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

Kenta Endo; Shota Inoue

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arXiv · 2004.00823

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

Kenta Endo·Shota Inoue

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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.

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Kenta Endo, Shota Inoue. 2020-04-02. On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: denseness. https://arxiv.org/abs/2004.00823

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