Relations between theta-functions Hardy sums Eisenstein and Lambert series in the transformation formula of log ηg,h(z)

In this paper. by using generalized logarithms of Dedekind eta-functions, generalized logarithms of theta-functions are obtained. Applying these functions, the relations between Hardy sums and Theta-functions are found. The special cases of these relations give Berndt's Theorems 6.1-8.1 (J. Reine Angew. Math. 303,304 (1978) 332) and explicit formulae of Hardy sums. Using derivative of logarithms of the Dedekind eta-function, relations between logarithm of the theta-functions and Eisenstein series are given, Applying connection between Lambert series and generalized Dedekind Sums. the relation between theta-functions and Lambert series are obtained. (C) 2002 Elsevier Science (USA). All rights reserved.