition-indexed series. The starting point is the Espinosa--Riemann
Equivalence Theorem (Theorem~\ref{teo:damian_final}) established in \cite{espinosa-part2}, which reduces the
conjecture to an inequality of the form
\begin{equation}\label{eq:espinosa-intro}
\sum_{k \in \mathbb{N}} \frac{H_n^k}{k!} \, a_k \ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N},
\end{equation}
for an admissible sequence $\{a_k\}_{k \ge 1}$ satisfying three explicit
conditions: a prescribed asymptotic behaviour of the associated generating
function, a lower bound $a_k > H_k - \gamma$, and a finite base validation for
$1 \le n \le 60$. The central contribution of all these works is the reformulation of
\eqref{eq:espinosa-intro} in the language of integer partitions and Young's Lattice. Using the identity
\begin{equation}\label{eq:partition-expansion-intro}
\frac{H_n^k}{k!} \ = \
\sum_{\boldsymbol{\lambda} \, \vdash \, k}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!},
\end{equation}
where $H_n^{\boldsymbol{\lambda}}$ denotes the generalized harmonic number
associated with a partition $\boldsymbol{\lambda} \vdash k$ and
$\boldsymbol{\lambda}!$ is the partition factorial, one obtains the Riemann
Hypothesis Young's Lattice Equivalence: the hypothesis holds if and only if
\begin{equation}\label{eq:young-lattice-intro}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k
\ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N}.
\end{equation}
The passage from \eqref{eq:espinosa-intro} to \eqref{eq:young-lattice-intro}
is a direct application of \eqref{eq:partition-expansion-intro}; the content
of the equivalence lies in the combinatorial structure that emerges from the
partition decomposition, not in the algebraic manipulation itself.

In particular, two of the explicit equivalences stated in
Section~\ref{subsec:new_eq} can be rewritten in this partition-based form in
a particularly elegant way. The first, with logarithmic
coefficients, was established in \cite{espinosa-part3}, while the
second, with fully explicit harmonic coefficients, was established in
\cite{espinosa-part1}. For the logarithmic coefficients $a_k =
\log(k+1)$ and for the fully explicit coefficients
$a_k = \frac{1}{k} + \cdots + \frac{1}{k^2}$, the corresponding
partition-indexed inequalities read
\begin{equation}\label{eq:young-lattice-log1}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \log(k+1)
\ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N},
\end{equation}
\begin{equation}\label{eq:young-lattice-harm}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!}
\left( \tfrac{1}{k} + \cdots + \tfrac{1}{k^2} \right)
\ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N},
\end{equation}
respectively. Both inequalities are equivalent to the Riemann Hypothesis, and
both are obtained by the same direct substitution
\eqref{eq:partition-expansion-intro} into the corresponding logarithmic and
harmonic forms of the equivalence. This shows that the partition-based
reformulation is not only a restatement of the original result, but a
systematic framework that accommodates the different realizations of the
coefficients $a_k$ in a unified way.
The structural analysis of \eqref{eq:young-lattice-intro} is carried out
through the Generalized Assembly Theorem
(Theorem~\ref{teo:generalized_ensemble}), which extends the classical
Assembly Theorem to arbitrary summation intervals. For all
$n, m, s \in \mathbb{N}$ with $s \le n$, the generalized assembly sum
satisfies
\begin{equation}\label{eq:assembly-intro}
\mathfrak{S}_{s,n}^{m} \ := \
\sum_{k=s}^{n} \ \sum_{\boldsymbol{\lambda} \in \wp_{k-s+1}^{k-s+m}}
H_{s,n}^{\boldsymbol{\lambda}}
\ = \ \frac{n - s + 1}{s^m},
\end{equation}
where $\wp_{k-s+1}^{k-s+m}$ denotes the Assembly Set of all partitions of
$k - s + 1 + m$ into exactly $k - s + 1$ parts, and
$H_{s,n}^{\boldsymbol{\lambda}}$ is the generalized harmonic number over the
restricted index range $[s, n]$. In the particular case $s = 1$, the identity
reduces to the classical Assembly Theorem,
\begin{equation}\label{eq:classical-assembly-intro}
\mathfrak{S}_{n}^{m} \ = \ n
\qquad \text{for every } m \in \mathbb{N},
\end{equation}
revealing a hidden invariance: independently of the excess parameter $m-1$, the
ensemble of harmonic weights always reconstructs the count of indices
$\{1, \dots, n\}$.

\begin{figure}[H]
    \centering
    \includegraphics[width=1\linewidth]{1.pdf}
    \caption{The truncated Young's Lattice up to rank $N = 11$, restricted
    to partitions with at most $n = 6$ parts. Each node is a partition
    $\boldsymbol{\lambda} \vdash_{j} j + m - 1$ belonging to the Assembly Set
    $\wp_{j}^{j+m-1}$, drawn as a transposed Young diagram: the number of
    columns equals the number of summands $j = \ell(\boldsymbol{\lambda})$,
    and the height of each column equals the value of the corresponding part.
    The nodes are colored by the excess parameter $m - 1$,
    and the leaders of each level, that is, the partitions without any part
    equal to $1$ together with the partition $(1)$, are highlighted with a
    double rectangular frame. The edges in marker style trace the chains of
    partitions that remain within the same $m$ as $j$ increases in the Assembly Set $\wp_{j}^{j+m-1}$.
    This is precisely the combinatorial structure that underlies the
    Generalized Assembly Theorem~\ref{teo:generalized_ensemble}: each level
    $m$ corresponds to one row of the lattice, and the accumulated
    contribution of all levels reconstructs the value
    $\frac{n - s + 1}{s^m}$ that appears in
    equation~\eqref{eq:assembly-theorem}.}
    \label{fig:placeholder1}
\end{figure}

The Generalized Assembly Theorem provides the combinatorial mechanism through
which the divisors of $n$ are assembled by the partition-indexed series. Each
level $m$ of the assembly contributes exactly $n$ to the total sum, so that
the series \eqref{eq:young-lattice-intro} can be re-indexed by the excess
parameter $m-1$:
\begin{equation}\label{eq:levels-intro}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k
\ = \
\sum_{m \ge 1} \left( \sum_{k=1}^{n}
\sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_{k+m-1} \right).
\end{equation}
The family $\wp_k^{k+m-1}$ corresponds to the partitions of $k + m - 1$ into
exactly $k$ parts, and each level $m$ reconstructs the count of indices
$\{1, \dots, n\}$ via the Assembly Theorem. Since $a_k \ge 1$ for every
$k \in \mathbb{N}$ and all terms are positive, the first two levels $m = 1, 2$
already yield the coarse bound
\begin{equation}\label{eq:coarse-intro}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k
\ > \
\frac{n}{1!} + \frac{n}{2!},
\end{equation}
and the accumulated contributions of the remaining levels $m \ge 3$ refine
this bound. The realization $\frac{n}{\boldsymbol{\Lambda}!} = d$ reconstructs
$\sum_{d \mid n} d$ exactly, not asymptotically. It allows for the successive extraction of the divisors of the \enquote{peaks numbers}. Since the most general and natural \enquote{peak numbers}, are the highly abundant numbers $n$, for which
$\sum_{d \mid n} d$ exceeds the value of $\sum_{d \mid m} d$ for every
$m < n$, the points of maximal value of the divisor function, it
suffices to verify the inequality at those integers.
\section{Main Framework}
\label{sec:framework}

\subsection{A Theorem for a new family of equivalences of the Riemann Hypothesis}\label{sec:nueva-version}

\subsubsection{On $\sum_{k \in \mathbb{N}} (H_k - \gamma) \frac{x^k}{k!}$}\label{sec:series1}

We first establish the following, which will provide the analytic backbone for the equivalence theorem.

\begin{proposition}\label{prop:digamma-series}
For every $x \in \mathbb{R}^+$,
\begin{equation}\label{eq:digamma-series}
\sum_{k \in \mathbb{N}_0} \psi(k + 1) \frac{x^k}{k!}  \ = \ e^x \log x + E_1(x),
\end{equation}
where $\psi(k) = \frac{\Gamma'(k)}{\Gamma(k)}$ and $E_1(x) = \int_x^{\infty} \tfrac{e^{-t}}{t}\,dt$.
\end{proposition}
\begin{proof}
See \cite[Ch.~III, p.~37]{dingle}.
\end{proof}

\begin{proposition}\label{prop:digamma-harmonic}
For every $n \in \mathbb{Z}_{\ge 1}$, $\psi(n) = -\gamma + H_{n-1}$.
\end{proposition}
\begin{proof}
See \cite[Eq. 6.3.1, p.~258]{abramowitz}.
\end{proof}

Then a two-sided bound of $e^{x} E_1(x)$:

\begin{proposition}[Bounds for the Exponential Integral]\label{prop:expint-bounds}
For every $x \in \mathbb{R}^+$ with $x \neq 0$,
\begin{equation}\label{eq:expint-bounds}
\tfrac{1}{2}\log\!\left(1 + \tfrac{2}{x}\right) \ < \ e^{x} E_1(x) \ < \ \log\!\left(1 + \tfrac{1}{x}\right),
\end{equation}
where $E_1(x) = \int_x^{\infty} \tfrac{e^{-t}}{t}\,dt$.
\end{proposition}
\begin{proof}
See \cite[Eq. 5.1.20, p.~229]{abramowitz}.
\end{proof}

Finally, combining the previous three results we obtain a two-sided bound and the asymptotic expansion of the series $\sum_{k \in \mathbb{N}} (H_k - \gamma) \frac{x^k}{k!}$. 

\begin{proposition}\label{prop:serie-modificada}
For every $x \in \mathbb{R}^+$,
\begin{equation}\label{eq:serie-modificada}
\sum_{k \in \mathbb{N}} (H_k - \gamma) \frac{x^k}{k!}  \ = \ e^x \log x + \gamma + \mathcal{O}\!\left(\frac{1}{x}\right) \qquad (x \to \infty).
\end{equation}
\end{proposition}
\begin{proof}
By Proposition~\ref{prop:digamma-harmonic}, $H_k - \gamma = \psi(k+1)$ for $k \ge 1$, so
\[
\sum_{k\ge1}(H_k - \gamma)\frac{x^k}{k!} \ = \ \sum_{k\ge0}\psi(k+1)\frac{x^k}{k!} - \psi(1) \ = \ e^{x}\log x + E_1(x) + \gamma,
\]
using Proposition~\ref{prop:digamma-series} and $\psi(1) = -\gamma$. Since $E_1(x) = \mathcal{O}(\frac{e^{-x}}{x})$ as $x \to \infty$ \cite[Eq.~5.1.51, p.~231]{abramowitz}, the claim follows.
\end{proof}

\begin{proposition}\label{prop:cotas-serie-modificada}
For every $x \in \mathbb{R}^+$,
\begin{equation}\label{eq:cotas-serie-modificada}
e^x \log x + \tfrac{1}{2} \log\!\left(1 + \tfrac{2}{x}\right) + \gamma
\ < \ \sum_{k \in \mathbb{N}} (H_k - \gamma) \frac{x^k}{k!} \
< \ e^x \log x + \log\!\left(1 + \tfrac{1}{x}\right) + \gamma.
\end{equation}
\end{proposition}
\begin{proof}
From the proof of Proposition~\ref{prop:serie-modificada},
$\sum_{k\ge1}(H_k-\gamma)\frac{x^k}{k!} = e^{x}\log x + e^{x}E_1(x) + \gamma$.
Substituting the two-sided bound \eqref{eq:expint-bounds} of Proposition~\ref{prop:expint-bounds} for $e^{x}E_1(x)$ yields \eqref{eq:cotas-serie-modificada} verbatim.
\end{proof}
\subsubsection{On elementary versions of the Riemann Hypothesis}\label{sec:equivalencias}

We now recall the two criteria that translate the Riemann Hypothesis into elementary inequalities involving the sum-of-divisors function. These will serve as the arithmetic bridge between the analytic results of the previous section and the equivalence theorem below.

