On Generalized Hardy Sums $s_5(h, k)$

Abstract

The aim of this paper is to study generalized Hardy sums $s_5(h, k)$. By using mediants and the adjacent difference of Farey fractions, we establish a relationship between $s_5(h, k)$ and Farey fractions. Using generalized Dedekind sums and a generalized periodic Bernoulli function, we define generalized Hardy sums $s_5(h, k)$. A relationship between $s_5(h, k)$ and the Hurwitz zeta function is established. By using the definitions of Lambert series and cotπz, we establish a relationship between $s_5(h, k)$ and Lambert series.