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 integral(t)(0) log zeta(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.