On the sum of two squares function

Key Points

• The study aims to derive an asymptotic formula for the sum of two squares function.
• Derivation of the asymptotic formula for the sum of two squares function.
• Summation over integers where the representation function of the next integer is non-zero.
• The asymptotic formula is expressed as $$ rac{r(n)}{r(n+1)} \sim {x}{( ext{log } x)^{-3/4}}(c+o(1)) $$ as $x \to +\infty$.
• Establishes a relationship involving the number of representations of integers as sums of two squares.