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 $kb^2+da^2 = N$ or the positive integer solutions $(a,b)$ of the Diophantine equation $kb^2-da^2 = N$ are derived. As one of the consequences, we propose a possible improvement of the Robin-Lagarias criteria for the Riemann hypothesis.