Analytic expressions pertaining to certain arithmetical functions

We demonstrate the general outlines of a method for obtaining analytic expressions for certain types of general arithmetical sums. In particular, analytical expressions for a general arithmetical sum whose terms are summed over either the positive integer solutions $(a,b)$ of the Diophantine equation $k{b}^2+d{a}^2=N$ or the positive integer solutions $(a,b)$ of the Diophantine equation $k{b}^2-d{a}^2=N$ are derived. As one of the consequences, we propose a possible improvement of the Robin-Lagarias criteria for the Riemann hypothesis.