A uniform asymptotic formula for the second moment of primitive L-functions on the critical line

Abstract

Abstract. We prove an asymptotic formula for the second moment of automorphic L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift and twist.

Averages of long Dirichlet polynomials with modular coefficients

“…In this paper, we focus on averages of Hecke 𝐿-functions at the central point † , in the weight aspect; in particular, we consider the normalized Hecke basis 𝐻 𝑘 = 𝐻 𝑘 (1) of the space of holomorphic cusp forms of level 1 and (even) weight 𝑘 ⩾ 6 and we study the 𝑟th moment of the associated 𝐿-functions, that is,…”

Averages of long Dirichlet polynomials with modular coefficients

“…where the superscript h indicates the usual harmonic weight arising from Petersson norm and involving the symmetric square of the L-function. For small integer values of r, this problem has been studied in various works [1,7,8,2,3,19] but for an arbitrary integer r it is currently out of reach. However, very precise conjectures have been formulated; L-functions associated to primitive cusp forms of weight k form an orthogonal family in the sense of Katz and Sarnak [24], thus one expects M r (k) to be asymptotic, as k goes to infinity, to a certain (explicit) polynomial of degree r(r−1) 2 .…”

Non-vanishing of automorphic L-functions of prime power level

“…We follow the notations of [2] and let The Lerch zeta function with parameters α, β ∈ R is defined by…”

“…The value c = 1/4 is a natural barrier. Iwaniec and Sarnak [12] showed that if inequality (1.7) holds for some c > 1/4 with an additional lower bound L(1/2, f ) ≥ 1/(log N) 2 , there are no Landau-Siegel zeros for Dirichlet L-functions of real primitive characters.…”

“…Asymptotic formula for the second moment, which is the hardest part of our arguments, is derived in [2]. The proof is based on techniques developed by Kuznetsov and Bykovskii, see [4,5,6,16].…”