The "Riemann Hypothesis" is True for Period Polynomials of Almost All\n Newforms

2016/07/15 by Yang P. Liu, Peter S. Park, Liu, Yang P. +3
Mathematics · #11F11 #11F67 #Advanced Algebra and Geometry #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1607.04699

openalex publication_date 2016/07/15 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28

Abstract

The period polynomial r_f(z) for a weight k \≥ 3 newform f \∈\nS_k(\Γ₀(N),\χ) is the generating function for special values of\nL(s,f). The functional equation for L(s, f) induces a functional equation\non r_f(z). Jin, Ma, Ono, and Soundararajan proved that for all newforms f\nof even weight k \≥ 4 and trivial nebetypus, the "Riemann Hypothesis" holds\nfor r_f(z): that is, all roots of r_f(z) lie on the circle of symmetry |z|\n=1/\√(N). We generalize their methods to prove that this phenomenon holds\nfor all but possibly finitely many newforms f of weight k \≥ 3 with any\nnebentypus. We also show that the roots of r_f(z) are equidistributed if N\nor k is sufficiently large.\n

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