Value distribution for the derivatives of the logarithm of L-functions from the Selberg class in the half-plane of absolute convergence

摘要

In the present paper, we show that, for every delta > 0, the function (log L(s))((m)), where m is an element of NU {0} and L(s) := Sigma(infinity)(n=1) a(n)n(-s) is an element of the Selberg class S, takes any value infinitely often in the strip 1 < Re(s) < 1 + delta, provided Sigma(p <= x) vertical bar a(p)vertical bar(2) similar to kappa pi(x) for some kappa > 0. In particular, L(s) takes any non-zero value infinitely often in the strip 1 < Re(s) < 1 + delta, and the first derivative of L(s) has infinitely many zeros in the half-plane Re(s) > 1. (C) 2015 Elsevier Inc. All rights reserved.