Zeros of the Dirichlet series of even zeta values

Abstract

We give a complete unconditional description of the zero set of the Dirichlet series $\mathbf{D}(s)={\sum }_{n\ge 1}\zeta (2n)n^{-s}$, which continues meromorphically to $\mathbb{C}$ with a single simple pole at $s=1$. The series possesses neither an Euler product nor a self-dual functional equation, and descriptions with this level of completeness are exceedingly rare for such series. The key input is an exact functional equation of Hecke type, obtained from the Lipschitz summation formula, which expresses $\mathbf{D}$ in the left half-plane as a gamma factor times a dual series over the complex logarithms of the perfect squares; Riemann's functional equation appears as a single column of the dual series. The zeros fall into four families. The half-plane $\sigma \ge {\sigma }_0=1.5001\dots$ is zero-free, and $\mathbf{D}$ has a unique real zero ${\rho }_0=0.2004\dots$. The zeros in the critical strip are perturbed $a$-points of $\zeta$ for values of $a$ near $-(\zeta (2)-1)$, and their counting function obeys a Riemann--von Mangoldt law. The remaining zeros form two complex-conjugate strings that recede into the left half-plane along explicit rays, are eventually simple, and satisfy an asymptotic with geometrically decaying error. The string geometry is governed by interference between the two smallest frequencies of the dual series, $2\log 2$ contributed by the entire part of $\mathbf{D}$ and $\pm 2\pi i$ contributed by $\zeta$. No hypothesis of Riemann type is assumed at any point.

v2: Conjecture 4.3 of v1 is now Proposition 4.3, proved following an observation of Qiping Zhou; two references added. No other result is affected. 26 pages, 4 figures, 3 tables