Li Coefficients for Automorphic L-Functions

Details Li Coefficients for Automorphic L-Functions Li Coefficients for Automorphic L-Functions

2004-04-21

arxiv.org/abs/math/0404394v4

Annales Institut Fourier 57 (2007), 1689-1740.

Mathematics

Number Theory

42 pages; minor corrections; references updated; correction to remainder term in Theorem 1.1

Scientific paper

Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients lambda_n, indexed by the integers. We define similar coefficients attached to principal automorphic L-functions over GL(N). We relate these coefficients to values of Weil's quadratic functional for the associated automorphic representation, and deduce a Riemann hypothesis criterion in terms of positivity of the real parts of these coefficients. We determine asymptotics of the coefficients, both unconditionally and on the Riemann hypothesis, as n increases. We show the existence of an entire function of exponential type that interpolates the generalized Li coefficients at integer values.

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