TRANSCENDENTAL SUMS RELATED TO THE ZEROS OF ZETA FUNCTIONS. Issue 3 (1st August 2018) - Gun, Sanoli; Murty, M. Ram; Rath, Purusottam
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 ∑ ρ R ( ρ ) x ρ, where the sum is over the non‐trivial zeros ρ of ζ ( s ), R ( x ) ∊ 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 ) = ∑ ρ x ρ ρ has infinitely many zeros in ( 1, ∞ ), 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 ⩾ 1, f ( π d x ) has a rational zero in ( 0, 1 / π d ), then L ′ ( 1, χ − d ) ≠ 0, where χ − d is the quadratic character associated with the imaginary quadratic field K : = Q ( − 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.