An extension of Motohashi’s observation on the zero-free region of the Riemann zeta-function

Abstract

We give an extension of Yoichi Motohashi's theorem saying that if the Riemann zeta-function on the line Re s = 1 attains very small values, then Vinogradov's zero-free region can be improved.

A consequence of Littlewood's conditional estimates for the Riemann zeta-function

“…Introduction. The approach of Y. Motohashi [1] to the zero-free region of the Riemann zeta-function extended by the author in [2] may be modified to give regions free of large values of some products, which contain finite products j ζ(s j ). On the Riemann hypothesis, one can obtain upper bounds for such products for s j = 1 + t j using the method of Littlewood.…”