Abstract
Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of the set of coefficientsλn = P ρ 1 − � 1 − 1 � n , (n = 1,2, ...), in which ρ runs over the nontrivial zeros of the Riemann zeta function. We define similar coefficientsλn(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(s, π) to hold. We derive an unconditional asymptotic formula for the coefficients λn(π), in terms of the zeros of L(s, π). Assuming the Riemann hypothesis for L(s, π), we deduce that λn(π) = N 2 nlog n + C1(π)n + O( √ nlog n), where C1(π) is a real-valued constant and the implied constant in the remainder term depends on π. We also show that there exists a entire function Fπ(z) of exponential type that interpolates the generalized Li coefficients at integer values. Assuming the Riemann hypothesis there is an (essentially) unique interpolation function having exponential type at most π, and this function restricted to the real axis has a (tempered) distributional Fourier transform whose support is a countable set in [−π, π] having 0 as its only limit point.
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