Li coefficients for automorphic L-functions

Abstract

Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of the set of coefficients λ n = ρ 1 − 1 − 1 ρ n , (n = 1, 2, ...), in which ρ runs over the nontrivial zeros of the Riemann zeta function. We define similar coefficients λ n (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 λ n (π), in terms of the zeros of L(s, π). Assuming the Riemann hypothesis for L(s, π), we deduce thatwhere C 1 (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 F π (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.

Sharpenings of Li's Criterion for the Riemann Hypothesis

“…In the former case, the connection (16) ⇒ (15) often makes the asymptotic form of λ n fully explicit without any calculation, and it also ensures R −2 ≥ 0 in line with the corresponding generalized Li's criterion [2,14,18,19].…”

“…As we ended up with two mutually exclusive large-n behaviors for the λ n , (14) and (15), together they provide a sharp equivalence result. For the Riemann zeros, using the explicit values (6):…”

“…If both conditions (5) and (16) hold, then the polar coefficients of Z(σ) in (5) have to be precisely the R −j from (16). All of that is realized in many cases including, but not limited to, Dedekind zeta functions [15,13,27] and some Dirichlet L-functions [5,27,18]; see also [14] (discussed at end); in all those instances, δN(T ) = O(log T ). For the corresponding λ n , our saddle-point evaluation then always yields: either (16) ⇒ (15) if all the zeros have Re ρ = 1 2 -or the immediately general result (14) otherwise.…”

“…More recently and in a broader setting (the λ n for automorphic L-functions), Lagarias presented an alternative approach to estimate the λ n with greater accuracy, mod O( √ n log n) under GRH [14]: in the notations of (20)- (21), he directly proves that S n obeys (15) mod O(1) unconditionally (thm 5.1), then that S n = O( √ n log n) under GRH (thm 6.1). We note that his leading λ n -behavior remains tied to the large-T behavior of a counting function according to the rule (16) ⇒ (15) [14, Equations (2.11-12),(1.18),(5.2)].…”

On the asymptotic criterion for the zero-free regions of certain $L$-functions

“…Generalized coefficients for the Dirichlet and Hecke L -functions were treated in [8], for automorphic Lfunctions in [6], for the class S ♯♭ that contains both the Selberg class S and the class of automorphic L-functions in [11], and for Rankin-Selberg L -functions in [9].…”