\begin{theorem}[Lagarias, 2002]\label{teo:lagarias}
The \textbf{Riemann Hypothesis} is equivalent to
\begin{equation}\label{eq:lagarias-main}
\sum_{d \mid n} d \ \le \ H_n + e^{H_n} \log H_n \qquad \text{for every } n \in \mathbb{N},
\end{equation}
where $H_n = \sum_{i=1}^{n} \frac{1}{i}$.
Equivalently, the \textbf{Riemann Hypothesis} is equivalent to
\begin{equation}\label{eq:lagarias-small}
\sum_{d \mid n} d \ < \ e^{H_n} \log H_n \qquad \text{for every } n > 60.
\end{equation}
\end{theorem}
\begin{proof}
See \cite[Thm.~1.1 and Cor.~1.2]{lagarias}.
\end{proof}

\begin{theorem}[Robin, 1984]\label{teo:robin1}
The following two statements hold.
\begin{enumerate}[label=\textup{(\roman*)}]
    \item If the \textbf{Riemann Hypothesis} is true, then
    \begin{equation}\label{eq:robin-rh}
    \sum_{d \mid n} d \ < \ e^{\gamma} \, n \log \log n \qquad \text{for every } n \ge 5041 .
    \end{equation}
    \item If the \textbf{Riemann Hypothesis} is false, then
    \begin{equation}\label{eq:robin-norh}
    \sum_{d \mid n} d \ \ge \ e^{\gamma} \, n \log \log n
    \end{equation}
    for infinitely many $n \in \mathbb{N}$.
\end{enumerate}
\end{theorem}
\begin{proof}
See \cite[Thm.~1 and Thm.~2, pp.~192--196]{robin}.
\end{proof}

\begin{theorem}[Robin, 1984, quantitative form]\label{teo:robin2}
If the \textbf{Riemann Hypothesis} is false, then there exist absolute constants
$C > 0$ and $\beta \in (0, \tfrac{1}{2})$ such that
\begin{equation}\label{eq:robin-quant}
\sum_{d \mid n} d \ \ge \ e^{\gamma} \, n \log \log n \ + \ \frac{C \, n \log \log n}{(\log n)^{\beta}}
\end{equation}
for infinitely many $n \in \mathbb{N}$.
\end{theorem}
\begin{proof}
See \cite[Thm.~2, p.~194 and \S4, pp.~201--207]{robin}.
\end{proof}
\subsubsection{On the Espinosa-Riemann equivalence theorem}\label{sec:espinosa-riemann}

Given the preceding sections, and the two preceding families of criteria wich reduce the Riemann Hypothesis to inequalities involving the sum-of-divisors function. We now present a general framework, in the spirit of these criteria, that produces new elementary equivalences in a systematic and compact way. The starting point is the following auxiliary result.

\begin{proposition}\label{prop:damian_inequality}
Let $\mathfrak{T}: \mathbb{R}^+ \to \mathbb{R}$ be a function such that, for every
$k \in \mathbb{N}$,
\begin{equation}\label{eq:T-lower}
\mathfrak{T}(k) \ > \ H_k - \gamma,
\end{equation}
and let $\mathfrak{O}: \mathbb{R}^+ \to \mathbb{R}$ be defined by the everywhere convergent series
\begin{equation}\label{eq:O-def}
\mathfrak{O}(x) \ := \ \sum_{k \in \mathbb{N}} \frac{x^k}{k!}\, \mathfrak{T}(k).
\end{equation}
Then, for every $x \in \mathbb{R}^+$,
\begin{equation}\label{eq:O-lower}
\mathfrak{O}(x) \ > \ e^x \log x.
\end{equation}
\end{proposition}
\begin{proof}
Fix $x > 0$. By \eqref{eq:T-lower}, multiplying both sides by the positive
quantity $\frac{x^k}{k!}$ and summing over $k \in \mathbb{N}$ gives
\[
\sum_{k \in \mathbb{N}} (H_k - \gamma)\frac{x^k}{k!} \ < \ \mathfrak{O}(x),
\]
in view of \eqref{eq:O-def}. Proposition~\ref{prop:cotas-serie-modificada} provides the lower bound
\[
e^x \log x + \tfrac{1}{2}\log\!\left(1 + \tfrac{2}{x}\right) + \gamma
\ < \ \sum_{k \in \mathbb{N}} (H_k - \gamma)\frac{x^k}{k!}.
\]
Combining both inequalities,
\[
e^x \log x + \tfrac{1}{2}\log\!\left(1 + \tfrac{2}{x}\right) + \gamma \ < \ \mathfrak{O}(x).
\]
Since $\tfrac{1}{2}\log(1 + \tfrac{2}{x}) > 0$ for all $x > 0$ and $\gamma > 0$, the
additional terms on the left are strictly positive. Hence
$e^x \log x < \mathfrak{O}(x)$.
\end{proof}

With this auxiliary inequality in hand, we can now state the central result of section \ref{sec:espinosa-riemann}. The following theorem in \cite{espinosa-part2} is stated in terms of a function $\mathfrak{O}$ satisfying three explicit conditions, and it produces a new family of equivalences of the Riemann Hypothesis by varying the choice of $\mathfrak{T}$.

\begin{theorem}[Espinosa, 2025, Espinosa--Riemann Equivalence Theorem]\label{teo:damian_final}
The \textbf{Riemann Hypothesis} is equivalent to
\begin{equation}\label{eq:espinosa-main}
\mathfrak{O}(H_n) \ = \ \sum_{k \in \mathbb{N}} \frac{H_n^k}{k!}\,\mathfrak{T}(k) \ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N},
\end{equation}
where $H_n = \sum_{i=1}^{n} \frac{1}{i}$ and the functions
$\mathfrak{O}, \mathfrak{T} : \mathbb{R}^+ \to \mathbb{R}$ satisfy:
\begin{enumerate}[label=\textup{(\roman*)}]
    \item \textbf{Asymptotic behaviour of $\mathfrak{O}$:}
    \begin{equation}\label{eq:O-asymp}
    \mathfrak{O}(x) \ = \ e^x \log x + \mathcal{O}\!\left(\frac{e^x}{x}\right)
    \qquad (x \to \infty).
    \end{equation}
    \item \textbf{Inner condition on $\mathfrak{T}$:}
    \begin{equation}\label{eq:T-cond}
    \mathfrak{T}(k) \ > \ H_k - \gamma \qquad \text{for every } k \in \mathbb{N}.
    \end{equation}
    \item \textbf{Base validation:} Inequality \eqref{eq:espinosa-main} holds for every
    integer $1 \le n \le 60$.
\end{enumerate}
\end{theorem}
\begin{proof}
\textbf{(Riemann Hypothesis $\Rightarrow$ inequality \eqref{eq:espinosa-main}).}
Assume Riemann Hypothesis. By Theorem~\ref{teo:lagarias},
\[
\sum_{d \mid n} d \ < \ e^{H_n} \log H_n \qquad \text{for every } n > 60.
\]
Condition \eqref{eq:T-cond} and Proposition~\ref{prop:damian_inequality} give
$\mathfrak{O}(x) > e^x \log x$ for all $x \in \mathbb{R}^+$. Evaluating at
$x = H_n$ yields
\[
\sum_{d \mid n} d \ < \ e^{H_n} \log H_n \ < \ \mathfrak{O}(H_n).
\]
Together with the base validation, \eqref{eq:espinosa-main} holds for every
$n \in \mathbb{N}$.

\smallskip
\textbf{(Inequality \eqref{eq:espinosa-main} $\Rightarrow$ Riemann Hypothesis).}
Suppose \eqref{eq:espinosa-main} holds for every $n \in \mathbb{N}$ but Riemann Hypothesis is
false. By Theorem~\ref{teo:robin2}, there exist absolute constants $C > 0$ and
$\beta \in (0, \tfrac{1}{2})$ such that
\begin{equation}\label{eq:robin-quant-local}
\sum_{d \mid n} d \ \ge \ e^{\gamma} n \log \log n  +  \frac{C \, n \log \log n}{(\log n)^{\beta}}
\qquad \text{for infinitely many } n \in \mathbb{N}.
\end{equation}
On the other hand, by \eqref{eq:O-asymp} there exist $M > 0$ and $n_0 \in \mathbb{N}$
such that, for all $n \ge n_0$,
\[
\mathfrak{O}(H_n) \ \le \ e^{H_n} \log H_n  +  M \, \frac{e^{H_n}}{H_n}.
\]
Combined with \eqref{eq:espinosa-main}, this gives, for every $n \ge n_0$,
\[
\sum_{d \mid n} d \ < \ e^{H_n} \log H_n  +  M \, \frac{e^{H_n}}{H_n}.
\]
Lagarias' elementary bounds \cite[Ineqs.~(3.11), (3.9), (3.5)]{lagarias}
state that, for $n \ge 3$,
\begin{align}
e^{H_n} \log H_n \ &\le \ e^{\gamma} n \log \log n + \frac{3 e^{\gamma} n}{2 \log n},
\label{eq:lag-1}\\
e^{H_n} \ &\le \ e^{\gamma}(n + 1),
\label{eq:lag-2}\\
H_n \ &> \ \log n + \gamma \ > \ \log n.
\label{eq:lag-3}
\end{align}
From \eqref{eq:lag-3}, $\frac{1}{H_n}<\frac{1}{\log n}$. Multiplying by \eqref{eq:lag-2} and by
$M > 0$ yields
\begin{equation}\label{eq:lag-4}
M \, \frac{e^{H_n}}{H_n} \ < \ M \, e^{\gamma} \, \frac{n + 1}{\log n}
\ \le \ \frac{2 M e^{\gamma} n}{\log n} \qquad (n \ge 1).
\end{equation}
Adding \eqref{eq:lag-1} and \eqref{eq:lag-4}, we obtain, for
$n \ge n_1 := \max\{n_0, 3\}$,
\[
\sum_{d \mid n} d \ < \ e^{\gamma} n \log \log n  + \frac{3 e^{\gamma} n}{2 \log n}
 +  \frac{2 M e^{\gamma} n}{\log n}
\ = \ e^{\gamma} n \log \log n  +  S \, \frac{n}{\log n},
\]
where $S := \tfrac{3 e^{\gamma}}{2} + 2 M e^{\gamma} > 0$ is a constant.
Since $\beta < \tfrac{1}{2}$, we have $(\log n)^{-\beta} > (\log n)^{-\frac{1}{2}}$ for
$n \ge 3$, and
$C \, n \log \log n \, (\log n)^{-\frac{1}{2}} > S \, \frac{n}{\log n}$ for all sufficiently
large $n$. Hence, for $n \ge n_2$,
\[
\sum_{d \mid n} d \ < \ e^{\gamma} n \log \log n  +  \frac{C \, n \log \log n}{(\log n)^{\frac{1}{2}}}.
\]
Equivalently, the reverse inequality
\[
\sum_{d \mid n} d \ \ge \ e^{\gamma} n \log \log n  + \frac{C \, n \log \log n}{(\log n)^{\frac{1}{2}}}
\]
can hold only for finitely many $n$, namely $1 \le n < n_2$.
But by \eqref{eq:robin-quant-local} and $\beta < \tfrac{1}{2}$,
\[
\sum_{d \mid n} d \ > \ e^{\gamma} n \log \log n  + \frac{C \, n \log \log n}{(\log n)^{\frac{1}{2}}}
\qquad \text{for infinitely many } n \in \mathbb{N},
\]
which contradicts the finiteness just established. Therefore Riemann Hypothesis must be true and the both implications together prove the equivalence.
\end{proof}

The inner condition on $\mathfrak{T}(k)$ can be relaxed without affecting the equivalence, provided the weaker lower bound is imposed directly on $\mathfrak{O}(x)$. This yields the following corollary, which strictly generalizes the theorem

\begin{corollary}[Espinosa, 2025, Espinosa--Riemann Equivalence Corollary]\label{cor:damian_final}
The \textbf{Riemann Hypothesis} is equivalent to \eqref{eq:espinosa-main} under
conditions \textup{(i)} and \textup{(iii)} of Theorem~\ref{teo:damian_final},
with condition \textup{(ii)} replaced by the weaker
\begin{equation}\label{eq:O-strong}
\mathfrak{O}(x) \ > \ e^x \log x \qquad \text{for every } x \in \mathbb{R}^+.
\end{equation}
\end{corollary}
\begin{proof}
The proof is identical to that of Theorem~\ref{teo:damian_final}, except that
condition \eqref{eq:T-cond} on $\mathfrak{T}$ is replaced by the direct
lower bound \eqref{eq:O-strong} on $\mathfrak{O}$; the paragraph invoking
Proposition~\ref{prop:damian_inequality} is therefore omitted. Since
\eqref{eq:T-cond} implies \eqref{eq:O-strong} (by
Proposition~\ref{prop:damian_inequality}), but not conversely, the corollary
strictly generalizes the theorem.
\end{proof}

This concludes the construction of the equivalence. The framework is now sufficiently general to accommodate a wide range of explicit sequences $\{a_k\}_{k \ge 1}$, some of which are exhibited in the next section \ref{subsec:new_eq}.

