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The asymptotic formulae of sums of two smooth squares for divisor function (2303.08391v3)

Published 15 Mar 2023 in math.NT

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.

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