Fourier–Dedekind sums and an extension of Rademacher reciprocity

Rademacher sums, Moonshine and Gravity

In 1939 Rademacher demonstrated how to express Klein’s modular invariant as a sum over elements of the modular group. In this article we generalize Rademacher’s approach so as to construct bases for the spaces of automorphic integrals of arbitrary even integer weight, for an arbitrary group commensurable with the modular group. Our methods provide explicit expressions for the Fourier expansions...

Lévy’s inequality, Rademacher sums, and Kahane’s inequality

Let (Ω,A , P ) be a probability space. A random variable is a Borel measurable function Ω→ R. For a random variable X, we denote by X∗P the pushforward measure of P by X. X∗P is a Borel probability measure on R, called the distribution of X. A random variable X is called symmetric when the distribution of X is equal to the distribution of −X. Because the collection {(−∞, a] : a ∈ R} generates t...

Elliptic Apostol Sums and Their Reciprocity Laws

We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter τ having positive imaginary part. When τ → i∞, these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable τ . We obtain a ...

On the Tail Estimation of the Norm of Rademacher Sums

The main aim of this paper is to prove a bilateral inequality for P [‖ n ∑ 1 akrk‖ > t ] , where t > 0, (ak) are elements of a normed space, while (rk) are Rademacher functions. Then this inequality is applied for estimation of E‖ n ∑ 1 akrk‖. As another corollary we give a maximal inequality for exchangeable random variables that recently has been published in [4]. 2000 Mathematics Subject Cla...

A. Quadratic Reciprocity Via Gauss Sums