Density theorems for Riemann’s zeta-function near the line Re s =1

Pintz, János (2023) Density theorems for Riemann’s zeta-function near the line Re s =1. ACTA ARITHMETICA. ISSN 0065-1036

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Abstract

We prove a series of density theorems for Riemann’s zeta-function for the number of zeros lying near the boundary line Re s = 1 of the critical strip. In particular, we improve the constant appearing in the exponent of the Hal ́asz–Tur ́an density theorem. The proof uses the relatively recent strong estimate for the zeta-function near the line Re s = 1 showed by Heath-Brown. The necessary exponential sums were estimated by Heath-Brown via the new results of Wooley and of Bourgain, Demeter and Guth on Vinogradov’s mean value integral.

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