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?



Sum of number of divisors function
Estimates on these quantities are used in Elsholtz and Tao's work on the [Erdos-Straus conjecture](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Straus_conjecture). See their paper "[Counting the number of solutions to the Erdos-Straus equation on unit fractions](https://arxiv.org/abs/1107.1010)" and Tao's [blog post](https://terrytao.wordpress.com/2011/07/31/counting-the-number-of-solutions-to-the-erdos-straus-equation-on-unit-fractions/) on the paper.


