On Li’s coefficients for the Rankin–Selberg L-functions

Abstract

We define a generalized Li coefficient for the L-functions attached to the Rankin-Selberg convolution of two cuspidal unitary automorphic representations π and π of GL m (A F ) and GL m (A F ). Using the explicit formula, we obtain an arithmetic representation of the nth Li coefficient λ π,π (n) attached to L(s, π f × π f ). Then, we deduce a full asymptotic expansion of the archimedean contribution to λ π,π (n) and investigate the contribution of the finite (non-archimedean) term. Under the generalized Riemann hypothesis (GRH) on non-trivial zeros of L(s, π f × π f ), the nth Li coefficient λ π,π (n) is evaluated in a different way and it is shown that GRH implies the bound towards a generalized Ramanujan conjecture for the archimedean Langlands parameters μ π (v, j ) of π. Namely, we prove that under GRH for L(s, π f × π f ) one has | Re μ π (v, j )| ≤ 1 4 for all archimedean places v at which π is unramified and all j = 1, . . . , m.

Euler-Stieltjes constants for the Rankin-Selberg L-function and weighted Selberg orthogonality

“…Combining Theorem 2 and Theorem 4 from [27] it is easy to see that, under the generalized Riemann hypothesis S NA (n, π, π ′ ) = o(n), as n → ∞. In general, it is very difficult to control the growth of S NA (n, π, π ′ ).…”

“…In a general case of a number field E of degree l = [E : Q] , it is proved in [27] that coefficients λ π,π ′ (n) are well defined for all integers n and that the generalized Riemann hypothesis for the Rankin-Selberg L−function L(s, π × π ′ ) is equivalent to non-negativity of numbers Reλ π,π ′ (n), for all n ∈ N. Furthermore, an arithmetic expression for λ π,π ′ (n) is obtained, as stated in the following theorem.…”

“…, m ′ ). An asymptotic expression for the archimedean contribution S ∞ (n, π, π ′ ) to the generalized Li coefficient λ π,π ′ (−n) is obtained in [27,Theorem 2], up to a term O(n −k ), for arbitrary k ∈ N, as n → ∞, where it was proved that S ∞ (n, π, π ′ ) grows as 1 2 lmm ′ n log n, as n → ∞. Derivation of asymptotic behavior as n → ∞ of the non-archimedean (finite) contribution…”

“…For example, coefficients γ π,π ′ (k) appear in the arithmetic formula for the Li coefficients attached to L(s, π × π ′ ) that is used to formulate the Li criterion for the generalized Riemann hypothesis for the Rankin-Selberg convolution in [27]. Expressing coefficients γ π,π ′ (k) in terms of the Satake and Langlands parameters related to π and π ′ enables one to relate the generalized Riemann hypothesis to non-negativity of sequence of numbers expressed only in terms of arithmetic parameters related to corresponding representations, see Corollary 3.5. below.…”

On Li's criterion for the Riemann hypothesis for the Selberg class

“…However, all that is unconditionally known at present is the bound Re κ j (π ) > −1/2, proved by Z. Rudnick and P. Sarnak in [30,Appendix], and the bound |Re κ j (π )| 1/2 − 1/(n 2 + 1) proved in [19,Theorem 2], for the representation π unramified at the archimedean place. Under the Generalized Riemann Hypothesis, exploiting properties of the Li coefficients attached to the Rankin-Selberg L-functions, the last bound is improved to the bound |Re κ j (π )| 1/4 (for π that is unramified at the archimedean place) in [23,Corollary 3].…”