TRANSCENDENTAL SUMS RELATED TO THE ZEROS OF ZETA FUNCTIONS

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 where the sum is over the non-trivial zeros of , is a rational function over algebraic numbers and is a real algebraic number. In particular, we show that the function has infinitely many zeros in , at most one of which is algebraic. The transcendence tools required for studying in the range seem to be different from those in the range . For , we have the following non-vanishing theorem: If for an integer , has a rational zero in, then where is the quadratic character associated with the imaginary quadratic field . Finally, we consider analogous questions for elements in the Selberg class. Our proofs rest on results from analytic as well as transcendental number theory.

Locations

• arXiv (Cornell University) - View - PDF
• Mathematika - View