Density theorems for Riemann’s zeta-function near the line ${\rm Re}\,s = 1$
Volume 208 / 2023
Acta Arithmetica 208 (2023), 1-13
MSC: Primary 11M26; Secondary 11M06.
DOI: 10.4064/aa210824-10-5
Published online: 26 June 2023
Abstract
We prove a series of density theorems for Riemann’s zeta-function for the number of zeros lying near the boundary line ${\rm Re}\,s = 1$ of the critical strip. In particular, we improve the constant appearing in the exponent of the Halász–Turán density theorem. The proof uses the relatively recent strong estimate for the zeta-function near the line ${\rm Re}\,s = 1$ 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.
