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arXiv · 1601.03114

The Riemann Hypothesis For Period Polynomials Of Modular Forms

Seokho Jin·Wenjun Ma·Ken Ono·Kannan Soundararajan
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Abstract

The period polynomial $r_f(z)$ for an even weight $k\geq 4$ newform $f\in S_k(Γ_0(N))$ 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.

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Seokho Jin, Wenjun Ma, Ken Ono, Kannan Soundararajan. 2016-02-09. The Riemann Hypothesis For Period Polynomials Of Modular Forms. https://doi.org/10.1073/pnas.1600569113

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