|
This article is cited in 3 scientific papers (total in 3 papers)
Bounds and behaviour of the quantities $P(x)$, $\Delta(x)$ on short intervals
D. A. Popov A. N. Belozersky Institute of Physico-Chemical Biology, M. V. Lomonosov Moscow State University
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.
Received: 21.01.2015 Revised: 02.02.2015
Citation:
D. A. Popov, “Bounds and behaviour of the quantities $P(x)$, $\Delta(x)$ on short intervals”, Izv. RAN. Ser. Mat., 80:6 (2016), 230–246; Izv. Math., 80:6 (2016), 1213–1230
Linking options:
https://www.mathnet.ru/eng/im8341https://doi.org/10.1070/IM8341 https://www.mathnet.ru/eng/im/v80/i6/p230
|
Statistics & downloads: |
Abstract page: | 303 | Russian version PDF: | 40 | English version PDF: | 7 | References: | 46 | First page: | 20 |
|