# The -Riemann Hypothesis- is true for period polynomials of almost all newforms

Research in the Mathematical Sciences

, 3:31

First Online: 11 December 2016Received: 18 July 2016Accepted: 15 August 2016

Abstract

The period polynomial fz\ for a weight \k \ge 3\ and level N newform \f \in S k\varGamma 0N,\chi \ is the generating function for special values of Ls, f. The functional equation for Ls, f induces a functional equation on fz\. Jin, Ma, Ono, and Soundararajan proved that for all newforms f of even weight \k \ge 4\ and trivial nebentypus, the -Riemann Hypothesis- holds for fz\: that is, all roots of fz\ 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 fz\ are equidistributed if N or k is sufficiently large.

Author: Yang Liu - Peter S. Park - Zhuo Qun Song