Abstract:
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
($|P(x)-P(n)|<B|P(n)|$, $B<1$, if $|x-n|<Cn^{1/2-\varepsilon}$), then
$|P(n)|>Cn^{1/4+\varepsilon}$.
Keywords:circle problem and divisor problem, Voronoi–Hardy and Landau formulae, short intervals.
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