\subsection{Three equivalences of the Riemann Hypothesis}
\label{subsec:new_eq}
In Theorem~\ref{teo:damian_final} the function $\mathfrak{T} : \mathbb{R}^+ \to \mathbb{R}$
is only required to satisfy conditions \textup{(i)}--\textup{(iii)}. Its restriction
to the natural numbers yields an admissible sequence
\begin{equation}\label{eq:ak-sequence}
a_k \ := \ \mathfrak{T}\big|_{\mathbb{N}}(k) \ = \ H_k - \gamma + \varepsilon_k,
\qquad \varepsilon_k > 0, \qquad k \in \mathbb{N},
\end{equation}
and each choice of such a sequence produces an explicit equivalence of the Riemann Hypothesis. In what follows we exhibit three natural
realizations of $\{a_k\}_{k \ge 1}$. This reduces the
conjecture to an inequality of the form
\begin{equation}\label{eq:espinosa-intro1}
\sum_{k \in \mathbb{N}} \frac{H_n^k}{k!} \, a_k \ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N},
\end{equation}
for an admissible sequence $\{a_k\}_{k \ge 1}$ satisfying the three explicit conditions in Theorem \ref{teo:damian_final}.
These three equivalences are the simplest explicit instances of the Espinosa–Riemann Equivalence Theorem. Each provides a direct combinatorial bridge to Young's Lattice, as developed in the next section.

\begin{theorem}[Espinosa, 2025, First equivalence: logarithmic coefficients]\label{teo:eq1}
The \textbf{Riemann Hypothesis} is equivalent to
\begin{equation}\label{eq:eq1}
\sum_{k \in \mathbb{N}} \frac{H_n^k}{k!} \log(k+1) \ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N}.
\end{equation}
\end{theorem}
\begin{proof}
Apply Theorem~\ref{teo:damian_final} with $a_k := \log(k+1)$. It is
straightforward to verify that this sequence satisfies the three conditions
of the theorem:
\begin{enumerate}[label=\textup{(\roman*)}]
    \item 
    $\sum_{k \in \mathbb{N}} \frac{H_n^k}{k!} \log(k+1) = e^x \log x + \mathcal{O}(\frac{e^x}{x})$ as $x \to \infty$;
    
    Substituting Proposition~\ref{prop:digamma-harmonic}, $H_k - \gamma = \psi(k+1)$ for $k \ge 1$ in \cite[Eq.~6.3.18, p.~259]{abramowitz} we obtain:
    \[
    H_k - \gamma \ < \ \log(k+1) \ < \ H_k - \gamma + \frac{1}{k} \qquad (k \ge 1).
    \]
    Taking the upper bound, multiplying by \(\frac{x^k}{k!} > 0\) and summing over \(k \in \mathbb{N}\) yields
    \begin{equation}\label{eq:r}
    0 
    \ < \ \sum_{k\ge1} \frac{x^k}{k!} \log(k+1) 
    \ < \ \sum_{k\ge1} (H_k - \gamma)\frac{x^k}{k!} + \sum_{k\ge1} \frac{x^k}{k!}\frac{1}{k},
    \end{equation}
    finally applying $\sum_{k\ge1} \frac{x^k}{k!}\frac{1}{k} = \mathcal{O}\!\left(\frac{e^x}{x}\right)$ by:
    \[
    0 \ < \ \sum_{k\ge1} \frac{x^k}{k!}\frac{1}{k} 
    \ = \ \frac{1}{x} \sum_{k\ge1} \frac{x^{k+1}}{(k+1)!}\left(1 + \frac{1}{k}\right) 
    \ \le \ \frac{2}{x} \sum_{k\ge1} \frac{x^{k+1}}{(k+1)!} \ = \ \frac{2}{x} (e^x - 1 - x) \ = \ \mathcal{O}\!\left(\frac{e^x}{x}\right)
    \]
    and the Proposition~\ref{prop:serie-modificada} in \eqref{eq:r}, we obtain the desired result:
    \[
    0
    \ < \ \sum_{k\ge1} \frac{x^k}{k!} \log(k+1) 
    \ < \ e^x \log x + \gamma + \mathcal{O}\!\left(\frac{1}{x}\right) + \mathcal{O}\!\left(\frac{e^x}{x}\right) \ = \ e^x \log x + \mathcal{O}\!\left(\frac{e^x}{x}\right) \quad (x \to \infty).
    \]
    \item the lower bound $\log(k+1) > H_k - \gamma$ holds for every
    $k \in \mathbb{N}$;
    \item the base inequality \eqref{eq:eq1} is verified directly for
    $1 \le n \le 60$.
\end{enumerate}
The equivalence follows. See \cite[Thm.~3]{espinosa-part3}.
\end{proof}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{4c.pdf}
        \caption{Plot of the left-hand side of \eqref{eq:eq1} against the
        right-hand side $\sum_{d \mid n} d$, for small values of $n$. The
        logarithmic coefficients $\log(k+1)$ produce a sequence that
        dominates the divisor sum from the very first integers, illustrating
        the base validation of condition \textup{(iii)}.}
        \label{fig:eq1-left}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{4e.pdf}
        \caption{Comparison of the coefficients $\log(k+1)$ with the lower
        bound $H_k - \gamma$, for $k$ ranging over the first integers. The
        logarithmic growth of $\log(k+1)$ lies strictly above
        $H_k - \gamma$, confirming condition \textup{(ii)} of the
        theorem.}
        \label{fig:eq1-right}
    \end{subfigure}
    \caption{Graphical verification of the First Equivalence. \\ \textbf{(a)}
    The $\sum_{k \in \mathbb{N}} \frac{H_n^k}{k!} \log(k+1)$ series on the left-hand side of \eqref{eq:eq1}
    dominates the divisor sum for small $n$, validating the base case.
    \textbf{(b)} The coefficients $\log(k+1)$ lie strictly above the lower
    bound $H_k - \gamma$, confirming the inner condition of the theorem.}
    \label{fig:eq1-figures}
\end{figure}
\begin{theorem}[Espinosa, 2025, Second equivalence: alternating generalized-harmonic coefficients]\label{teo:eq2}
The \textbf{Riemann Hypothesis} is equivalent to
\begin{equation}\label{eq:eq2}
\sum_{k \in \mathbb{N}} \frac{H_n^k}{k!} \, a_k \ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N},
\end{equation}
where
\begin{equation}\label{eq:eq2-ak}
a_k \ := \ \sum_{m=1}^{k} \tfrac{(-1)^{m+1}}{m} \, H_{k-m+1}^{(m)},
\qquad
H_p^{(m)} \ := \ \sum_{j=1}^{p} \tfrac{1}{j^m}.
\end{equation}
\end{theorem}
\begin{proof}
Apply Theorem~\ref{teo:damian_final} with $a_k$ defined as in \eqref{eq:eq2-ak}. It is straightforward to verify that this sequence
satisfies the three conditions of the theorem:
\begin{enumerate}[label=\textup{(\roman*)}]
    \item $\sum_{k \in \mathbb{N}} \frac{H_n^k}{k!} \, a_k = e^x \log x + \mathcal{O}(\frac{e^x}{x})$ as $x \to \infty$;
    \item the lower bound $a_k > H_k - \gamma$ holds for every
    $k \in \mathbb{N}$;
    \item the base inequality \eqref{eq:eq2} is verified directly for
    $1 \le n \le 60$.
\end{enumerate}
The equivalence follows. See \cite[Thm.~4]{espinosa-part4}.
\end{proof}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{3c.pdf}
        \caption{Plot of the left-hand side of \eqref{eq:eq2} against the
        right-hand side $\sum_{d \mid n} d$, for small values of $n$. The
        alternating generalized-harmonic coefficients $a_k$ defined in
        \eqref{eq:eq2-ak} produce a sequence that dominates the divisor sum
        from the very first integers, illustrating the base validation of
        condition \textup{(iii)}.}
        \label{fig:eq2-left}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{3e.pdf}
        \caption{Comparison of the coefficients $a_k$ defined in
        \eqref{eq:eq2-ak} with the lower bound $H_k - \gamma$, for $k$
        ranging over the first integers. The alternating harmonic layers
        accumulate so that $a_k$ lies strictly above $H_k - \gamma$,
        confirming condition \textup{(ii)} of the theorem.}
        \label{fig:eq2-right}
    \end{subfigure}
    \caption{Graphical verification of the Second Equivalence.
    \\ \textbf{(a)} The series on the left-hand side of \eqref{eq:eq2} dominates the divisor sum for small $n$, validating the
    base case. \textbf{(b)} The coefficients $a_k$ lie strictly above the
    lower bound $H_k - \gamma$, confirming the inner condition of the
    theorem.}
    \label{fig:eq2-figures}
\end{figure}

\begin{theorem}[Espinosa, 2025, Third equivalence: fully explicit coefficients]\label{teo:eq3}
The \textbf{Riemann Hypothesis} is equivalent to
\begin{equation}\label{eq:eq3}
\sum_{k \in \mathbb{N}} \frac{\left( \frac{1}{1} + \cdots + \frac{1}{n} \right)^k}{k!}
\left( \tfrac{1}{k} + \cdots + \tfrac{1}{k^2} \right) \ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N}.
\end{equation}
\end{theorem}
\begin{proof}
Apply Theorem~\ref{teo:damian_final} with
$a_k := \frac{1}{k} + \cdots + \frac{1}{k^2}$. It is straightforward to
verify that this sequence satisfies the three conditions of the theorem, the proof is exactly the same of Theorem~\ref{teo:eq1} replacing with $H_k - \gamma < \frac{1}{k} + \cdots + \frac{1}{k^2} < H_k - \gamma + \frac{1}{k}$ for $k \ge 1$, the equivalence follows. See \cite[Thm.~7]{espinosa-part1}.
\end{proof}

\begin{figure}[H]
    \centering
    \includegraphics[width=.47\linewidth]{10000NEGRO5-42.pdf}
    \caption{Third equivalence of the Riemann Hypothesis, with $a_k=\tfrac{1}{k} + \cdots + \tfrac{1}{k^2}$.}
    \label{fig:placeholder}
\end{figure}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.5\linewidth}
        \centering
        \includegraphics[width=\linewidth]{Ingles3.pdf}
        \caption{Plot of the left-hand side of \eqref{eq:eq3} against the
        right-hand side $\sum_{d \mid n} d$, for small values of $n$. The
        fully explicit coefficients
        $a_k = \frac{1}{k} + \cdots + \frac{1}{k^2}$ produce a sequence that
        dominates the divisor sum from the very first integers,
        illustrating the base validation of condition \textup{(iii)}.}
        \label{fig:eq3-left}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.5\linewidth}
        \centering
        \includegraphics[width=\linewidth]{Ingles5.pdf}
        \caption{Comparison of the coefficients
        $a_k = \frac{1}{k} + \cdots + \frac{1}{k^2}$ with the lower bound
        $H_k - \gamma$, for $k$ ranging over the first integers. The
        harmonic series of coefficients accumulates so that $a_k$ lies
        strictly above $H_k - \gamma$, confirming condition \textup{(ii)}
        of the theorem.}
        \label{fig:eq3-right}
    \end{subfigure}
    \caption{Graphical verification of the Third Equivalence. \\ \textbf{(a)} 
    The series on the left-hand side of \eqref{eq:eq3}
    dominates the divisor sum for small $n$, validating the base case.
    \textbf{(b)} The coefficients $a_k$ lie strictly above the lower bound
    $H_k - \gamma$, confirming the inner condition of the theorem.}
    \label{fig:eq3-figures}
\end{figure}

\subsection{On the generalized harmonic number of a partition $\boldsymbol{\lambda}$ and a $k$-tuple $\lambda$}
\label{subsec:harmonic_partition}

\subsubsection{About $H_n^{\boldsymbol{\lambda}}$ and $H_n^{\lambda}$}\label{sec:series2}

We now develop the partition-theoretic machinery that underlies the combinatorial reformulation of the Riemann Hypothesis. The central objects are the generalized harmonic numbers associated with partitions and tuples, which extend the classical harmonic numbers to the multi-index setting, see \cite{espinosa-part6}. We begin by establishing the formal language of partitions.

\begin{definition}[Partition of an integer]\label{def:partitioninteger}
Let $m \in \mathbb{N}$. A \emph{\textbf{partition}} $\boldsymbol{\lambda}$ of $m$ of length $k$
is a $k$-tuple of integers $\lambda_i \in \mathbb{N}$ for $1 \le i \le k$ such that
\[
\boldsymbol{\lambda} \ = \ (\lambda_1, \lambda_2, \dots, \lambda_k),
\qquad
\lambda_1 \ \ge \ \lambda_2 \ \ge \ \dots \ \ge \ \lambda_k \ \ge \ 1,
\qquad
\sum_{r=1}^{k} \lambda_r \ = \ m.
\]
We write $\boldsymbol{\lambda} \vdash m$ to denote that $\boldsymbol{\lambda}$ is a
partition of $m$.
\end{definition}

A crucial aspect of our proposal is the distinction between a partition (where order
is fixed) and the set of its possible arrangements. To capture the full symmetry of
the harmonic power, we define the set of all distinct permutations as follows.

