The Riemann Hypothesis For Period Polynomials Of Modular Forms

Abstract

The period polynomial $r_f(z)$ for an even weight $k\ge 4$ 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.

11 pages. We added the one sentence explanation for the $2n^{(k-1)/2}$ bound for ${\Lambda }_f(n)$. This paper will be appearing in Proceedings of the National Academy of Sciences, USA