The first moment of cusp form $L$-functions in weight aspect on average

Abstract

We study the asymptotic behaviour of the twisted first moment of central L-values associated to cusp forms in weight aspect on average. Our estimate of the error term allows extending the logarithmic length of mollifier ∆ up to 2. The best previously known result, due to Iwaniec and Sarnak, was ∆ < 1. The proof is based on a representation formula for the error in terms of Legendre polynomials.

Averages of long Dirichlet polynomials with modular coefficients

“…In this paper, we focus on averages of Hecke L-functions at the central point 1 , in the weight aspect; in particular, we consider the family H k = H k (1) of primitive cusp forms of level 1 and (even) weight k ≥ 6 and we study the r-th moment…”

“…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 .…”

“…Primary 11F11; Secondary 11M99. 1 We normalize the L-function so that the functional equation links values at s to 1 − s, so the central point is s = 1 2 .…”

“…(2. 15) We note that the above lemma provides a meromorphic continuation for F A (m, s) and m F A (m, s)χ(m)m −w in the region ℜ(s) > 1 2 , ℜ(w) > 0. Proof.…”

“…Lemma 2.2. For F A (m, s) and C(X; θ) defined in (2.5) and (2.11) respectively, ℜ(s) > 1 2 , m ∈ N, we have…”