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 λnn (n = 1,2, . . .). We define similar coefficients λn(π) associated to principal automorphic L-functions L(s, π) over GL(N). We relate these cofficients to values of Weil's quadratic functional associated to the representation π on a suitable set of test functions. The positivity of the real parts of these coefficients is a necessary and sufficient condition for the Riemann hypothesis for L(ε, π). Assuming the Riemann hypothesis for L(s, π), we show that λn(π) = N/2n log n + C1(π)n + O(√nlog n), where C1(π) is a real-valued constant. We construct an entire function Fπ(z) of exponential type that interpolates the generalized Li coefficients at integer values. Assuming the Riemann hypothesis for L(s, π), this function on the real axis has a Fourier transform that is a tempered distribution whose support is a countable set in [-π, π] having 0 as its only limit point.