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

For $v>1$, let
\[
 \beta(v)=\sum_{j=0}^{\infty}\frac{(-1)^j}{(2j+1)^v} =1-\frac1{3^v}+\frac1{5^v}-\cdots
\]
be the Dirichlet beta function. Its values at positive odd integers
are nonzero rational multiples of powers of $\pi$. The arithmetic
of its even values is much less understood. We study how many
irrational even values must occur in an interval whose starting
index is allowed to vary.

For an integer $B\ge1$, put
\begin{equation}\label{eq:closed-endpoint}
 D_B=\sum_{d=1}^B\varphi(d),\qquad
 s_B=2\left\lceil D_B\left(33+\frac{45}{2}\log B\right)\right\rceil,
\end{equation}
where $\varphi$ is Euler's totient function and $\log$ is the natural
logarithm.

\begin{theorem}\label{thm:intervals}
For every positive odd integer $m$ and every integer $B\ge1$,
at least $B$ of the numbers
\begin{equation}\label{eq:interval}
 \beta(m+1),\beta(m+3),\ldots,\beta\bigl(m(s_B+1)-1\bigr)
\end{equation}
are irrational. In particular, at least one of
\[
 \beta(m+1),\beta(m+3),\ldots,\beta(67m-1)
\]
is irrational for every positive odd $m$.
\end{theorem}

The upper endpoint grows linearly with the starting index,
with a multiplier depending only on $B$. Choosing the construction's
parameters for specific starting indices and counts gives the following
smaller upper endpoints (Section~\ref{sec:endpoints}), compared here
with those from Theorem~\ref{thm:intervals}:
\begin{center}
\begin{tabular}{@{}rrrr@{}}
\toprule
First even index&\multicolumn{2}{c}{Last even index}&\# irrational values\\
\cmidrule(lr){2-3}
&Specific parameters&Theorem~\ref{thm:intervals}&(at least)\\
\midrule
4&198&200&1\\
6&330&334&1\\
8&462&468&1\\
20&3080&3742&2\\
20&6950&8796&3\\
\bottomrule
\end{tabular}
\end{center}

\subsection{Comparison with earlier results}

For any fixed number $B$ of irrational values,
Theorem~\ref{thm:intervals} gives an upper endpoint that grows
linearly with the starting index $m+1$.

Let $I(s)$ count the irrational values among
$\beta(2),\beta(4),\ldots,\beta(s)$. Our earlier paper~\cite{Tcount}
proves that, for every even $s\ge160$,
\begin{equation}\label{eq:earlier-count}
 I(s)\ge\left\lfloor c\sqrt{\frac{s}{\log s}}\right\rfloor,
 \qquad c=\pi\sqrt{\frac{2}{135}}.
\end{equation}
To use this count for an interval starting at $\beta(m+1)$,
we must allow for all $(m-1)/2$ preceding values to be irrational.
We therefore need a count of at least
\[
 \frac{m-1}{2}+B
\]
to ensure that $B$ irrational values remain in the interval.
For fixed $B$, this target grows proportionally to $m$.
Reaching it with the square-root bound~\eqref{eq:earlier-count}
requires $s$ of order $m^2\log m$: quadratic growth in the
starting index, with an additional logarithmic factor.

For example, the earlier paper also proves the sharper count
$I(3828)\ge10$. At most nine of these irrational values can have
indices below $20$, so at least one of
$\beta(20),\beta(22),\ldots,\beta(3828)$ is irrational.
Theorem~\ref{thm:intervals} reduces this upper endpoint to $1272$.
The table compares endpoints guaranteeing one irrational value
from each listed starting index:
\begin{center}
\begin{tabular}{@{}rrr@{}}
\toprule
First even index&Endpoint from~\cite{Tcount}&Endpoint here\\
\midrule
4&160&198\\
6&364&330\\
20&3828&1272\\
100&187\,340&6632\\
\bottomrule
\end{tabular}
\end{center}
The middle column follows from the bounds $I(160)\ge2$,
$I(364)\ge3$, $I(3828)\ge10$ and $I(187\,340)\ge50$
in~\cite{Tcount}. Discarding the first $1$, $2$, $9$ and $49$
even beta values, respectively, leaves at least one irrational
value in each interval.

