Riemann hypothesis for period polynomials of modular forms.
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- Record sourced from PubMed, PMID 26903628.
- Also identified by DOI 10.1073/pnas.1600569113 and PMC identifier 4790983.
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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.