\begin{definition}[Set of distinct permutations]\label{def:psi}
For a partition $\boldsymbol{\lambda} = (\lambda_1, \dots, \lambda_k) \vdash m$, let
$\Psi(\boldsymbol{\lambda})$ denote the \emph{\textbf{set of all distinct rearrangements of $\boldsymbol{\lambda}$}}, equivalently, the set of all distinct permutations of the $k$-tuple $(\lambda_1, \dots, \lambda_k)$:
\[
\Psi(\boldsymbol{\lambda}) \ := \
\left\{
(\sigma_1, \sigma_2, \ldots, \sigma_k) \in \mathbb{N}^k :
\forall j \in \mathbb{N}, \
\operatorname{card}\{r : \sigma_r = j\} = \operatorname{card}\{r : \lambda_r = j\}
\right\}.
\]
Each $\Theta = (\sigma_1, \ldots, \sigma_k) \in \Psi(\boldsymbol{\lambda})$ is a
$k$-tuple representing a distinct rearrangement of the parts of $\boldsymbol{\lambda}$.
\end{definition}

Once the partition is defined, it is necessary to introduce the algebraic weighting
associated with it. The following definitions for the factorial and the multinomial
coefficient of a partition are essential for our final formula.

\begin{definition}[Partition factorial]\label{def:partitionfactorial}
For a partition $\boldsymbol{\lambda} = (\lambda_1, \dots, \lambda_k) \vdash m$, we
define the \emph{\textbf{partition factorial}}:
\[
\boldsymbol{\lambda}! \ := \ \lambda_1! \, \lambda_2! \cdots \lambda_k!.
\]
\end{definition}

\begin{definition}[Multinomial coefficient for partitions]\label{def:multcoeff}
For a partition $\boldsymbol{\lambda} = (\lambda_1, \dots, \lambda_k) \vdash m$, we
define the \emph{\textbf{partition multinomial coefficient}}:
\[
\binom{m}{\boldsymbol{\lambda}} \ := \ \binom{m}{\lambda_1, \dots, \lambda_k}
\ = \ \frac{m!}{\lambda_1! \, \lambda_2! \cdots \lambda_k!}
\ = \ \frac{m!}{\boldsymbol{\lambda}!}.
\]
\end{definition}

With these tools at hand, we introduce the two core objects of this study: the
partition-based harmonic number $H_n^{\boldsymbol{\lambda}}$
(Definition~\ref{def:partition}), which exploits the full symmetry of the
partition, and the tuple-based harmonic number $H_n^{\lambda}$
(Definition~\ref{def:ktuple}), whose dependence on the order of the
exponents is essential.

\begin{definition}[Generalized harmonic number of a partition]\label{def:partition}
Let $\boldsymbol{\lambda} = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ be a partition of
length $k$ with $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 1$.
For $n \in \mathbb{N}$ with $n \ge k$, the \emph{\textbf{$n$-th generalized harmonic number of order
$\lambda_1 + \lambda_2 + \ldots + \lambda_k$ associated with the partition
$\boldsymbol{\lambda}$}} is defined as
\[
H_n^{\boldsymbol{\lambda}} \ := \
\sum_{\Theta \in \Psi(\boldsymbol{\lambda})}
\sum_{1 \leq i_1 < i_2 < \cdots < i_k \leq n}
\prod_{r=1}^{k} \tfrac{1}{i_r^{\sigma_r}},
\]
where $\Psi(\boldsymbol{\lambda})$ is as in Definition~\ref{def:psi}.
For $n < k$, we set $H_n^{\boldsymbol{\lambda}} := 0$.
\end{definition}

\begin{definition}[Generalized harmonic number for a $k$-tuple]\label{def:ktuple}
Let $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ be a $k$-tuple of positive
integers. For $n \in \mathbb{N}$ with $n \ge k$, the \emph{\textbf{$n$-th generalized harmonic number
of order $\lambda_1 + \lambda_2 + \ldots + \lambda_k$ associated with the $k$-tuple
$\lambda$}} is defined as
\[
H_n^{\lambda} \ := \
\sum_{1 \leq i_1 < i_2 < \cdots < i_k \leq n}
\prod_{r=1}^{k} \frac{1}{i_r^{\lambda_r}}.
\]
In this definition the \emph{\textbf{order of the exponents matters}}: distinct orderings of
the same multiset generally yield distinct harmonic numbers.
For $n < k$, we set $H_n^{\lambda} := 0$.
\end{definition}

The elegance of this framework lies in the direct correspondence between these two
definitions: the harmonic number of a partition is the sum of the harmonic numbers of
all its distinct tuple arrangements.

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{2.pdf}
        \caption{Incremental contributions to the generalized harmonic
        numbers $H_n^{\boldsymbol{\lambda}}$ for partitions of the form
        $(1,1,1,\dots)$, that is, partitions with all parts equal to $1$. Each bar
        represents the increment
        $\Delta H_n^{\boldsymbol{\lambda}} = H_n^{\boldsymbol{\lambda}} - H_{n-1}^{\boldsymbol{\lambda}}$
        added by the index $n$ to the partial sum, so that the accumulated
        value $H_n^{\boldsymbol{\lambda}}$ is recovered by stacking the bars
        of colors $1, 2, \dots, n$. The horizontal axis lists the partitions
        $\boldsymbol{\lambda} = (1)$, $(1,1)$, $\dots$, $(1,1,1,1,1,1,1,1)$, from right to
        left; the depth axis shows the upper limit $n$ from $1$ to $8$; and
        the height of each bar is the contribution of the level $n$ to the
        total sum. The colors follow the natural order of $n$: blue for
        $n = 1$, red for $n = 2$, green for $n = 3$, yellow for $n = 4$,
        purple for $n = 5$, dark red for $n = 6$, pink for $n = 7$, and
        gray for $n = 8$.}
        \label{fig:harmonic-1k}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{3.pdf}
        \caption{Incremental contributions to the generalized harmonic
        numbers $H_n^{\boldsymbol{\lambda}}$ for general partitions,
        including the forms $(2)$, $(2,1)$, $(2,1,1)$, $(2,1,1,1)$, and so on. As in panel \textup{(a)}, each bar represents
        the increment
        $\Delta H_n^{\boldsymbol{\lambda}} = H_n^{\boldsymbol{\lambda}} - H_{n-1}^{\boldsymbol{\lambda}}$
        contributed by the index $n$, and the total value $H_n^{\boldsymbol{\lambda}}$
        is recovered by stacking the bars of colors $1, 2, \dots, n$. The
        additional partitions with parts greater than $1$ generate
        contributions that cannot be captured by the one-part partition
        $(k)$ alone, and they become significant at higher levels of $n$.
        The color encoding follows the same convention as in panel
        \textup{(a)}.}
        \label{fig:harmonic-general}
    \end{subfigure}
    \caption{Incremental contributions to the generalized harmonic numbers
    $H_n^{\boldsymbol{\lambda}}$ for different partitions $\boldsymbol{\lambda}$
    and different values of the upper limit $n$. \textbf{(a)} Partitions of
    the form $(1,1,1,\dots)$. \textbf{(b)} Partitions of
    the form $(2,1,1,\dots)$. In both panels, each bar represents the increment
    $\Delta H_n^{\boldsymbol{\lambda}} = H_n^{\boldsymbol{\lambda}} - H_{n-1}^{\boldsymbol{\lambda}}$
    added by the index $n$ to the partial sum $H_{n-1}^{\boldsymbol{\lambda}}$,
    so that the accumulated value at level $n$ is obtained by stacking the
    bars of colors $1, 2, \dots, n$. The horizontal axis lists the
    partitions, the depth axis shows the upper limit $n$ from $1$ to $8$,
    and the height of each bar is the contribution of the level $n$ to the
    total sum. The colors encode the value of $n$: blue for $n = 1$, red
    for $n = 2$, green for $n = 3$, yellow for $n = 4$, purple for $n = 5$,
    dark red for $n = 6$, pink for $n = 7$, and gray for $n = 8$.}
    \label{fig:harmonic-pair}
\end{figure}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{4.pdf}
        \caption{Family $(2,1,1,\dots)$: partitions of the form $(2,1,1,\dots)$. Each bar represents the generalized harmonic number
        $H_n^{\Theta}$ associated with a distinct permutation
        $\Theta \in \Psi(\boldsymbol{\lambda})$ of the partition
        $\boldsymbol{\lambda}$, decomposed into its tuple-based harmonic
        components according to Proposition~\ref{prop:relacion}.
        The permutations are ordered from largest to smallest value of the
        tuple, so that $(2,1,1,1,1,1)$ appears at the right and
        $(1,1,1,1,1,2)$ at the left. The horizontal axis lists the
        partitions; the depth axis shows the upper limit $n$ from $1$ to
        $8$; and the height of each bar is the value of $H_n^{\Theta}$.
        The colors encode the value of $n$ at which each contribution is
        added to the accumulated sum, so that stacking the bars of colors
        $1, 2, \dots, n$ recovers the total harmonic number
        $H_n^{\boldsymbol{\lambda}} = \sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_n^{\Theta}$.}
        \label{fig:harmonic-2s}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{5.pdf}
        \caption{Family $(3,1,1,\dots)$: partitions of the form $(3,1,1,\dots)$. Each bar represents the generalized harmonic number
        $H_n^{\Theta}$ associated with a distinct permutation
        $\Theta \in \Psi(\boldsymbol{\lambda})$ of the partition
        $\boldsymbol{\lambda} = (3)$, $(3,1)$, $(3,1,1)$, $\dots$. As in panel
        \textup{(a)}, the permutations are ordered from largest to smallest
        tuple value, and the color encoding follows the same convention,
        with the cumulative sum at level $n$ recovering the full harmonic
        number $H_n^{\boldsymbol{\lambda}}$. The additional partitions with
        parts equal to $3$ generate contributions of a different magnitude,
        and the visualization makes transparent how each tuple-based
        harmonic accumulates at successive levels of $n$.}
        \label{fig:harmonic-3s}
    \end{subfigure}
    \caption{Generalized harmonic numbers $H_n^{\Theta}$ for the tuple-based
    permutations $\Theta \in \Psi(\boldsymbol{\lambda})$ of the partitions
    in the families $(2,1,1,\dots)$ and $(3,1,1,\dots)$, evaluated at different values of
    the upper limit $n$. \textbf{(a)} Family $(2,1,1,\dots)$. \textbf{(b)} Family
    $(3,1,1,\dots)$. In both panels, the horizontal axis lists the partitions, the
    depth axis shows the upper limit $n$ from $1$ to $8$, and the height of
    each bar is the value of $H_n^{\Theta}$. The colors encode the value of
    $n$: blue for $n = 1$, red for $n = 2$, green for $n = 3$, yellow for
    $n = 4$, purple for $n = 5$, dark red for $n = 6$, pink for $n = 7$,
    and gray for $n = 8$. Each level $n$ contributes the incremental amount
    added to the partial sum, so that the accumulated value at level $n$ is
    obtained by stacking the bars of colors $1, 2, \dots, n$. This follows
    directly from Proposition~\ref{prop:relacion}, which expresses each
    partition-based harmonic number as the sum of the tuple-based harmonic
    numbers of its distinct permutations.}
    \label{fig:harmonic-pair-23}
\end{figure}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{6.pdf}
        \caption{Partition blocks for the families $(2,2,1,1,1,\dots)$ and $(3,1,1,1,\dots)$: the
        partitions obtained by adding trailing $1$'s to the base partitions
        $(2,2)$ and $(3)$. Each bar represents the full generalized harmonic
        number $H_n^{\boldsymbol{\lambda}}$ associated with the partition
        $\boldsymbol{\lambda}$, without decomposing it into its tuple-based
        permutations. The horizontal axis lists the partitions; the depth
        axis shows the upper limit $n$ from $1$ to $8$; and the height of
        each bar is the value of $H_n^{\boldsymbol{\lambda}}$. The colors
        encode the value of $n$ at which each contribution is added to the
        partial sum, so that stacking the bars of colors $1, 2, \dots, n$
        recovers the accumulated value
        $\sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_n^{\Theta} = H_n^{\boldsymbol{\lambda}}$
        at level $n$. This panel shows the harmonic numbers directly, as
        they appear in the partition-based formulation of the Riemann
        Hypothesis.}
        \label{fig:harmonic-blocks-22-3}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{7.pdf}
        \caption{Tuple-based decompositions for the same two families
        $(2,2,1,1,1,\dots)$ and $(3,1,1,1,\dots)$. Each bar represents the generalized harmonic
        number $H_n^{\Theta}$ associated with a distinct permutation
        $\Theta \in \Psi(\boldsymbol{\lambda})$ of the partition
        $\boldsymbol{\lambda}$. The permutations are ordered from largest
        to smallest tuple value, so that the tuple with the largest parts
        first appears at the right. The height of each bar is $H_n^{\Theta}$
        and the colors encode the value of $n$, following the same
        convention as in panel \textup{(a)}. The cumulative stacking of
        colors $1, 2, \dots, n$ recovers the full harmonic number
        $H_n^{\boldsymbol{\lambda}} = \sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_n^{\Theta}$,
        as stated in Proposition~\ref{prop:relacion}. This panel exposes
        the individual contributions of each tuple arrangement, showing how
        the harmonic number of a partition is assembled from its constituent
        permutations.}
        \label{fig:harmonic-perms-22-3}
    \end{subfigure}
    \caption{Two complementary visualizations of the generalized harmonic
    numbers for the partition families $(2,2,1,1,1,\dots)$ and $(3,1,1,1,\dots)$. \textbf{(a)}
    Partition blocks: the harmonic numbers $H_n^{\boldsymbol{\lambda}}$ are
    plotted directly for each partition $\boldsymbol{\lambda}$, without
    decomposition. \textbf{(b)} Tuple-based decompositions: the harmonic
    numbers are decomposed into their constituent permutations
    $\Theta \in \Psi(\boldsymbol{\lambda})$ according to
    Proposition~\ref{prop:relacion}, and the permutations are ordered from
    largest to smallest tuple value. In both panels, the horizontal axis
    lists the partitions, the depth axis shows the upper limit $n$ from
    $1$ to $8$, and the height of each bar is the value of the corresponding
    harmonic number. The colors encode the value of $n$: blue for $n = 1$,
    red for $n = 2$, green for $n = 3$, yellow for $n = 4$, purple for
    $n = 5$, dark red for $n = 6$, pink for $n = 7$, and gray for $n = 8$.
    The two panels are complementary: the first shows the harmonic numbers
    as they appear in the partition-based formulation of the Riemann
    Hypothesis, while the second exhibits the internal structure that
    produces them, via the tuple-based decomposition of
    Proposition~\ref{prop:relacion}.}
    \label{fig:harmonic-pair-blocks-perms}
\end{figure}

