First moments of Fourier coefficients of GL(r) cusp forms

Abstract

Let π be an irreducible cuspidal automorphic representation of GL(r, A) where r ≥ 2 and A is the adele ring of Q. Let a π (n) denote the nth Dirichlet series coefficient of the L-function associated to π. The main goal of this paper is to obtain strong bounds for the first moment n≤x a π (n) as x → ∞. The bounds we obtain are better than all previously obtained bounds for the higher rank situation when r ≥ 3 and π is not a symmetric power of a GL(2, A) cuspidal automorphic representation.

Oscillations of coefficients of Dirichlet series attached to automorphic forms

“…We apply the above theorem to deduce the following result on sign changes of coefficients of L-series attached to certain automorphic forms. However, the bounds of Golfeld and Sengupta [4] for the first moment of the coefficients of the automorphic L-functions are enough to prove the following result. Theorem 1.3.…”

“…The method of proof of the above theorem is same as in [4], but we use a result of Lü [9] which ultimately follows from the result of Chandrasekharan and Narasimhan [1]. The difference in our proof is that we estimate differently which is similar to Lü [9] and get a better bound in the general case too.…”

On sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$

“…which is absolutely convergent for ℜ(s) > 1 due to (7) and defines a holomorphic function on this half-plane. Since f is a Hecke eigenform, the coefficients A(m, 1, ..., 1) are multiplicative by using (4).…”

“…Note that the condition y < (nY /2) n is satisfied by the choice of Y . The contribution coming from the error terms of Corollary 5 can be estimated as follows by using partial summation together with (7) and recalling the fact that Y ≍ X (1+θ)/n : Let us fix a smooth compactly supported non-negative weight function w majorising the characteristic function of the interval [1,2]. Now we simply compute:…”

“…Concerning the higher rank analogue, Goldfeld and Sengupta [7] have recently shown that for the Fourier coefficients of a GL(n) Maass cusp form, the upper bound m≤x A(m, 1, ..., 1) ≪ ε x (n 3 −1)/(n 3 +n 2 +n+1)+ε holds for any n ≥ 3. Again the trivial bound for the sum is ≪ ε x 1+ε .…”