Li Coefficients for Automorphic L-Functions
Abstract
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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- April 2004
- DOI:
- 10.48550/arXiv.math/0404394
- arXiv:
- arXiv:math/0404394
- Bibcode:
- 2004math......4394L
- Keywords:
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- Mathematics - Number Theory;
- 11M26 22E55
- E-Print:
- 42 pages