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

TL;DR

This paper investigates the denseness of iterated integrals of the logarithm of the Riemann zeta-function, establishing new results on their distribution on horizontal lines and linking the denseness on the critical line to the Riemann Hypothesis.

Contribution

It proves the denseness of iterated integrals on horizontal lines and shows the equivalence of denseness on the critical line to the Riemann Hypothesis.

Findings

Denseness of iterated integrals on horizontal lines established.

Denseness of integrals on the critical line linked to the Riemann Hypothesis.

Equivalence between denseness of m-times iterated integrals and the Riemann Hypothesis for m ≥ 2.

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 $\int_{0}^{t} \log \zeta(1/2 + it')dt'$ 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.