Geodesic
Bounds and behaviour of the quantities $P(x)$, $\Delta(x)$ on short intervals
Izvestiya. Mathematics , Tome 80 (2016) no. 6, pp. 1213-1230.

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We study the dependence of upper bounds for the quantity $|P(n)|$ on certain properties of the behaviour of $|P(x)|$ in a neighbourhood of the point $x=n$. In particular, it is proved that, if $n$ is a point of local maximum of the quantity $|P(x)|$, where $|P(n)|>Cn^{1/4}$ and the maximum is broad ($|P(x)-P(n)|$, $B1$, if $|x-n|$), then $|P(n)|>Cn^{1/4+\varepsilon}$.
Keywords: circle problem and divisor problem, Voronoi–Hardy and Landau formulae, short intervals. 
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D. A. Popov. Bounds and behaviour of the quantities $P(x)$, $\Delta(x)$ on short intervals. Izvestiya. Mathematics , Tome 80 (2016) no. 6, pp. 1213-1230. http://geodesic.mathdoc.fr/item/IM2_2016_80_6_a10/

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