On the behavior of the logarithm of the Riemann zeta-function

Abstract

The purpose of the present paper is to reveal the relation between the behavior of the logarithm of the Riemann zeta-function $\log{\zeta(s)}$ and the distribution of zeros of the Riemann zeta-function. We already know some examples for the relation by some previous works. For example, Littlewood showed an upper bound of $\log{\zeta(1/2 + it)}$ by assuming the Riemann Hypothesis in 1924. One of our results reveals that Littlewood's upper bound can be proved without assuming a hypothesis as strong as the Riemann Hypothesis.