Even and Odd Integer Values of the Secondary Zeta Function

Abstract

The secondary zeta function is defined by $$ Z(s)=\sum_{γ>0}\frac{1}{γ^{s}}, $$ where $\tfrac12+iγ$ 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 $ξ$-function. Define $$ L(x)=\log\left(\frac{ξ(x+\tfrac12)}{ξ(\tfrac12)}\right), \qquad \mathcal{E}=x\frac{\mathrm{d}}{\mathrm{d}x}, \qquad \mathcal{D}_k=\mathcal{E}\prod_{j=1}^{k}(\mathcal{E}-2j). $$ Then, for $k\geq1$, $$ Z(2k)=\frac{(-1)^k}{2^k(k-1)!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+1}}\,\mathrm{d}x $$ and $$ Z(2k+1)=\frac{(-1)^k}{π\,(2k-1)!!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+2}}\,\mathrm{d}x. $$ In the last part, we write these formulas out explicitly for $Z(k)$, $2\leq k\leq 7$, both in terms of $L(x)$ and in terms of $ξ(x+\tfrac12)$ for reference. We also derive a limit formula for $γ_1$, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.