On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness

Abstract

We consider iterated integrals of {\log\zeta(s)} on certain vertical and horizontal lines. Here, the function {\zeta(s)} is the Riemann zeta-function. It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of {\int_{0}^{t}\log\zeta(\frac{1}{2}+it^{\prime})\,dt^{\prime}} under the Riemann Hypothesis. Moreover, we show that, for any {m\geq 2}, the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.

Universality for the iterated integrals of logarithms of $L$-functions in the Selberg class

“…The problem that the set {ζ(1 2 + it) ∶ t ∈ R} is dense or not in C is still open. For this reason, Endo and Inoue [7] considered the following functions ηm (s) defined by…”

Large deviations for values of $L$-functions attached to cusp forms in the level aspect

“…Finally, an analogue of (1.11) for Φ 1,T (σ, τ ) was achieved by Lamzouri-Lester-Radziwi l l [16,Theorem 1.3]. See also [8] for a refinement.…”

On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: denseness

“…From these backgrounds, the authors adopted the method not involving the probability density function as the first step of the study of the value-distribution of iterated integrals of the logarithm of the Riemann zetafunction. We shall have further deep arguments including the probability density function in [5].…”