\begin{proposition}[Relation between the two definitions]\label{prop:relacion}
For any partition $\boldsymbol{\lambda} = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ of
length $k$ and any $n \in \mathbb{N}$,
\[
H_n^{\boldsymbol{\lambda}} \ = \ \sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_n^{\Theta}.
\]
\end{proposition}
\begin{proof}
By Definition~\ref{def:partition}, for $n \ge k$,
\[
H_n^{\boldsymbol{\lambda}}
\ = \ \sum_{\Theta \in \Psi(\boldsymbol{\lambda})}
\sum_{1 \le i_1 < \cdots < i_k \le n}
\prod_{r=1}^{k} \frac{1}{i_r^{\sigma_r}}
\ = \ \sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_n^{\Theta},
\]
where the last equality uses Definition~\ref{def:ktuple}. For $n < k$ both sides
vanish by definition, so the identity holds trivially.
\end{proof}

\subsubsection{On $\frac{H_n^k}{k!}$}\label{sec:series3}

The expansion of the harmonic power $H_n^k$ relies on the classical multinomial theorem, which we recall first.

\begin{theorem}[Multinomial Theorem]\label{teo:multteo}
For $m \in \mathbb{N}$ and commuting variables $x_1, \dots, x_n$,
\[
(x_1 + x_2 + \cdots + x_n)^m
\ = \ \sum_{\substack{\alpha_1, \dots, \alpha_n \ge 0 \\
\alpha_1 + \cdots + \alpha_n = m}}
\binom{m}{\alpha_1, \dots, \alpha_n}
\, x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n},
\]
where $\binom{m}{\alpha_1, \dots, \alpha_n} = \dfrac{m!}{\alpha_1! \cdots \alpha_n!}$
is the multinomial coefficient.
\end{theorem}
\begin{proof}
See \cite[Sec.~1.4]{stanley-ec1}.
\end{proof}

We now present the central decomposition of this section: the normalized harmonic
power $\frac{H_n^k}{k!}$ as a weighted sum over partitions of $k$.

\begin{proposition}[Partition expansion of the harmonic power I]\label{prop:partition-expansion}
For all $n, k \in \mathbb{N}$,
\[
\frac{H_n^k}{k!} \ = \ \sum_{\boldsymbol{\lambda} \, \vdash \, k}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!}.
\]
\end{proposition}
\begin{proof}
By the Multinomial Theorem (Theorem~\ref{teo:multteo}) applied to
$x_i = \frac{1}{i}$,
\[
H_n^k \ = \ \sum_{\substack{\alpha_1, \dots, \alpha_n \ge 0 \\
\alpha_1 + \cdots + \alpha_n = k}}
\frac{k!}{\alpha_1! \cdots \alpha_n!} \prod_{j=1}^{n} \frac{1}{j^{\alpha_j}}.
\]
Group the terms of this sum according to the partition
$\boldsymbol{\lambda} = (\lambda_1, \dots, \lambda_m) \vdash k$ formed by the positive
exponents $\alpha_j$, arranged in non-increasing order. For a fixed partition
$\boldsymbol{\lambda}$ and a fixed set of indices $1 \le j_1 < \cdots < j_m \le n$,
each assignment of $\lambda_1, \dots, \lambda_m$ to these positions is a permutation
$\Theta = (\sigma_1, \dots, \sigma_m) \in \Psi(\boldsymbol{\lambda})$, whose
multinomial coefficient satisfies
\[
\frac{k!}{\sigma_1! \cdots \sigma_m!} \ = \ \frac{k!}{\boldsymbol{\lambda}!}.
\]
Summing over $\Theta \in \Psi(\boldsymbol{\lambda})$ and over all strictly increasing
$m$-tuples recovers $\frac{k!}{\boldsymbol{\lambda}!} H_n^{\boldsymbol{\lambda}}$ by
Definition~\ref{def:partition}. Summing over all partitions yields
$H_n^k = \sum_{\boldsymbol{\lambda} \vdash k} \frac{k!}{\boldsymbol{\lambda}!}
H_n^{\boldsymbol{\lambda}}$, and dividing by $k!$ proves the claim.
\end{proof}

Substituting the relation between partition-based and tuple-based harmonic numbers
(Proposition~\ref{prop:relacion}) gives a more granular expansion.

\begin{proposition}[Partition expansion of the harmonic power II]\label{prop:partition-expansion-1}
For all $n, k \in \mathbb{N}$,
\[
\frac{H_n^k}{k!} \ = \ \sum_{\boldsymbol{\lambda} \, \vdash \, k}
\frac{1}{\boldsymbol{\lambda}!} \sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_n^{\Theta}.
\]
\end{proposition}
\begin{proof}
Combine Proposition~\ref{prop:partition-expansion} with
Proposition~\ref{prop:relacion} and substitute
$H_n^{\boldsymbol{\lambda}} = \sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_n^{\Theta}$.
\end{proof}
The raw power $H_n^k$ admits an analogous form in terms of the partition multinomial
coefficient of Definition~\ref{def:multcoeff}.

\begin{proposition}[Partition expansion of the harmonic power III]\label{prop:partition-expansion-2}
For all $n, k \in \mathbb{N}$,
\[
H_n^k \ = \ \sum_{\boldsymbol{\lambda} \, \vdash \, k}
\binom{k}{\boldsymbol{\lambda}} \, H_n^{\boldsymbol{\lambda}}.
\]
\end{proposition}
\begin{proof}
Multiply the identity of Proposition~\ref{prop:partition-expansion} by $k!$ and apply
Definition~\ref{def:multcoeff}.
\end{proof}

Finally, the most detailed decomposition is obtained by combining
Propositions~\ref{prop:partition-expansion-2} and~\ref{prop:relacion}.

\begin{proposition}[Partition expansion of the harmonic power IV]\label{prop:partition-expansion-3}
For all $n, k \in \mathbb{N}$,
\[
H_n^k \ = \ \sum_{\boldsymbol{\lambda} \, \vdash \, k}
\binom{k}{\boldsymbol{\lambda}}
\sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_n^{\Theta}.
\]
\end{proposition}
\begin{proof}
Substitute $H_n^{\boldsymbol{\lambda}} = \sum_{\Theta \in \Psi(\boldsymbol{\lambda})}
H_n^{\Theta}$ into Proposition~\ref{prop:partition-expansion-2}.
\end{proof}

\subsubsection{A Partition Decomposition of the Multinomial Theorem}
\label{sec:series4}

We close this section with a partition-symmetric restatement of the multinomial theorem. This form makes the underlying combinatorial structure transparent and provides the natural bridge to Young's Lattice.

\begin{theorem}[Partition-symmetric form of the Multinomial Theorem]
\label{thm:multinomial-partition}
Let \(m,n\in\mathbb{N}\) and let \(x_1,\dots,x_n\) be commuting variables. Then
\begin{equation}
\label{eq:multinomial-partition}
(x_1+\cdots+x_n)^m
\ = \
\sum_{\substack{\boldsymbol{\lambda}\vdash m\\ \ell(\boldsymbol{\lambda})\le n}}
\binom{m}{\boldsymbol{\lambda}}
\sum_{\Theta\in\Psi(\boldsymbol{\lambda})}
\sum_{1\le j_1<\cdots<j_k\le n}
\prod_{r=1}^{k}x_{j_r}^{\sigma_r},
\end{equation}
where \(k=\ell(\boldsymbol{\lambda})\), \(\Theta=(\sigma_1,\dots,\sigma_k)\), and the inner sums run over all strictly increasing \(k\)-tuples and all distinct permutations \(\Theta\in\Psi(\boldsymbol{\lambda})\).
\end{theorem}

\begin{proof}
By the classical Multinomial Theorem,
\[
(x_1+\cdots+x_n)^m
\ = \ \sum_{\substack{\alpha_1,\dots,\alpha_n\ge0\\\alpha_1+\cdots+\alpha_n=m}}
\frac{m!}{\alpha_1!\cdots\alpha_n!}
\prod_{j=1}^n x_j^{\alpha_j}.
\]
Group the summands according to the partition determined by their positive exponents. For a fixed multi-index \((\alpha_1,\dots,\alpha_n)\) let
\[
\{j_1,\dots,j_k\} \ = \ \{j:\alpha_j>0\},\qquad j_1<\cdots<j_k,
\]
and set \(\mu_r=\alpha_{j_r}\). Arranging the positive integers \(\mu_1,\dots,\mu_k\) in non-increasing order yields a partition \(\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_k)\vdash m\) with length \(k\le n\). The assignment of these values to the ordered positions \(j_1,\dots,j_k\) is precisely a permutation
\[
\Theta \ = \ (\sigma_1,\dots,\sigma_k)\in\Psi(\boldsymbol{\lambda}).
\]
Since \(\Theta\) is a rearrangement of \(\boldsymbol{\lambda}\), we have
\[
\frac{m!}{\alpha_1!\cdots\alpha_n!}
\ = \ 
\frac{m!}{\sigma_1!\cdots\sigma_k!}
\ = \ 
\frac{m!}{\lambda_1!\cdots\lambda_k!}
\ = \
\binom{m}{\boldsymbol{\lambda}}.
\]
The corresponding monomial is \(\prod_{r=1}^k x_{j_r}^{\sigma_r}\).

The map
\[
(\alpha_1,\dots,\alpha_n)
\;\longleftrightarrow\;
\bigl(\boldsymbol{\lambda}\vdash m,\;(j_1,\dots,j_k),\;\Theta\in\Psi(\boldsymbol{\lambda})\bigr)
\]
is bijective: given the triple one recovers the multi-index by setting \(\alpha_{j_r}=\sigma_r\) and \(\alpha_j=0\) otherwise. Summing over all such triples therefore exhausts the multinomial expansion. Because the coefficient \(\binom{m}{\boldsymbol{\lambda}}\) depends neither on \(\Theta\) nor on the choice of indices, it factors out of the two inner sums, yielding \eqref{eq:multinomial-partition}.
\end{proof}
\subsection{The Riemann Hypothesis Young's Lattice Equivalence}
\label{subsec:young_lattice}

Building on the partition decomposition of the harmonic power
(Proposition~\ref{prop:partition-expansion}) and the equivalence theorem of
Section~\ref{sec:espinosa-riemann}, we can now restate the Riemann
Hypothesis as a growth condition indexed by Young's Lattice \cite{espinosa-part8b}.

