On the sum-of-squares function

Abstract

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

81 pages, the paper accepted for publication in the journal Izvestiya RAN