The Riemann Hypothesis For Period Polynomials Of Modular Forms

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The period polynomial rf(z) for an even weight k≥ 4 newform f∈ Sₖ(Γ₀(N)) is the generating function for the critical values of L(f,s). It has a functional equation relating rf(z) to rf(-1/Nz). We prove the Riemann Hypothesis for these polynomials: that the zeros of rf(z) lie on the circle |z|=1√N. We prove that these zeros are equidistributed when either k or N is large.