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Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and Applications
January 30, 2016 14:00–14:25, Dorodnitsyn Computing Centre, Department of Mechanics and
Mathematics of Lomonosov Moscow State University., 119991, GSP-1, Moscow, Leninskie Gory, 1, Main Building, Department of Mechanics and Mathematics, 16 floor, Lecture hall 16-10
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On the fractional [parts connecter with the function $N/x$
A. V. Shubin Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
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Abstract:
The Dirichlet divisor problem is closely related to the sum of fractional parts
$$
\sum_{n\le N}\left\{\frac{x}{n}\right\}\,=\,cN\,+\,O\bigl(x^{\alpha+\varepsilon}\bigr).
$$
In the case $N \le x^\beta$, $0.5\le \beta < 1$, the fractional parts are uniformly distributed over $[0, 1)$
and therefore $c=\tfrac{1}{2}$. However, in the case $N = x$ the distribution is not uniform, since $c=1-\gamma=0.422784\ldots$ ($\gamma$ is Euler constant).
In the talk, we will consider some asymptotic formulas for general sums
$$
\sum_{\substack{n \le x \\ n \in \mathcal{A}}} f\left(\left\{\frac{x}{n}\right\}\right), \quad
\sum_{n \le x}g(n)f\left(\left\{\frac{x}{n}\right\}\right),
$$
where $\mathcal{A}$ denotes some subset of natural numbers, and $f$, $g$ are real-valued functions satisfying some natural conditions.
We also give some applications of these formulas.
Language: Russian and English
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