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

Abstract

In the present paper, we show that, for every δ > 0, the function (log L(s)) (m) , where m ∈ N ∪ {0} and L(s) := ∞ 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 + δ, provided p≤x |a(p)| 2 ∼ κπ(x) for some κ > 0. In particular, L(s) takes any non-zero value infinitely often in the strip 1 < Re(s) < 1 + δ, and the first derivative of L(s) has infinitely many zeros in the half-plane Re(s) > 1.2010 Mathematics Subject Classification. Primary 11M06, 11M26.

Zeros of $L(s)+L(2s)+\cdots+L(Ns)$ in the region of absolute convergence

“…Note that in all aforementioned papers it was crucial to have a combination of L-functions with an abscissa of absolute convergence equal to 1. Combinations of two L-functions, where at least one term has an abscissa of absolute convergence 1 were partially investigated by Nakamura and Pańkowski in [11]. Let us notice that the case of a combination of only two terms in much easier as it is sufficient to consider the value-distribution of a quotient of two L-functions, which has good property like an Euler product.…”