DOI: 10.4213/im9681e ISSN: 1064-5632
On the sum of two squares function
Vitalii Victorovich IudelevichWe derive the following asymptotic formula $$ \mathop{{\sum}'}_{n\leqslant x}\frac{r(n)}{r(n+1)} = {x}{(\log x)^{-3/4}}(c+o(1)),\qquad x \to +\infty, $$ where $r(n)$ is the number of representations of $n$ as a sum of two squares, $c$ is a positive constant, and the prime indicates summation over those $n$ for which $r(n+1)\neq 0$.