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...

We provide systematic evaluations, in terms of binary quadratic representations of 4p, for the p-th Fourier coefficients of each member f of an infinite class C of CM eigenforms. As an application, previously conjectured evaluations of three algebro-geometric character sums can now be formulated explicitly without reference to eigenforms. There are several non-CM newforms that appear to share s...

These notes describe some conjectures and results related to the distribution of Fourier coefficients of modular forms. This is a rough draft and these notes should forever be considered incomplete.

Let {ej} be an orthonormal basis of Laplace eigenfunctions a compact Riemannian manifold (M, g). H ⊂ M submanifold and {ψk} with the induced metric. We obtain joint asymptotics for Fourier coefficients $${\left\langle {{\gamma _H}{e_j},{\psi _k}} \right\rangle _{{L^2}(H)}} = {\int_H {{e_j}\overline \psi } _k}d{V_H}$$ restrictions γhej ej to H. In particular, we sums norm-squares over spectrum $...

Abstract We are looking at families of functions or measures on the torus which specified by a finite number parameters N . The task, for given family, is to look small Fourier coefficients object, set locations that predetermined and may depend only , determine object. (a) indicator most intervals (b) sums complex point masses multidimensional torus. In first case we reprove theorem Courtney s...