The “Riemann Hypothesis” is true for period polynomials of almost all newforms

@article{Liu2016TheH,
title={The “Riemann Hypothesis” is true for period polynomials of almost all newforms},
author={Yang Liu and Peter S. Park and Zhuo Qun Song},
journal={Research in the Mathematical Sciences},
year={2016},
volume={3},
pages={1-11}
}
• Published 16 July 2016
• Mathematics
• Research in the Mathematical Sciences
The period polynomial $$r_f(z)$$rf(z) for a weight $$k \ge 3$$k≥3 and level N newform $$f \in S_k(\varGamma _0(N),\chi )$$f∈Sk(Γ0(N),χ) is the generating function for special values of L(s, f). The functional equation for L(s, f) induces a functional equation on $$r_f(z)$$rf(z). Jin, Ma, Ono, and Soundararajan proved that for all newforms f of even weight $$k \ge 4$$k≥4 and trivial nebentypus, the “Riemann Hypothesis” holds for $$r_f(z)$$rf(z): that is, all roots of $$r_f(z)$$rf(z) lie on the…

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