The "Riemann Hypothesis" is True for Period Polynomials of Almost All Newforms

TL;DR

This paper proves that for almost all newforms of weight at least 3, the roots of their period polynomials satisfy a Riemann Hypothesis-like property, and these roots become equidistributed as parameters grow large.

Contribution

It generalizes previous results to all but finitely many newforms of weight ≥3 with any nebentypus, establishing the Riemann Hypothesis for their period polynomials.

Findings

Roots of period polynomials lie on the symmetry circle for almost all newforms.

Roots are equidistributed when the level or weight is large.

The Riemann Hypothesis holds for all but finitely many newforms of specified weight and nebentypus.

Abstract

The period polynomial $r_f(z)$ for a weight $k \geq 3$ newform $f \in S_k(\Gamma_0(N),\chi)$ is the generating function for special values of $L(s,f)$. The functional equation for $L(s, f)$ induces a functional equation on $r_f(z)$. Jin, Ma, Ono, and Soundararajan proved that for all newforms $f$ of even weight $k \ge 4$ and trivial nebetypus, the "Riemann Hypothesis" holds for $r_f(z)$: that is, all roots of $r_f(z)$ lie on the circle of symmetry $|z| =1/\sqrt{N}$. We generalize their methods to prove that this phenomenon holds for all but possibly finitely many newforms $f$ of weight $k \ge 3$ with any nebentypus. We also show that the roots of $r_f(z)$ are equidistributed if $N$ or $k$ is sufficiently large.