Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms

Abstract

Let ν f (n) be the nth normalized Fourier coefficient of a Hecke-Maass cusp form f for SL(2, Z) and let α be a real number. We prove strong oscillations of the argument of ν f (n)µ(n) exp(2πinα) as n takes consecutive integral values.

Quadratic forms connected with Fourier coefficients of holomorphic and Maass cusp forms

“…where c is a suitable positive constant. We claim that, taking into account the absence of exceptional zeros proved by Hoffstein-Ramakrishnan and a consequent estimate of [1], a stronger bound can be obtained. More precisely, our purpose is to prove the following statement.…”

“…We will not give the details of the proof. The bound can be easily deduced from [1,Theorem 4.1]. In particular, it follows with standard methods by formula (28), which is essentially based on the zero-free region proved by Hoffstein-Ramakrishnan (see [2, Theorem C, part (3)]) and the consequent absence of exceptional zeros.…”

On exponential sums involving Fourier coefficients of cusp forms over primes

“…Hence it suffices to consider the sum on the right-hand side. Let 0 ≤ φ(n) ≤ 1 be a C ∞ function supported on (1 − (1/16N )), 2 + (3/16N )), which is identically equal to one on [1,2] and satisfies φ (j) N j for any j ≥ 0. We then have N ≤n≤2N…”

“…As a connection between (1.6) and the normalized Fourier coefficient of a primitive holomorphic form (or Maass cusp form) f , Fouvry and Ganguly [2] proved a uniform bound that for any…”

An Extension of the Bourgain–sarnak–ziegler Theorem With Modular Applications

“…The proof of this result is nowadays rather standard thanks to the non-existence of the Siegel zeros for the twisted Hecke L-functions associated with the cusp form f , proved by Hoffstein-Ramakrishnan [5] in 1995. Indeed, one may follow the arguments in Perelli [15], plugging in this extra information, or use those in Sections 4 and 7 of Fouvry-Ganguly [3], already incorporating the Hoffstein-Ramakrishnan theorem. Proof.…”