Unconditional representations of the Euler-Mascheroni constant and the Bernoulli numbers via the Taylor coefficients of the Riemann ξ-function, and a hierarchy of exact relations at the central point

Abstract

We give a self-contained, numerically verified derivation of three equivalent unconditional representations of the Euler--Mascheroni constant $\gamma$ as a rapidly convergent series in the Taylor coefficients $a_{2n}$ of the Riemann $\xi$-function about its centre of symmetry $s=1/2$ (equivalently, in the Jensen coefficients $c_n$ or in the Csordas--Norfolk--Varga Turan moments ${\hat{b}}_n$). We also record the corresponding convergent formula for the even-indexed Bernoulli numbers $B_{2r}$, obtained by evaluating the (everywhere convergent) Taylor series of $\xi$ at $s=2r$. We then develop, from the evenness of $\Xi (x)=\xi (1/2+ix)$, an infinite hierarchy of exact relations at $s=1/2$: the even-order relations reduce the central logarithmic derivatives of $\zeta$ to closed forms in $\pi$ and the odd zeta values (base case $2{\zeta }^{\prime }(1/2)/\zeta (1/2)=\gamma +\pi /2+\log (8\pi )$), and the odd-order relations evaluate the power sums over the nontrivial zeros and link back to the coefficients $a_{2n}$. All of these identities hold whether or not the Riemann Hypothesis (RH) is true; consequently they carry no evidential weight for or against RH, and we make the correct, unconditional relationship of these coefficient sequences to the Jensen--Polya program precise. This note is a corrected version of an earlier preprint [arXiv:2112.11228v1]: we retain only the results that are correct, we remove a divergent series of Bernoulli numbers that was previously used as though it were convergent, we correct several attributions, and we withdraw the earlier claim that these computations support RH. A summary of the corrections is given in Section 7.

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