Algebraic integers as values of elliptic functions

Algebraic Independence of Values of Elliptic Functions

Chromatic roots as algebraic integers

A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the “α + n conjecture” and the “nα conjecture”. These say, respectively, that given any algebraic integer α there is a natural number n such ...

Algebraic values of analytic functions

Given an analytic function of one complex variable f , we investigate the arithmetic nature of the values of f at algebraic points. A typical question is whether f(α) is a transcendental number for each algebraic number α. Since there exist transcendental entire functions f such that f (t)(α) ∈ Q[α] for any t ≥ 0 and any algebraic number α, one needs to restrict the situation by adding hypothes...

Algebraic Values of Transcendental Functions at Algebraic Points

In this paper, the authors will prove that any subset of Q can be the exceptional set of some transcendental entire function. Furthermore, we could generate this theorem to a much more general version and present a unified proof.

Dedekind Sums and Values of L-functions at Positive Integers

In this paper, we study Dedekind sums and we connect them to the mean values of Dirichlet L-functions. For this, we introduce and investigate higher order dimensional Dedekind-Rademacher sums given by the expression Sd( −→ a0 , −→ m0) = 1 a0 0 a0−1 ∑