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

Ray, Swapnil (2025) Levinson’s theorem and its generalization for Dirichlet L-functions. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Swapnil Ray (20MS102)) 20MS102_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (467kB)

Abstract

In this report, we shall present a proof of Levinson’s theorem, published by Matthew P. Young in 2010, which states that one-third of the non-trivial zeroes of the Riemann-zeta function ζ(s) lie on the critical line, i.e. the line Re(s) = 1/2, using a mollified second moment of the zeta-function. Later, we shall present a generalized result for Dirichlet’s L-function presented by Xiaosheng Wu in 2018, using the method of Levinson that more than two-fifths of the non-trivial zeroes of the Dirichlet’s L-functions are on the critical line and in addition, more than two-fifths of the non-trivial zeroes are simple and on the critical line, using a longer mollifier than the one used in the proof of Levinson’s theorem. This is a generalization of a result deduced by Conrey in 1989 than the Riemann zeta-function has atleast two-fifth zeroes on the critical line.

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