\begin{theorem}[Espinosa, 2025, Riemann Hypothesis Young's Lattice Equivalence]\label{teo:young-lattice}
The \textbf{Riemann Hypothesis} is equivalent to
\begin{equation}\label{eq:young-lattice}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k
\ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N},
\end{equation}
where $\boldsymbol{\lambda} \vdash k$ denotes that $\boldsymbol{\lambda}$ is a
partition of $k$, $H_n^{\boldsymbol{\lambda}}$ is the $n$-th generalized harmonic
number of order $\lambda_1 + \cdots + \lambda_m = k$ associated with the partition
$\boldsymbol{\lambda} = (\lambda_1, \dots, \lambda_m) \vdash k$,
$\boldsymbol{\lambda}! = \lambda_1! \, \lambda_2! \cdots \lambda_m!$ is the
partition factorial associated with $\boldsymbol{\lambda} \vdash k$, and
$a_k = \frac{1}{k} + \dots + \frac{1}{k^2}$.
\end{theorem}
\begin{proof}
By Proposition~\ref{prop:partition-expansion}, the inner sum satisfies
\[
\sum_{\boldsymbol{\lambda} \, \vdash \, k}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!}
\ = \ \frac{H_n^k}{k!}.
\]
Substituting this into \eqref{eq:young-lattice} yields
\[
\sum_{k \in \mathbb{N}} \frac{H_n^k}{k!} \, a_k \ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N},
\]
which, for $a_k = \frac{1}{k} + \cdots + \frac{1}{k^2}$, is the Third Equivalence
(Theorem~\ref{teo:eq3}).
\end{proof}

\begin{theorem}[Espinosa, 2025, Riemann Hypothesis Young's Lattice Equivalence --- Logarithmic Form]\label{teo:young-lattice-log}
The \textbf{Riemann Hypothesis} is equivalent to
\begin{equation}\label{eq:young-lattice-log}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, \log(k+1)
\ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N},
\end{equation}
where $\boldsymbol{\lambda} \vdash k$ denotes that $\boldsymbol{\lambda}$ is a
partition of $k$, $H_n^{\boldsymbol{\lambda}}$ is the $n$-th generalized harmonic
number of order $\lambda_1 + \cdots + \lambda_m = k$ associated with the partition
$\boldsymbol{\lambda} = (\lambda_1, \dots, \lambda_m) \vdash k$, and
$\boldsymbol{\lambda}! = \lambda_1! \, \lambda_2! \cdots \lambda_m!$ is the
partition factorial associated with $\boldsymbol{\lambda} \vdash k$.
\end{theorem}
\begin{proof}
By Proposition~\ref{prop:partition-expansion}, the inner sum satisfies
\[
\sum_{\boldsymbol{\lambda} \, \vdash \, k}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!}
\ = \ \frac{H_n^k}{k!}.
\]
Substituting this into \eqref{eq:young-lattice-log} yields
\[
\sum_{k \in \mathbb{N}} \frac{H_n^k}{k!} \, \log(k+1) \ > \ \sum_{d \mid n} d
\qquad \text{for every } n \in \mathbb{N},
\]
which is the First Equivalence (Theorem~\ref{teo:eq1}).
\end{proof}

\subsection{The Generalized Assembly Theorem}
\label{subsec:assembly_theorem}

We close the combinatorial part of this work with the structural identity that
governs the assembly of natural numbers by partition-indexed harmonic weights.
This result in \cite{espinosa-part8c} extends the classical Assembly Theorem in \cite{espinosa-part7} to arbitrary summation
intervals and connects the harmonic objects of the previous sections to
Young's Lattice in a quantitative way.
We first fix the restricted partition by length and to analyze the specific configurations required by the Assembly Theorem, we define a restricted subspace of partitions where the length is fixed, leading to the construction of the following Assembly Set.

\begin{definition}[Restricted partition by length]\label{def:restrictedpartition}
Let $m, n \in \mathbb{N}$ with $n \le m$. A \emph{\textbf{partition of $m$ into
exactly $n$ parts}} is a tuple
$\boldsymbol{\lambda} = (\lambda_1, \dots, \lambda_n)$ with
\[
\lambda_1 \ \ge \ \lambda_2 \ \ge \ \dots \ \ge \ \lambda_n \ \ge \ 1,
\qquad \sum_{i=1}^{n} \lambda_i \ = \ m.
\]
We write $\boldsymbol{\lambda} \vdash_n m$ to denote that $\boldsymbol{\lambda}$
is a partition of $m$ into exactly $n$ parts.
\end{definition}

\begin{definition}[Assembly Set]\label{def:assemblyset}
Let $m, n \in \mathbb{N}$ with $n \le m$. The \emph{\textbf{Assembly Set}} $\wp_n^m$ is
the collection of all distinct partitions $\boldsymbol{\lambda}$ of $m$ into
exactly $n$ parts:
\[
\wp_n^m \ := \ \left\{ \boldsymbol{\lambda} = (\lambda_1, \dots, \lambda_n) \in
\mathbb{N}^n : \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n \ge 1, \
\sum_{i=1}^{n} \lambda_i = m \right\}.
\]
Equivalently, $\wp_n^m = \{\boldsymbol{\lambda} : \boldsymbol{\lambda} \vdash_n m\}$.
\end{definition}

We now introduce the generalized harmonic numbers over an arbitrary index interval $[s,n]$.

\begin{definition}[Generalized harmonic number of a partition over an closed interval]\label{def:partition1}
Let $\boldsymbol{\lambda} = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ be a partition
of length $k$ with $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 1$.
For $n, s \in \mathbb{N}$ with $s \le n$ and $n \ge k$, the \emph{\textbf{$[s,n]$-th generalized
harmonic number of order $\lambda_1 + \lambda_2 + \cdots + \lambda_k$ associated
with the partition $\boldsymbol{\lambda}$}} is defined as
\begin{equation}\label{eq:H-partition-interval}
H_{s,n}^{\boldsymbol{\lambda}} \ := \
\sum_{\Theta \in \Psi(\boldsymbol{\lambda})}
\sum_{s \leq i_1 < i_2 < \cdots < i_k \leq n}
\prod_{r=1}^{k} \frac{1}{i_r^{\sigma_r}},
\end{equation}
where $\Psi(\boldsymbol{\lambda})$ is the set of distinct rearrangements of
$\boldsymbol{\lambda}$ given in Definition~\ref{def:psi}. 

Each
$\Theta = (\sigma_1, \sigma_2, \ldots, \sigma_k) \in \Psi(\boldsymbol{\lambda})$
is a $k$-tuple representing a distinct rearrangement of the parts of
$\boldsymbol{\lambda}$. For $n < k$ or $s > n$, we set
$H_{s,n}^{\boldsymbol{\lambda}} := 0$.
\end{definition}

\begin{definition}[Generalized harmonic number for a $k$-tuple over an closed interval]\label{def:ktuple1}
Let $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ be a $k$-tuple of
positive integers. For $n, s \in \mathbb{N}$ with $s \le n$ and $n \ge k$, the
\emph{\textbf{$[s,n]$-th generalized harmonic number of order
$\lambda_1 + \lambda_2 + \cdots + \lambda_k$ associated with the $k$-tuple
$\lambda$}} is defined as
\begin{equation}\label{eq:H-tuple-interval}
H_{s,n}^{\lambda} \ := \
\sum_{s \leq i_1 < i_2 < \cdots < i_k \leq n}
\prod_{r=1}^{k} \frac{1}{i_r^{\lambda_r}}.
\end{equation}
In this definition the order of the exponents matters: distinct orderings of
the same multiset generally yield distinct harmonic numbers. For $n < k$ or
$s > n$, we set $H_{s,n}^{\lambda} := 0$.
\end{definition}

The following remark clarifies the relation between the interval-based notation and the unshifted notation introduced in the previous section.

\begin{remark}[Relation with the unshifted notation]\label{rem:unshifted}
When $s = 1$, the $[1, n]$-th generalized harmonic numbers
$H_{1,n}^{\boldsymbol{\lambda}}$ and $H_{1,n}^{\lambda}$ coincide with the
unshifted harmonic numbers $H_n^{\boldsymbol{\lambda}}$ and $H_n^{\lambda}$
introduced in Definitions~\ref{def:partition1} and~\ref{def:ktuple1}. Throughout
the paper we use the unshifted notation unless an arbitrary starting point
$s$ is explicitly required.
\end{remark}

The elegance of this framework lies in the direct correspondence between the
two representations. The harmonic number of a partition is precisely the
aggregate of the harmonic numbers of its constituent tuple arrangements.

\begin{proposition}[Relation between the two definitions]\label{prop:relacion1}
For any partition $\boldsymbol{\lambda} = (\lambda_1, \lambda_2, \ldots, \lambda_k)$
of length $k$ and any $n, s \in \mathbb{N}$ with $s \le n$,
\begin{equation}\label{eq:relacion-interval}
H_{s,n}^{\boldsymbol{\lambda}} \ = \ \sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_{s,n}^{\Theta}.
\end{equation}
\end{proposition}
\begin{proof}
By Definition~\ref{def:partition1}, for $n \ge k$ and $s \le n$,
\[
H_{s,n}^{\boldsymbol{\lambda}}
\ = \ \sum_{\Theta \in \Psi(\boldsymbol{\lambda})}
\sum_{s \le i_1 < \cdots < i_k \le n}
\prod_{r=1}^{k} \frac{1}{i_r^{\sigma_r}}
\ = \ \sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_{s,n}^{\Theta},
\]
where the last equality uses Definition~\ref{def:ktuple1}. For $n < k$ or
$s > n$ both sides vanish by definition, so the identity holds trivially.
\end{proof}

To state the theorem, we define the following sum:

\begin{definition}[Generalized Assembly Sum]\label{def:assemblysum}
For $n, m, s \in \mathbb{N}$ with $s \le n$, the \emph{\textbf{generalized assembly
sum}} is
\begin{equation}\label{eq:assembly-sum}
\mathfrak{S}_{s,n}^{m} \ := \
\sum_{k=s}^{n} \ \sum_{\boldsymbol{\lambda} \in \wp_{k-s+1}^{k-s+m}}
H_{s,n}^{\boldsymbol{\lambda}},
\end{equation}
where $\boldsymbol{\lambda} \vdash_{k-s+1} k-s+m$ and
$H_{s,n}^{\boldsymbol{\lambda}}$ is the $[s, n]$-th generalized harmonic number
of order $\lambda_1 + \dots + \lambda_{k-s+1}$ associated with
$\boldsymbol{\lambda}$.
\end{definition}

We can now state the central result of this section: $\mathfrak{S}_{s,n}^{m}$ is invariant with respect to the excess parameter $m-1$ and depends only on the interval endpoints. See \cite{espinosa-part8c}. Although we originally proposed a rigorous proof for this result using strong induction in \cite{espinosa-part8c}, a remarkably beautiful and short proof was subsequently suggested by Bîsceanu in \cite{513182}, which we present below.

\begin{theorem}[Generalized Assembly Theorem I]\label{teo:generalized_ensemble}
For all $n, m, s \in \mathbb{N}$ with $s \le n$,
\begin{equation}\label{eq:assembly-theorem}
\mathfrak{S}_{s,n}^{m} \ := \
\sum_{k=s}^{n} \ \sum_{\boldsymbol{\lambda} \in \wp_{k-s+1}^{k-s+m}}
H_{s,n}^{\boldsymbol{\lambda}}
\ = \ \frac{n - s + 1}{s^m}.
\end{equation}
\end{theorem}
\begin{proof}
Fix $m \ge 1$ and put $j = k - s + 1$, so that
$\boldsymbol{\lambda} = (\lambda_1, \dots, \lambda_j)$ ranges over the
partitions of $j + m - 1$ into $j$ positive parts. For a fixed $j$, the
partitions of $j + m - 1$ into $j$ parts, together with all their distinct
permutations, enumerate each ordered composition $(\sigma_1, \dots, \sigma_j)$
with $\sum_{r=1}^{j} \sigma_r = j + m - 1$ exactly once.
Writing $\sigma_r = a_r + 1$ with $a_r \ge 0$, we obtain, as an identity of
formal power series in $t$,
\[
\sum_{m \ge 1} \mathfrak{S}_{s,n}^{m} \, t^{m-1}
\ = \ \sum_{\varnothing \neq I \subseteq \{s,\dots,n\}}
\ \prod_{i \in I} \sum_{a \ge 0} \frac{t^{a}}{i^{a+1}}.
\]
Each inner geometric series evaluates to
$\sum_{a \ge 0} \frac{t^{a}}{i^{a+1}} = \frac{1}{i - t}$, so
\[
\sum_{m \ge 1} \mathfrak{S}_{s,n}^{m} \, t^{m-1}
\ = \ \prod_{i=s}^{n} \left(1 + \frac{1}{i - t}\right) - 1
\ = \ \prod_{i=s}^{n} \frac{i + 1 - t}{i - t} - 1.
\]
The product telescopes,
$
\prod_{i=s}^{n} \frac{i + 1 - t}{i - t} = \frac{n + 1 - t}{s - t},
$
hence
\[
\sum_{m \ge 1} \mathfrak{S}_{s,n}^{m} \, t^{m-1}
\ = \ \frac{n + 1 - t}{s - t} - 1
\ = \ \frac{n - s + 1}{s - t}
\ = \ \frac{n - s + 1}{s} \cdot \frac{1}{1 - \frac{t}{s}}.
\]
Expanding $\frac{1}{1 - \frac{t}{s}} = \sum_{m \ge 1}
\left(\frac{t}{s}\right)^{m-1}$,
\[
\sum_{m \ge 1} \mathfrak{S}_{s,n}^{m} \, t^{m-1}
\ = \ \sum_{m \ge 1} \frac{n - s + 1}{s^{m}} \, t^{m-1}.
\]
Comparing coefficients of $t^{m-1}$ gives
$\mathfrak{S}_{s,n}^{m} = \frac{n - s + 1}{s^{m}}$, as claimed.
\end{proof}

