Even and Odd Integer Values of the Secondary Zeta Function

Abstract

The secondary zeta function is defined by $$Z(s)=\sum _{\gamma >0}\frac{1}{{\gamma }^s},$$ where $\frac{1}{2}+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of $Z(s)$ at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann $\xi$-function. Define $$L(x)=\log \left(\frac{\xi (x+\frac{1}{2})}{\xi (\frac{1}{2})}\right),\mathcal{E}=x\frac{\mathrm{d}}{\mathrm{d}x},{\mathcal{D}}_k=\mathcal{E}\prod _{j=1}^k(\mathcal{E}-2j).$$ Then, for $k\ge 1$, $$Z(2k)=\frac{(-1{)}^k}{2^k(k-1)!}{\int }_0^{\infty }\frac{{\mathcal{D}}_{k}L(x)}{x^{2k+1}}\mathrm{d}x$$ and $$Z(2k+1)=\frac{(-1{)}^k}{\pi (2k-1)!!}{\int }_0^{\infty }\frac{{\mathcal{D}}_{k}L(x)}{x^{2k+2}}\mathrm{d}x.$$ In the last part, we write these formulas out explicitly for $Z(k)$, $2\le k\le 7$, both in terms of $L(x)$ and in terms of $\xi (x+\frac{1}{2})$ for reference. We also derive a limit formula for ${\gamma }_1$, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.

arXiv comment14 pages