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 Uspekhi Mat. Nauk, 2019, Volume 74, Issue 5(449), Pages 145–162 (Mi umn9911)

Circle problem and the spectrum of the Laplace operator on closed 2-manifolds

D. A. Popov

Lomonosov Moscow State University, Belozerskii Research Institute for Physical and Chemical Biology

Abstract: In this survey the circle problem is treated in the broad sense, as the problem of the asymptotic properties of the quantity $P(x)$, the remainder term in the circle problem. A survey of recent results in this direction is presented. The main focus is on the behaviour of $P(x)$ on short intervals. Several conjectures on the local behaviour of $P(x)$ which lead to a solution of the circle problem are presented. A strong universality conjecture is stated which links the behaviour of $P(x)$ with the behaviour of the second term in Weyl's formula for the Laplace operator on a closed Riemannian 2-manifold with integrable geodesic flow.
Bibliography: 43 titles.

Keywords: circle problem, Voronoi's formula, short intervals, quantum chaos, universality conjecture.

DOI: https://doi.org/10.4213/rm9911

Full text: PDF file (573 kB)
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English version:
Russian Mathematical Surveys, 2019, 74:5, 909–925

Bibliographic databases:

UDC: 511.338
MSC: 11P21, 35P30, 58J51

Citation: D. A. Popov, “Circle problem and the spectrum of the Laplace operator on closed 2-manifolds”, Uspekhi Mat. Nauk, 74:5(449) (2019), 145–162; Russian Math. Surveys, 74:5 (2019), 909–925

Citation in format AMSBIB
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\by D.~A.~Popov
\paper Circle problem and the spectrum of the Laplace operator on closed 2-manifolds
\jour Uspekhi Mat. Nauk
\yr 2019
\vol 74
\issue 5(449)
\pages 145--162
\mathnet{http://mi.mathnet.ru/umn9911}
\crossref{https://doi.org/10.4213/rm9911}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4017577}
\transl
\jour Russian Math. Surveys
\yr 2019
\vol 74
\issue 5
\pages 909--925
\crossref{https://doi.org/10.1070/RM9911}