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

arXiv eprints
 Pub Date:
 July 2016
 arXiv:
 arXiv:1607.04699
 Bibcode:
 2016arXiv160704699L
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Complex Variables;
 11F11;
 11F67
 EPrint:
 11 pages, to appear in Res. Math. Sci