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The "Riemann Hypothesis" is True for Period Polynomials of Almost All Newforms (1607.04699v2)
Published 16 Jul 2016 in math.NT and math.CV
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.