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On Lehmer's problem and Dedekind sums

Xiaowei Pan, Wenpeng Zhang (2011)

Czechoslovak Mathematical Journal

Let p be an odd prime and c a fixed integer with ( c , p ) = 1 . For each integer a with 1 a p - 1 , it is clear that there exists one and only one b with 0 b p - 1 such that a b c (mod p ). Let N ( c , p ) denote the number of all solutions of the congruence equation a b c (mod p ) for 1 a , b p - 1 in which a and b ¯ are of opposite parity, where b ¯ is defined by the congruence equation b b ¯ 1 ( mod p ) . The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet L -functions to study the hybrid mean value problem involving...

On limit distribution of the Hurwitz zeta-function

Antanas Laurinčikas (2010)

Open Mathematics

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.

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