The last column uses the individual parameter choices in
Section~\ref{sec:endpoints} at the starting indices $4$ and $6$,
and the bound $67m-1$ from Theorem~\ref{thm:intervals}
at $20$ and $100$.

Sorokin~\cite[Theorem~S]{S07} treats shifted intervals directly.
For every sufficiently large odd $m$ and every even
$q>100m\log m$, he proves that at least one of
\[
 \beta(m+1),\beta(m+3),\ldots,\beta(m+q-1)
\]
is irrational. Huttner~\cite[Remark~5]{H12} gives the same
conclusion. Taking the smallest allowable $q$ gives an upper
endpoint of order $m\log m$. The case $B=1$ of
Theorem~\ref{thm:intervals} removes this logarithmic factor:
$q=66m$ suffices for every positive odd $m$.

Earlier finite results guaranteed an irrational value in a block
beginning at $\beta(2)$, with the endpoint reduced to $\beta(10)$
by Lai and Zhou~\cite[Theorem~6.1]{LZ22}. These statements are now
subsumed by the irrationality of $\beta(2)$~\cite{OAI26}.
At the starting index $4$, Krisman~\cite{K26} gives a sharper
result than either bound in the table: at least one of
$\beta(4),\beta(6),\ldots,\beta(36)$ is irrational.

\subsection{How the proof locates the irrational values}

We construct $B$ sequences of linear forms, all supported on the
desired interval. The integer $m$ fixes its starting point, and
$B$ fixes the count; the sequence index $n$ tends to infinity.
Their beta coefficients have the common shape
\[
 F_{n,b}=q_b(n)+
   \sum_{\substack{m+1\le\nu\le s\\\nu\text{ even}}}
           b^\nu\rho_\nu(n)\beta(\nu),\qquad 1\le b\le B.
\]
The dependence on the grid index $b$ is through the powers $b^\nu$.
If at most $B-1$ of these beta values were irrational, a fixed
integer combination of the $B$ forms would cancel their coefficients
at every $n$. This would produce a sequence of rational numbers.
We arrange that clearing their denominators gives integers tending
to zero, and that infinitely many of them are nonzero. Since a
nonzero integer has absolute value at least $1$, that is impossible.

To remove the values below $\beta(m+1)$, we differentiate $m-1$
times. For example, normalized second differentiation changes
\[
 \frac{a_i}{(t+4h)^i}
 \quad\hbox{into}\quad
 \binom{i+1}{2}\frac{a_i}{(t+4h)^{i+2}}.
\]
Reflection removes odd indices, so the first surviving even
exponent is $4$. The zeros used to control rational denominators
must survive this operation; we therefore start with triple zeros.
The general construction pairs $(m-1)$st differentiation with
zeros of multiplicity $m$. The second-derivative and triple-zero
device is also used in the beta setting by Krisman~\cite{K26}.

The remaining question is whether the rational constants still
provide enough freedom after differentiation. We express the
constants at the seed index $n=1$ as a Gram matrix of independent
polynomials against a positive weight. Differentiating the kernels
in the Gram construction of~\cite[Section~3]{Tcount} preserves this positive
representation. It lets us prescribe a nonzero seed constant that
survives the final cancellation. Prime dilation then propagates
that constant to infinitely many indices.

Our starting point is the construction in~\cite{Tcount}. We extend
its polynomial multipliers, grouped denominator argument and
prime-dilation calculation to the differentiated functions. The
rational-grid construction and coefficient elimination follow the
methods used for odd zeta values by Lai and Yu~\cite[Sections~2
and~5]{LY20} and adapted to real Dirichlet characters by Kumar
and Tosi~\cite[Section~2]{KT26}.

Section~\ref{sec:construction} defines the functions and linear
forms. Section~\ref{sec:gram} constructs the seed, and
Sections~\ref{sec:arithmetic}--\ref{sec:dilation} prove the
denominator, decay and nonvanishing estimates.
Section~\ref{sec:contradiction} completes the contradiction;
Section~\ref{sec:endpoints} chooses the parameters.