\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{8.pdf}
    \caption{The assembly process for the classical Assembly Theorem
    $\mathfrak{S}_n^m = n$ with $s = 1$ and excess parameter $m -1 = 0$. Each
    panel shows the partial sum of the generalized harmonic numbers
    $H_n^{\boldsymbol{\lambda}}$ over the Assembly Set $\wp_k^k$, as the
    level $k$ increases from $1$ to $n$. For each $n \in \{1, 2, \dots, 8\}$,
    the bars represent the cumulative contributions
    $\sum_{k=1}^{j} \sum_{\boldsymbol{\lambda} \in \wp_k^k} H_n^{\boldsymbol{\lambda}}$
    for $j = 1, \dots, n$, and the horizontal axis lists the levels $k$. The
    colors distinguish the contributions of the successive levels, so that
    each new layer added at level $k$ refines the partial sum. The identity
    $\mathfrak{S}_n^1 = n$ is recovered as the total height of each column
    at level $k = n$, confirming that the ensemble of harmonic weights
    over the Assembly Set $\wp_k^k$ always reconstructs the count of
    indices $\{1, \dots, n\}$, independently of the excess parameter $m-1$.}
    \label{fig:assembly-process}
\end{figure}

The figure above visualizes the accumulation process underlying the
classical Assembly Theorem
\begin{equation*}
\mathfrak{S}_n^m \ = \ n
\qquad \text{for every } m \in \mathbb{N},
\end{equation*}
in the particular case $m = 1$, corresponding to the branch $m=1$ of
Young's Lattice. For each value $n = 1, 2, \dots, 8$, the panel displays the
partial sums
\[
\sum_{k=1}^{j} \ \sum_{\boldsymbol{\lambda} \in \wp_k^k} H_n^{\boldsymbol{\lambda}},
\qquad j = 1, 2, \dots, n,
\]
where $\wp_k^k$ denotes the Assembly Set of partitions of $k$ into exactly
$k$ parts, that is, the singleton $\{(1, 1, \dots, 1)\}$ corresponding to
the partition with $k$ parts all equal to $1$. The horizontal axis lists
the levels $k$, the depth of the bars reflects the cumulative contribution
of each level, and the colors distinguish the successive layers that are
added as $k$ increases. Each panel exhibits a staircase structure in which
the total height at level $k = n$ recovers the value $\mathfrak{S}_n^1 = n$,
in agreement with the theorem.
This visualization makes transparent the two key features of the Assembly
Theorem. First, the accumulated sum at level $k = n$ always reconstructs the
count of indices $\{1, \dots, n\}$, regardless of the particular path taken
through the assembly. Second, the incremental contributions of each level
$k$ are precisely the harmonic weights $H_n^{(1, 1, \dots, 1)}$ of the
Assembly Set $\wp_k^k$, which by Proposition~\ref{prop:relacion1} coincide
with the tuple-based harmonic numbers $H_n^{\Theta}$ for the unique
permutation $\Theta = (1, 1, \dots, 1)$ of the partition $(1,1, \dots, 1)$. The
staircase structure of each panel therefore encodes the progressive
refinement of the assembly sum as new levels are incorporated, illustrating
the combinatorial mechanism that makes the identity $\mathfrak{S}_n^m = n$
hold for every value of $n$ and $m$. Finally the tuple-based version of the theorem follows by direct application of the relation between partition-based and tuple-based harmonic numbers.

\begin{corollary}[Generalized Assembly Theorem II]\label{cor:generalized_ensemble_v2}
For all $n, m, s \in \mathbb{N}$ with $s \le n$,
\begin{equation}\label{eq:assembly-theorem-v2}
\mathfrak{S}_{s,n}^{m} \ = \
\sum_{k=s}^{n} \ \sum_{\boldsymbol{\lambda} \in \wp_{k-s+1}^{k-s+m}}
\ \sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_{s,n}^{\Theta}
\ = \ \frac{n - s + 1}{s^m}.
\end{equation}
\end{corollary}
\begin{proof}
By Proposition~\ref{prop:relacion1},
$\sum_{\Theta \in \Psi(\boldsymbol{\lambda})} H_{s,n}^{\Theta} =
H_{s,n}^{\boldsymbol{\lambda}}$ for every partition $\boldsymbol{\lambda}$.
Substituting this identity into \eqref{eq:assembly-theorem-v2} reduces the
left-hand side to $\mathfrak{S}_{s,n}^m$, and
Theorem~\ref{teo:generalized_ensemble} then gives
$\frac{n - s + 1}{s^m}$.
\end{proof}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.76\linewidth]{9.pdf}
    \caption{Assembly Sums $\mathfrak{S}_{1,n}^{m}$ for $n = 6$. Each bar is
    the generalized harmonic number $H_n^{\boldsymbol{\lambda}}$ of one
    partition $\boldsymbol{\lambda}$ in the Assembly Set
    $\wp_k^{k+m-1}$. The bars of one color form a single branch, that is,
    one Assembly Sum
    $
    \mathfrak{S}_{1,n}^{m}
     = 
    \sum_{k=1}^{n} \ \sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}}
    H_n^{\boldsymbol{\lambda}}
     =  n,
    $
    where the equality follows from the Generalized Assembly
    Theorem~\ref{teo:generalized_ensemble}. The depth axis  lists all the partitions
    \(\boldsymbol{\lambda} \vdash (k+m-1)\), The horizontal axis encodes the parameter $k+m-1$, the
    vertical axis shows the value $H_n^{\boldsymbol{\lambda}}$ and the red lines connect partitions.
    Each branch independently reconstructs the count of indices
    $\{1, \dots, n\}$.}
    \label{fig:assembly-sums}
\end{figure}

The figure above displays the Assembly Sums for $n = 6$ with $s = 1$. Each
bar is the generalized harmonic number $H_n^{\boldsymbol{\lambda}}$ of one
partition $\boldsymbol{\lambda}$ in the Assembly Set $\wp_k^{k+m-1}$, and
the bars of one color form the Assembly Sum
\[
\mathfrak{S}_{1,n}^{m}
\ = \
\sum_{k=1}^{n} \ \sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}}
H_n^{\boldsymbol{\lambda}}
\ = \ n,
\]
by the Generalized Assembly Theorem~\ref{teo:generalized_ensemble}. Two features are apparent. First, the contributions of the
partitions are ordered by the excess parameter $m-1$. Second, the
accumulated value of every branch, obtained by stacking the bars of that
color, recovers the count $\mathfrak{S}_{1,n}^{m} = n$ for every $m$.

\begin{figure}[H]
    \centering
    \includegraphics[width=.9\linewidth]{10.pdf}
    \caption{Young's Lattice representation of the assembly sets
    \(\wp_k^{k+m-1}\) for \(n = 6\). The Assembly Sums $\mathfrak{S}_{1,n}^{m}$ for $n = 6$, with
    $s = 1$. Each bar is the generalized harmonic number
    $H_n^{\boldsymbol{\lambda}}$ of one partition $\boldsymbol{\lambda}$ in the truncated lattice (partitions with at
    most \(n\) summands) and the Assembly Set $\wp_k^{k+m-1}$. The bars are arranged in horizontal levels: each level
    corresponds to a fixed value of \(k+m-1\), so that all partitions
    \(\boldsymbol{\lambda} \vdash (k+m-1)\) lie on the same line. The bars of a single color form one
    Assembly Sum (one branch of the assembly), namely
    $
    \mathfrak{S}_{1,n}^{m}
     = 
    \sum_{k=1}^{n} \ \sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}}
    H_n^{\boldsymbol{\lambda}}.
    $
    The horizontal axis lists the partitions of \(k+m-1\), the
    vertical axis shows the value $H_n^{\boldsymbol{\lambda}}$, and the
    depth axis encodes the parameter $k+m-1$. By the Generalized Assembly
    Theorem~\ref{teo:generalized_ensemble}, every branch satisfies
    $\mathfrak{S}_{1,n}^{m} = n$, so each Assembly Sum reconstructs the
    count of indices $\{1, \dots, n\}$ independently of the excess
    parameter $m-1$.}
    \label{fig:assembly-sums-1}
\end{figure}



\subsection{Hierarchical expansion of \(e^\gamma\) via Multiple Zeta Values}
\label{subsec:mzvs}

We close the main framework with a expansion of the exponential of the Euler–Mascheroni constant in terms of multiple zeta values. This intuition is a natural consecuence of thinking like in the Assembly Theorem, by with the weights $\frac{1}{\boldsymbol{\lambda}!}$, because $e^{\gamma} n= e^{\gamma + \ln(n)} \approx  e^{H_n} = \sum_{k \in \mathbb{N}} \frac{H_n^k}{k!} -1 = \sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} -1$, we take in the Young's Lattice the partitions without $1$'s, \enquote{the leaders}, and factoring the partitions which are their \enquote{extension} of each leader by adding $1$'s, these are the partitions wich have the same partition factorial $\frac{1}{\boldsymbol{\lambda}!}$, because $1!$ do not affect the product. A sketch of a proof of this expansion was provided by Hucht in \cite{509300}.

\begin{theorem}[Hierarchical Expansion of \(e^\gamma\)]
\label{teo:egamma-mzv}
The constant \(e^\gamma\) admits the hierarchical expansion
\begin{align}
e^\gamma \ &= \ 
1 \ 
+ \ 
\sum_{a\ge2}
\frac1{a!}
\sum_{m\ge0}
(-1)^m\zeta^{\mathrm{A}}(a+m)
\notag\\
&\quad
+ \ 
\sum_{a,b\ge2}
\frac1{a!\,b!}
\sum_{m,n\ge0}
(-1)^{m+n}\zeta^{\mathrm{A}}(a+m,b+n)
\notag\\
&\quad
+ \ 
\sum_{a,b,c\ge2}
\frac1{a!\,b!\,c!}
\sum_{m,n,p\ge0}
(-1)^{m+n+p}\zeta^{\mathrm{A}}(a+m,b+n,c+p) \ 
+ \ 
\cdots,
\label{eq:egamma-hierarchical}
\end{align}
or, equivalently,
\begin{equation}
\label{eq:egamma-Sk}
e^\gamma \ 
= \ 
\sum_{k=0}^\infty S_k,
\end{equation}
where the depth-\(k\) sum is defined by
\begin{equation}
\label{eq:Sk-def}
S_k \ 
= \ 
\sum_{\substack{a_1,\dots,a_k\ge2\\ m_1,\dots,m_k\ge0}}
\frac{(-1)^{m_1+\cdots+m_k}}{a_1!\cdots a_k!}
\,\zeta^{\mathrm{A}}(A_1,\dots,A_k),
\qquad
A_i:=a_i+m_i
\end{equation}
(with the convention \(S_0=1\)), and \(\zeta^{\mathrm{A}}\) denotes the Abel-regularized multiple zeta value
\begin{equation}
\label{eq:abel-def}
\zeta^{\mathrm{A}}(s_1,\dots,s_k) \ 
= \ 
\lim_{r\to1^-}
\sum_{n_1>n_2>\dots>n_k\ge1}
r^{n_1}\,n_1^{-s_1}\cdots n_k^{-s_k}.
\end{equation}
\end{theorem}

\begin{proof}
By definition, the right-hand side of \eqref{eq:egamma-hierarchical} is precisely \(\sum_{k=0}^\infty S_k\), where \(S_k\) is given by \eqref{eq:Sk-def}. Thus it is enough to prove \eqref{eq:egamma-Sk}. Define the auxiliary sums of ordinary Riemann zeta values
\begin{equation}
\label{eq:Sigma-def}
\Sigma_k \ 
= \ 
\sum_{\substack{a_1,\dots,a_k\ge2\\ m_1,\dots,m_k\ge0}}
\frac{(-1)^{m_1+\cdots+m_k}}{a_1!\cdots a_k!}
\,\zeta(A_1+\cdots+A_k),
\qquad
A_i:=a_i+m_i.
\end{equation}
The series defining each \(\Sigma_k\) converges absolutely (the general term is \(O(j^{-2k})\)).

