Sampling the Lindelöf hypothesis for Dirichlet $$L$$ L -functions by the Cauchy random walk

Abstract

We study the average value of Dirichlet L-function along vertical lines σ + i R inside the critical strip 1 N 1≤n≤N L(σ+i S n , χ), where S n is the sequence of partial sums of the infinite series of independent Cauchydistributed random variables and χ is a primitive character modulo q. By using Atkinson's formula and a proposition of Lifshits and Weber (Proc Lond Math Soc 98(1):241-270, 2009), we show that the expectation value of L(s, χ) on the Cauchy random walk s = σ + i S n equals one, which implies that the values of Dirichlet L-function are small on average.

An ergodic value distribution of certain meromorphic functions

“…This result implies that most of the values of ζ(s) on the critical line are quite small. Analogous to [LW09], T. Srichan investigated the value distributions of Dirichlet L-functions and Hurwitz zeta functions by using the Cauchy random walk in [Sri15].…”