ON ANALYTIC PROPERTIES AND CHARACTER ANALOGS OF HARDY SUMS

The aim of this paper is to define $Y(h,k)$ sum which is related to the Hardy's sums $s_{5}(h,k)$. On the semi-group $G$, matrix operation of this sum is defined. Substituting mediants of Farey fractions into the matrix operation, $Y(h,k)$ sum is generalized. By using contour integration, the reciprocity theorem of the $Y(h,k)$ sum is proved. Moreover, by using $% L(1,\chi )$ function and Gauss sums, generalized character analogs of the Hardy sums are found.