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

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 ${\int }_0^t\log \zeta (1\mathrm{/}2+i{t}^{\mathrm{\prime }})d{t}^{\mathrm{\prime }}$ under the Riemann Hypothesis. Moreover, we show that, for any $m\ge 2$, the denseness of the values of an $m$-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.