On the structure of the Selberg class, VI: non-linear twists

Abstract

We obtain the analytic properties of the standards twists of the L-functions in the Selberg class, i.e. meromorphic continuation, polar structure and polynomial growth on vertical lines. We also obtain uniform bounds on vertical strips. Moreover, as an application we improve certain estimates for exponential sums involving Fourier coefficients of modular forms obtained by Iwaniec-Luo-Sarnak. The results in this paper are important for the proof of the degree conjecture, when 1<d<2, obtained by the same authors in Annals of Math. 173 (2011), 1397-1441

Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GL m (ℤ)

“…For similar results concerning coefficients of L-functions for general Selberg class, see Kaczorowski and Perelli [18] [19].…”

Twists and Resonance ofL‐Functions, II

“…As a consequence, the part of (2.6) coming from H ν (s, z), which we denote by k X (s), is uniformly bounded in X for −R < σ < −R + δ. Moreover, the computations before (3.6) on p.334 of [4] and (2.4) show that the part of (2.6) coming from Γ (ν) 1−s…”

“…This phenomenon, along with the transformation formula (see Theorem 1) and the properties of the standard twist (see [4] and [7]), is the basis for the most interesting aspects of our twist theory. Indeed, in [3] and [5] we exploited it to prove the degree conjecture for S ♯ in the range 1 < d < 2, while in this paper it is used to obtain the analytic properties of a new class of nonlinear twists.…”

“…. , w N ) coincides with the residual function R (1) N (s, α) defined on p.333 of [4], after the following changes in the latter function:…”

“…We continue our investigations of the analytic properties of nonlinear twists of L-functions developed in [4], [5] and [7]. Let F (s) be an L-function of degree d. First we extend the transformation formula in [5], relating a twist F (s; f ) with leading exponent κ 0 > 1/d to its dual twist F (s; f * ).…”