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As a consequence of their work, Bruce C. Berndt, Ronald J. Evans, Larry Joel Goldstein and Michael Razar obtained a formula for the square of the class number of an imaginary quadratic number field in terms of Dedekind sums. We give a short proof of it and also express the relative class numbers of imaginary abelian number fields in terms of Dedekind sums. [ABSTRACT FROM AUTHOR]

Let q ≥ 2 be an integer and let s q (n) be the sum-of-digitsfunction of n in base q. The function s q (n) has been studied in many directions and many properties have been obtained on the distribution of s q (n) and s q (P (n)) , where P is a suitable polynomial. In this paper we derive the generatingfunctions of s p (n d mod p k) for d ≥ 2 and prime p ≥ 2 by using the properties of Dirichlet character, and study the correlation properties of the sequences ((- 1) n 2 mod p k ) n < p k . [ABSTRACT FROM AUTHOR]

For integers m, n, q, k, with q, k≧1 and Dirichlet characters $\chi, \chi' \text {\rm \;(mod}\,q)$ we define a generalized Kloosterman sum with a Dirichlet character and a Gauss sum G( a, χ′) as coefficient, where e( z)= e. The aim of this paper is to study the fourth power mean obtaining explicit formulas for M( q). [ABSTRACT FROM AUTHOR]