The asymptotic formulae of sums of two smooth squares for divisor function

Abstract

A natural number $n$ is $y$-smooth if the greatest prime factor of $n$ does not exceed $y$. Let $s_{1}$ and $s_{2}$ are $y$-smooth numbers. We consider sums of smooth squares of the binary Titchmarsh divisor problem and give asymptotic formulae for $\sum_{s_{1}^{2}+s_{2}^{2}\le x}\tau(s_{1}^{2}+s_{2}^{2}+1)$ for $(\log x)^{K}\le y<x^{\frac{1}{2}}$, where $K$ is large enough.