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

Abstract

Abstract We consider iterated integrals of log ⁡ ζ ⁢ ( s ) {\log\zeta(s)} on certain vertical and horizontal lines. Here, the function ζ ⁢ ( s ) {\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 ∫ 0 t log ⁡ ζ ⁢ ( 1 2 + i ⁢ t ′ ) ⁢ 𝑑 t ′ {\int_{0}^{t}\log\zeta(\frac{1}{2}+it^{\prime})\,dt^{\prime}} under the Riemann Hypothesis. Moreover, we show that, for any m ≥ 2 {m\geq 2} , the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.