Transcendental sums related to the zeros of zeta functions

Research paper by Sanoli Gun, M. Ram Murty, Purusottam Rath

Indexed on: 30 Jul '18Published on: 30 Jul '18Published in: arXiv - Mathematics - Number Theory

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.