Modular case of Levinson's theorem

TL;DR

This paper investigates the zeros of L-functions associated with primitive cusp forms, providing explicit results that show a positive proportion of these zeros lie on the critical line, advancing understanding of their distribution.

Contribution

It offers a new explicit positive proportion result for zeros on the critical line for L-functions of primitive cusp forms, using mollified second moments.

Findings

Established an explicit positive proportion of zeros on the critical line.

Analyzed the mollified second moment of L-functions.

Enhanced understanding of zero distribution for cusp form L-functions.

Abstract

We evaluate the integral mollified second moment of L-functions of primitive cusp forms and we obtain, for such L-function, an explicit positive proportion of zeros which lie on the critical line.