Transcendental sums related to the zeros of zeta functions

TL;DR

This paper investigates the transcendental nature of sums over the non-trivial zeros of the Riemann zeta function, revealing properties about zeros and connections to L-functions, using advanced number theory techniques.

Contribution

It introduces new results on the transcendence and zeros of sums involving zeta zeros, and links these properties to the non-vanishing of derivatives of L-functions.

Findings

The function f(x) has infinitely many zeros in (1, ∞), with at most one algebraic zero.

A non-vanishing theorem for f(π√d x) relates to the non-vanishing of L'(1, χ_{-d}).

Results extend to elements in the Selberg class, connecting zeros to transcendence properties.

Abstract

While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form $$ \sum_{\rho } R(\rho) x^{\rho}, $$ where the sum is over the non-trivial zeros $\rho$ of $\zeta(s)$, $R(x) \in \overline{\Q}(x) $ is a rational function over algebraic numbers and $x >0$ is a real algebraic number. In particular, we show that the function $$ f(x) = \sum_{\rho } \frac{x^{\rho}}{\rho} $$ has infinitely many zeros in $(1, \infty)$, at most one of which is algebraic. The transcendence tools required for studying $f(x)$ in the range $x<1$ seem to be different from those in the range $x>1$. For $x < 1$, we have the following non-vanishing theorem: If for an integer $d \ge 1$, $f(\pi \sqrt{d} x)$ has a rational zero in $(0,~1/\pi \sqrt{d})$, then $$ L'(1,\chi_{-d}) \neq 0, $$ where $\chi_{-d}$ is the quadratic character associated to the imaginary quadratic field $K:= \Q(\sqrt{-d})$. Finally, we consider analogous questions for elements in the Selberg class. Our proofs rest on results from analytic as well as transcendental number theory.