Almost all of the zeros of the Riemann zeta-function are on the critical line

TL;DR

This paper introduces a novel approach to studying the zeros of the Riemann zeta-function by focusing on a subset of the critical line and employing advanced approximation and integral techniques, aiming to prove that almost all zeros lie on the critical line.

Contribution

It proposes a new method that uses selective application of inequalities and integral estimates, leveraging the functional equation and mollifier construction to improve zero distribution analysis.

Findings

The approach suggests that almost all zeros are on the critical line.

Utilizes a refined mollifier construction based on Schwarz-Christoffel mapping.

Depends on zero-density estimates near the critical line.

Abstract

This is a reworked version of the paper. An idea that allows us to circumvent limitations of previous approaches is not to apply arithmetic-geometric mean inequality and the second moment asymptotics to the entire segment $[1/2-a/\log T+iT,1/2-a/\log T+i2T]$ but use them on a subset only, and use the integral of logarithm of the mollified function on the complement. Ultimately, the result depends on the exponent in the zero-density estimate near the critical line, which leads to the relation between the magnitude of $\widetilde{V}^{1/2}$ and the measure of the exceptional set in Theorem 4, Section 2.3. The exponents of Jutila and Conrey are enough for our purposes. We provide more details on an effective approximation of $1/z$ using the Schwarz-Christoffel mapping. This is needed in the construction of the mollifier. One observation on why the approach is feasible is that the functional equation established in the paper allows one to shift the segment of integration $[1/2-a/\log T+iT,1/2-a/\log T+i2T]$ to $[1/2+A/\log T+iT,1/2+A/\log T+i2T]$. A result of Selberg allows for a proof that almost all of zeros of our integrand are to the left of the shifted segment.