2018/07/30 by Sanoli Gun, M. Ram Murty, Gun, Sanoli +3 · 1 citation
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1807.11201
openalex publication_date 2018/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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) xρ, 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 \fracxρρ 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 to 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.