# Transcendental sums related to the zeros of zeta functions

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

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

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#### 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.