On Legendre numbers.
Let be an odd prime and a fixed integer with . For each integer with , it is clear that there exists one and only one with such that (mod ). Let denote the number of all solutions of the congruence equation (mod ) for , in which and are of opposite parity, where is defined by the congruence equation . The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet -functions to study the hybrid mean value problem involving...
The distribution of the vector (|ζ(s, α)|; ζ(s, α)), where ζ(s, α) is the Hurwitz zeta-function with transcendental parameter α, is considered and a probabilistic limit theorem is obtained. Also, the dependence between |ζ(s, α)| and ζ(s, α) in terms of m-characteristic transforms is discussed.
Given a subsequence of a uniformly distributed sequence, relations between the asymptotic densities of sets of its indices and the Lebesgue measure of the set of all its limit points are studied.