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

2020/04/02 by Endo, Kenta, Inoue, Shota
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2004.00823

Abstract

We consider iterated integrals of logζ(s) on certain vertical and horizontal lines. Here, the function ζ(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 ∫₀^t log ζ(1/2 + it')dt' under the Riemann Hypothesis. Moreover, we show that, for any m≥ 2, the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.

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