title: Riemann hypothesis for period polynomials of modular forms.
abstract: The period polynomial r(f)(z) for an even weight k≥4 newform f∈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)(-1/Nz). We prove the Riemann hypothesis for these polynomials: that the zeros of r(f)(z) lie on the circle |z|=1/√N . We prove that these zeros are equidistributed when either k or N is large.
authors.affiliation: School of Mathematics, Korea Institute for Advanced Study, Dongdaemun-gu, Seoul 130-722, Korea;
authors.forename: Seokho
authors.initials: S
authors.lastname: Jin
authors.affiliation: School of mathematics, Shandong University, Jinan, Shandong 250100, China;
authors.forename: Wenjun
authors.initials: W
authors.lastname: Ma
authors.affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322; ono@mathcs.emory.edu.
authors.forename: Ken
authors.initials: K
authors.lastname: Ono
authors.affiliation: Department of Mathematics, Stanford University, Stanford, CA 94305.
authors.forename: Kannan
authors.initials: K
authors.lastname: Soundararajan
keywords: Riemann hypothesis; modular forms; period polynomials