Closed Formulas for Sums over Riemann Zeros: S1, S2 and the Alternating Sum T1 via the V-Transform

Abstract

We present a novel application of the V-transform — introduced by the author in 1989 — to the theory of the Riemann zeta function. Assuming the Riemann Hypothesis (RH), we derive explicit closed-form expressions for the sums $$S_1=\sum_{k=1}^\infty\frac{1}{\gamma_k^2+1/4}, \qquad S_2=\sum_{k=1}^\infty\frac{1}{(\gamma_k^2+1/4)^2},$$ where $\gamma_k$ denotes the imaginary part of the $k$-th nontrivial zero of $\zeta(s)$. For $S_1$ we give a new proof of the classical formula first verified numerically by Keiper (1992). For $S_2$ we obtain the compact expression $$S_2 = 3 + (\ln 2\pi)^2 - 2\ln 2\pi + \ln\pi + \gamma - \frac{\pi^2}{24} + 2\zeta''(0),$$ with $\zeta''(0) = -2.006356\ldots$ the second derivative of $\zeta$ at $0$. This formula, while implicit in the sum-rule framework of Lehmer (1988) via the relation $S_2 = Z(2) + 2S_1$, does not appear explicitly in this form in the literature. Numerical verification against the first $10^5$ Odlyzko zeros confirms the result with a relative error of $0.04\%$. We further investigate the alternating analogues $$T_1=\sum_{k=1}^\infty\frac{(-1)^k}{\gamma_k^2+1/4}, \qquad T_2=\sum_{k=1}^\infty\frac{(-1)^k}{(\gamma_k^2+1/4)^2},$$ obtaining numerical values $T_1 = -0.003733\ldots$ and $T_2 = -0.000021774\ldots$ via direct summation. These lead naturally to two new constants $\eta_\pi'(0)$ and $\eta_\pi''(0)$, whose analytic nature — whether expressible in terms of known transcendental numbers or special values of $L$-functions — is left as an open problem. The method is entirely tabular: it reduces the Hadamard product factorisation of $\eta(s) = (s-1)\zeta(s)$ to elementary convolutions via the trinomial formula (PA-13) of the V-transform, requiring neither explicit knowledge of individual zeros nor the full Hadamard expansion.