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Let $d(n)$ denote the number of positive divisors of the positive integer $n$. Pick some positive $X,h \in \mathbb{R}$ and consider the sum

$$ S(X,h) := \sum_{X \leq n \leq X+ h} d(n).$$

In view of Huxley's results, we can give the correct asymptotic estimate (as $X \to \infty$) for $S(X,h)$ once $h \gg X^{0.315}$. If I recall correctly, Shiu proved the correct upper bound for $S(x,h)$ even when $h \sim x^\epsilon$.

For small $h$, say $h \sim X^{0.2}$, can we obtain the correct asymptotic for $S(X,h)$?

It seems to me that the methods (Poisson summation, etc.) used to estimate the full sum should be applicable here, so I would like to know whether anyone has already carried this out carefully.

I am less interested in bounds that are valid for "most" values of $X$.

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Perhaps it will be useful. https://core.ac.uk/download/pdf/11423881.pdf I did not go into details, my hypothesis is if we take a constant $c< \frac{131}{416}$, then the weakened condition in this article will still be preserved, and moreover $k$ can be limited, i.e. condition $\int_{X}^{X+Y}(\Delta (x+h(x))-\Delta (x))^{k}dx$ is valid for $k< C $ ($C$ - some constant).

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