math.NT — Number Theory

The Riemann Hypothesis on Young's Lattice

Contributed by José Damián Espinosa

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. $$e^\gamma = 1 + \sum_{a\ge2} \frac1{a!} \sum_{m\ge0} (-1)^m\zeta^{\mathrm{A}}(a+m) + \sum_{a,b\ge2} \frac1{a!\,b!} \sum_{m,n\ge0} (-1)^{m+n}\zeta^{\mathrm{A}}(a+m,b+n) + \cdots$$