Arithmetic means of Fourier coefficients

FOURIER COEFFICIENTS OF GLpNq AUTOMORPHIC FORMS IN ARITHMETIC PROGRESSIONS

Abstract. We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GLpNq Maass cusp forms for all N > 2, satisfy a central limit theorem in a suitable range, generalizing the case N “ 2 treated by É. Fouvry, S. Ganguly, E. Kowalski and P. Michel in [4]. Such universal Gaussian behaviour relies on a deep e...

Fourier Coefficients of Gl(n) Automorphic Forms in Arithmetic Progressions

Transformations of Fourier Coefficients

Introduction. In 1923 M. Fekete [2]2 introduced the concept of factor sequences that left invariant the class of a Fourier series. That is, Fekete investigated the conditions to which a sequence of constants (X„) must be subjected in order that (X„a„, X„i„) be Fourier coefficients of a function of the same class, K, as that of the function determined by (an, bn). Whenever (X„) has this property...

Subsets of Fpn without three term arithmetic progressions have several large Fourier coefficients

Suppose that f : Fpn → [0, 1] satisfies Σaf(m) = θF ∈ [F , F ], where F = |Fpn | = p. In this paper we will show the following: Let fj denote the size of the jth largest Fourier coefficient of f . If fj < θ j1/2+δF, for some integer j satisfying J0(δ, p) < j < F , then S = support(f) contains a non-trivial three-term arithmetic progression. Thus, the result is asserting that if the Fourier tran...

The Calculation of Fourier Coefficients

The Möbius inversion technique is applied to the Poisson summation formula. This results in expressions for the remainder term in the Fourier coefficient asymptotic expansion as an infinite series. Each element of this series is a remainder term in the corresponding Euler-Maclaurin summation formula, and the series has specified convergence properties. These expressions may be used as the basis...