The Riemann Hypothesis For Period Polynomials Of Modular Forms

TL;DR

This paper proves the Riemann Hypothesis for period polynomials of modular forms, showing their zeros lie on a specific circle and are equidistributed as parameters grow large.

Contribution

It establishes the zero distribution of period polynomials of modular forms, confirming the Riemann Hypothesis for these functions and analyzing their equidistribution properties.

Findings

Zeros lie on the circle |z|=1/√N

Zeros are equidistributed for large k or N

Confirms Riemann Hypothesis for these polynomials

Abstract

The period polynomial $r_f(z)$ for an even weight $k\geq 4$ newform $f\in S_k(\Gamma_0(N))$ is the generating function for the critical values of $L(f,s)$. It has a functional equation relating $r_f(z)$ to $r_f\left(-\frac{1}{Nz}\right)$. We prove the Riemann Hypothesis for these polynomials: that the zeros of $r_f(z)$ lie on the circle $|z|=\frac{1}{\sqrt{N}}$ . We prove that these zeros are equidistributed when either $k$ or $N$ is large.