Levinson's theorem and its generalization for Dirichlet L-functions

TL;DR

This paper proves Levinson's theorem for the Riemann zeta function and generalizes it to Dirichlet L-functions, showing a significant proportion of their zeros lie on the critical line, with implications for understanding their distribution.

Contribution

It provides a new proof of Levinson's theorem and extends the result to Dirichlet L-functions, demonstrating that over two-fifths of their zeros are on the critical line, including simple zeros.

Findings

Over one-third of Riemann zeta zeros are on the critical line.

More than two-fifths of Dirichlet L-function zeros are on the critical line.

A longer mollifier improves the zero distribution results.

Abstract

In this report, we present a proof of Levinson's theorem, following the ideas of Matthew P. Young in 2010, which states that one-third of the non-trivial zeros of the Riemann zeta function lie on the critical line, i.e. the line Re(s) = 1/2, using a mollified second moment of the zeta-function. Later, we present a generalized result for Dirichlet L-functions by Xiaosheng Wu in 2018, using Levinson's method, showing that more than two-fifths of the non-trivial zeros of Dirichlet L-functions are on the critical line. Moreover, more than two-fifths of the non-trivial zeros are simple and on the critical line, using a longer mollifier than in Levinson's original proof. This generalizes a result by Conrey from 1989 that the Riemann zeta-function has at least two-fifths of its zeros on the critical line.