Abel regularization of series is classical (Hardy~\cite{hardy1949divergent}). When applied to multiple zeta values via \eqref{eq:abel-def}, it yields a morphism of algebras that preserves both the shuffle product and the stuffle product (Ihara--Kaneko--Zagier~\cite{ihara2006derivation}; see also Hoffman~\cite{hoffman2003algebraic}). Consequently, the symmetrized shuffle relations imply that each \(S_k\) coincides with the complete Bell polynomial evaluated at \(\Sigma_1,\dots,\Sigma_k\). The first few polynomials may be written explicitly:
$S_1=\Sigma_1, S_2=\frac12\bigl(\Sigma_1^2-\Sigma_2\bigr), S_3=\frac1{6}\bigl(\Sigma_1^3-3\Sigma_1\Sigma_2+2\Sigma_3\bigr).$
These are precisely the complete Bell polynomials evaluated at the sequence \((\Sigma_m)_{m\ge 1}\) with the natural sign pattern. In general one has the identity
\[
S_k \ 
= \ 
\frac1{k!}
B_k\bigl(\Sigma_1,\,-1!\,\Sigma_2,\,2!\,\Sigma_3,\,\dots,\,(-1)^{k-1}(k-1)!\,\Sigma_k\bigr),
\]
where \(B_k\) denotes the \(k\)-th complete exponential Bell polynomial. Equivalently, the exponential generating function of the sequence \((S_k)_{k\ge 0}\) is given by the formal power series identity
\begin{equation}
\label{eq:gen-fn}
\sum_{k=0}^\infty S_k\,x^k \ 
= \ 
\exp\Biggl(
-\sum_{k=1}^\infty\Sigma_k\frac{(-x)^k}{k}
\Biggr).
\end{equation}
We now evaluate the exponent at \(x=1\). Substitute the Dirichlet series \(\zeta(s)=\sum_{j=1}^\infty j^{-s}\) into \eqref{eq:Sigma-def}:
\begin{align*}
\Sigma_k \ 
&= \ 
\sum_{\substack{a_1,\dots,a_k\ge2\\ m_1,\dots,m_k\ge0}}
\frac{(-1)^{m_1+\cdots+m_k}}{a_1!\cdots a_k!}
\sum_{j=1}^\infty j^{-(A_1+\cdots+A_k)}\\
&= \ 
\sum_{j=1}^\infty
\Biggl(
\sum_{a\ge2}\sum_{m\ge0}
\frac{(-1)^m}{a!}\,j^{-(a+m)}
\Biggr)^k.
\end{align*}
The inner double sum is elementary:
\begin{align*}
\sum_{a\ge2}\frac{j^{-a}}{a!}
\sum_{m\ge0}(-j^{-1})^m \ 
&= \ 
\frac{e^{\tfrac{1}{j}}-1-j^{-1}}{1+j^{-1}} \ 
= \ 
\frac{e^{\tfrac{1}{j}}}{1+\tfrac{1}{j}}-1.
\end{align*}
Thus we obtain the explicit formula
\begin{equation}
\label{eq:Sigma-explicit}
\Sigma_k \ 
= \ 
\sum_{j=1}^\infty
\Biggl(
\frac{e^{\tfrac{1}{j}}}{1+\tfrac{1}{j}}-1
\Biggr)^k.
\end{equation}
(The interchange of sums is justified by absolute convergence.)
Insert \eqref{eq:Sigma-explicit} into the series for the exponent:
\begin{align*}
\sum_{k=1}^\infty\Sigma_k\frac{(-1)^{k+1}}{k} \ 
&= \ 
\sum_{j=1}^\infty
\sum_{k=1}^\infty
\frac{(-1)^{k+1}}{k}
z_j^k,
\end{align*}
where we have set
\[
z_j \ 
:= \ 
\frac{e^{\tfrac{1}{j}}}{1+\tfrac{1}{j}}-1.
\]
For every \(j\ge1\) one has \(0<z_j<1\), so the Taylor series of the logarithm converges absolutely:
\[
\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k}z_j^k \ 
= \ 
\log(1+z_j).
\]
A direct computation gives
\begin{align*}
\log(1+z_j) \ 
&= \ 
\log\Biggl(\frac{e^{\tfrac{1}{j}}}{1+\tfrac{1}{j}}\Biggr) \ 
= \ 
\frac1j+\log j-\log(j+1).
\end{align*}
Therefore
\begin{equation}
\label{eq:g1}
\sum_{k=1}^\infty\Sigma_k\frac{(-1)^{k+1}}{k} \ 
= \ 
\sum_{j=1}^\infty\Biggl(\frac1j+\log j-\log(j+1)\Biggr).
\end{equation}
The right-hand side of \eqref{eq:g1} is the classical series representation of the Euler--Mascheroni constant (Abramowitz--Stegun~\cite{abramowitz}; originally due to Euler):
\[
\gamma \ 
= \ 
\sum_{j=1}^\infty\Biggl(\frac1j+\log j-\log(j+1)\Biggr).
\]
Hence the exponent evaluated at \(x=1\) equals \(\gamma\). Substituting into the generating function \eqref{eq:gen-fn} yields
\[
\sum_{k=0}^\infty S_k \ 
= \ 
e^\gamma,
\]
which is precisely \eqref{eq:egamma-Sk} (and therefore also \eqref{eq:egamma-hierarchical}).
\end{proof}
\section{Espinosa's branches and exact, not asymptotic, divisor realization}
\label{sec:branches_divisors}

\subsection{The path toward the proof of the Riemann Hypothesis: divisor assembly via the Young's Lattice}
\label{sec:path-proof}

The Young's Lattice equivalence \eqref{eq:young-lattice} is built from four
structural components that, taken together, produce the inequality. The
first two components are the lattice of partition factorials
$\frac{1}{\boldsymbol{\lambda}!}$ and the lattice of generalized harmonic numbers
$H_n^{\boldsymbol{\lambda}}$, whose interaction defines the assembly
mechanism over the assembly sets
(Figures~\ref{fig:struct-factorials} and~\ref{fig:struct-harmonics}). The
third component is the sequence of coefficients $a_k$, whose growth profile
is essentially logarithmic and controls the analytic behaviour of the
partition-indexed series (Figure~\ref{fig:struct-ak}). The fourth component
is $\sum_{d \mid n} d$: the sum-of-divisors function, which constitutes the arithmetic target of
the equivalence (Figure~\ref{fig:struct-divisors}). Each of these four
structures plays a distinct and indispensable role in the construction, and
the content of the equivalence lies not in any of them individually but in
their deep mutual interaction. we must study each component to be able to understand and the mechanism by which they combine produce
the inequality becomes transparent.

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.41\linewidth}
        \centering
        \includegraphics[width=\linewidth]{12.pdf}
        \caption{Generalized harmonic numbers $H_n^{\boldsymbol{\lambda}}$: The lattice of generalized harmonic numbers. Each node carries a harmonic value, and their aggregation over the assembly sets produces the assembly mechanism.}
        \label{fig:struct-harmonics}
    \end{subfigure}
    \hspace{.66cm}
    \begin{subfigure}[b]{0.41\linewidth}
        \centering
        \includegraphics[width=\linewidth]{a.pdf}
        \caption{Partition factorials $\frac{1}{\boldsymbol{\lambda}!}$: the
        normalization weights of the assembly. The lattice of partition factorials. These weights normalize the contribution of each partition to the assembled sum.}
        \label{fig:struct-factorials}
    \end{subfigure}
    \vspace{1em}
    \begin{subfigure}[b]{0.43\linewidth}
        \centering
        \includegraphics[width=\linewidth]{4e.pdf}
        \caption{Coefficients $a_k$: the smooth analytic growth profile of the
        series, essentially logarithmic matched by $\log(k+1)$.}
        \label{fig:struct-ak}
    \end{subfigure}
    \begin{subfigure}[b]{0.54\linewidth}
        \centering
        \includegraphics[width=\linewidth]{13.pdf}
        \caption{$\sum_{d \mid n} d$: the sum-of-divisors function, the arithmetic target of the equivalence.}
        \label{fig:struct-divisors}
    \end{subfigure}
    \caption{The four structural components of the Riemann Hypothesis Young's Lattice Equivalence: $\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k
 >  \sum_{d \mid n} d.$}
    \label{fig:struct-all}
\end{figure}

We now turn to the structural analysis of the partition-indexed series that appears in the Young's Lattice equivalence. The goal of this section is to exhibit the mechanism by which the divisors of $n$ are assembled through the levels of Young's Lattice, see \cite{espinosa-part8a}, using the Generalized Assembly Theorem of Section~\ref{subsec:assembly_theorem}. This analysis reveals how each level of the partition decomposition contributes to the divisor sum. We analyze the partition-indexed series
\begin{equation}\label{eq:path-series}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k,
\qquad
a_k \ = \ \frac{1}{k} + \dots + \frac{1}{k^2}.
\end{equation}
Since $a_k \ge 1$ for every $k \in \mathbb{N}$ and all terms are positive, we
decompose the series according to the excess parameter $m-1$ and the length $k$
of the partition:
\begin{equation}\label{eq:path-decomposition}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k
\ = \
\sum_{m \ge 1} \left( \sum_{k=1}^{n}
\sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_{k+m-1} \right).
\end{equation}
We isolate the first two levels $m = 1, 2$ from the tail $m \ge 3$:
\begin{align}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k \
& = \ \sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^k}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k 
\notag \\
&\quad + \ \sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^{k+1}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_{k+1}
\label{eq:path-levels-12} \\
&\quad + \ \sum_{m \ge 3} \sum_{k=1}^{n}
\sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_{k+m-1}.
\notag
\end{align}
For the first two levels, the partition factorial takes a single value. At
level $m = 1$, the Assembly Set $\wp_k^k$ consists of the sole partition
$(1, \dots, 1)$, so $\boldsymbol{\lambda}! = (1!)^k = 1$, and
\begin{equation}\label{eq:path-level-1}
\sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^k}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k
\ = \ \frac{1}{1!} \sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^k}
H_n^{\boldsymbol{\lambda}} \, a_k.
\end{equation}
At level $m = 2$, the Assembly Set $\wp_k^{k+1}$ consists of the sole partition
$(2, 1, \dots, 1)$, so $\boldsymbol{\lambda}! = 2! \cdot (1!)^{k-1} = 2$, and
\begin{equation}\label{eq:path-level-2}
\sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^{k+1}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_{k+1}
\ = \ \frac{1}{2!} \sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^{k+1}}
H_n^{\boldsymbol{\lambda}} \, a_{k+1}.
\end{equation}
Using $a_k \ge 1$ and $a_{k+1} \ge 1$, together with the positivity of the
tail $m \ge 3$ in \eqref{eq:path-levels-12}, we obtain
\begin{equation}\label{eq:path-partial-bound}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k
\ > \
\frac{1}{1!} \sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^k}
H_n^{\boldsymbol{\lambda}}
\ + \
\frac{1}{2!} \sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^{k+1}}
H_n^{\boldsymbol{\lambda}}.
\end{equation}
By the Generalized Assembly Theorem (\ref{teo:generalized_ensemble}) with
$s = 1$,
\begin{equation}\label{eq:path-assembly}
\mathfrak{S}_n^m \ = \
\sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}}
H_n^{\boldsymbol{\lambda}} \ = \ n
\qquad \text{for every } m \in \mathbb{N}.
\end{equation}
Substituting \eqref{eq:path-assembly} into \eqref{eq:path-partial-bound} for
$m = 1$ and $m = 2$,
\begin{equation}\label{eq:path-partial-sum}
\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}
\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k
\ > \
\frac{n}{1!} + \frac{n}{2!}.
\end{equation}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.66\linewidth]{15.pdf}
    \caption{Young's Lattice representation of the assembly
    branches. Each colored region groups the partitions
    \(\boldsymbol{\lambda} \in \wp_k^{k+m-1}\) that belong to the same single branch. The assembly sums along each colored branch reproduce the
    corresponding divisor of \(n\), making explicit the correspondence
    between the divisor branches of Figure~\ref{fig:lattice-colors} and assembly
    branches along Young's Lattice. Each branch assembles
    the divisors of \(n\) assembled at a fixed excess parameter \(m-1\), so that the first color is the first branch (\(m = 1\)) wich covers the divisor \(\frac{n}{1}\) (level
    \(m = 1\)) and the second color is the second, it covers the divisor \(\frac{n}{2}\) (level
    \(m = 2\)). The
    assembly sums along each colored branch reproduce the corresponding
    divisor of \(n\).}
    \label{fig:divisor-branches}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=.9\linewidth]{14.pdf}
    \caption{Divisor branches \(\frac{n}{k}\). Each branch collects the divisors of the form \(\frac{n}{k}\). The bars represent the contribution of each
    branch to the sum-of-divisors function of $n$. The first branch is \(\frac{n}{1}\), while the second branch is \(\frac{n}{2}\) and so on.}
    \label{fig:lattice-colors}
\end{figure}

\section{Concluding Remarks}
\label{sec:conclusion}

This unified framework bridges partition theory, permutations and symmetric functions, Multiple Zeta Values, and the Young's Lattice to the study of the Riemann Hypothesis.
The author firmly believe that the proof of the Riemann Hypothesis can very possibly be obtained by adding the divisors branches simultaneously with adding the assembly sums along the partitions of
Young's Lattice. By creating an association of branches of divisors of the form $\frac{n}{k}$—as illustrated in the figure—with assembly branches along the assembly sets weighted by coefficients in the lattice.

\section*{Responsibility Notice}
The author accepts full responsibility for the integrity, originality, and accuracy of all content, data, and text presented in this manuscript.

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