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.

The paper may have some faults, so I want to withdraw the paper