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TRANSCENDENTAL SUMS RELATED TO THE ZEROS OF ZETA FUNCTIONS

Part of: Diophantine approximation, transcendental number theory Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  01 August 2018

Sanoli Gun ,
M. Ram Murty  and
Purusottam Rath
Sanoli Gun
Affiliation:
Institute of Mathematical Sciences, Homi Bhabha National Institute, C.I.T Campus, Taramani, Chennai 600 113, India email [email protected]
M. Ram Murty
Affiliation:
Department of Mathematics, Queen’s University, Jeffrey Hall, 99 University Avenue, Kingston, ON K7L3N6, Canada email [email protected]
Purusottam Rath
Affiliation:
Chennai Mathematical Institute, Plot No H1, SIPCOT IT Park, Padur PO, Siruseri 603103, Tamil Nadu, India email [email protected]
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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

$$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D70C}}R(\unicode[STIX]{x1D70C})x^{\unicode[STIX]{x1D70C}},\end{eqnarray}$$
where the sum is over the non-trivial zeros $\unicode[STIX]{x1D70C}$ of $\unicode[STIX]{x1D701}(s)$, $R(x)\in \overline{\mathbb{Q}}(x)$ is a rational function over algebraic numbers and $x>0$ is a real algebraic number. In particular, we show that the function
$$\begin{eqnarray}f(x)=\mathop{\sum }_{\unicode[STIX]{x1D70C}}\frac{x^{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}}\end{eqnarray}$$
 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 $d\geqslant 1$, $f(\unicode[STIX]{x1D70B}\sqrt{d}x)$ has a rational zero in$(0,1/\unicode[STIX]{x1D70B}\sqrt{d})$, then
$$\begin{eqnarray}L^{\prime }(1,\unicode[STIX]{x1D712}_{-d})\neq 0,\end{eqnarray}$$
 where $\unicode[STIX]{x1D712}_{-d}$ is the quadratic character associated with the imaginary quadratic field $K:=\mathbb{Q}(\sqrt{-d})$. Finally, we consider analogous questions for elements in the Selberg class. Our proofs rest on results from analytic as well as transcendental number theory.

MSC classification

Primary: 11J81: Transcendence (general theory) 11J86: Linear forms in logarithms; Baker's method
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$
Type
Research Article
Information
Mathematika , Volume 64 , Issue 3 , 2018 , pp. 875 - 897
Copyright
Copyright © University College London 2018 

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Footnotes

Research of the second author was supported by an NSERC Discovery grant and a Simons Fellowship.

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