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TRANSCENDENTAL SUMS RELATED TO THE ZEROS OF ZETA FUNCTIONS
Author(s) -
Gun Sanoli,
Murty M. Ram,
Rath Purusottam
Publication year - 2018
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579318000293
Subject(s) - mathematics , transcendental number , transcendental function , algebraic extension , algebraic function , riemann zeta function , algebraic number , quadratic field , zero (linguistics) , pure mathematics , rational function , entire function , algebra over a field , discrete mathematics , quadratic equation , mathematical analysis , quadratic function , differential equation , ordinary differential equation , differential algebraic equation , linguistics , philosophy , geometry
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 functionf ( x ) = ∑ ρx ρ ρhas 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 ) , thenL ′ ( 1 , χ − d) ≠ 0 ,whereχ − dis 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.