Define $d(n)=\sum_{t|n}1$ (it is equal to $\sigma_0(n)=\tau(n)$). There are some estimates for upper bound of $$\sum_{x\leq N}d(f(x))$$ in the terms of $N$, where $f(x)$ is a polynomial, even from Erdos. So far I know for its application in Diophantine sets ($m$-tuples) when $f(x)=x^2-r^2$ for some integer $r$. Are there any other applications of such sums?
1 Answer
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Estimates on these quantities are used in Elsholtz and Tao's work on the Erdos-Straus conjecture. See their paper "Counting the number of solutions to the Erdos-Straus equation on unit fractions" and Tao's blog post on the paper.
answered Jan 18, 2018 at 17:28