Given a function f (n) periodic of period q ≥ 1 and an irrational number 0 < α ≤ 1, Chatterjee and Gun (cf. [4]) proved that the series F (s, f, α) = ∞ n=0 f (n) (n+α) s has infinitely many zeros for σ > 1 when α is transcendental and F (s, f, α) has a pole at s = 1, or when α is algebraic irrational and c = max f (n) min f (n) < 1.15. In this note, we prove that the result holds in full generality.
Given a degree 1 function F ∈ S and a real number α, we consider the linear twist F (s, α), proving that it satisfies a functional equation reflecting s into 1 − s, which can be seen as a Hurwitz-Lerch type of functional equation. We also derive some results on the distribution of the zeros of the linear twist.
In this work we prove a prime number type theorem involving the normalised Fourier coefficients of holomorphic and Maass cusp forms, using the classical circle method. A key point is in a recent paper of Fouvry and Ganguly, based on Hoffstein-Ramakrishnan's result about the non-existence of the Siegel zeros for GL(2) L-functions, which allows us to improve preceding estimates.
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