On the sum-of-squares function

In this paper, we derive the following asymptotic formula $${}_{n⩽x}{\displaystyle \frac{r(n)}{r(n+1)}}=x(\ln x{)}^{-3\mathrm{/}4}(c+o(1)),x\to +\mathrm{\infty },$$ 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 $r(n+1)\mathrm{\ne }0$.