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Russian Mathematical Surveys, 2019, Volume 74, Issue 5, Pages 909–925
DOI: https://doi.org/10.1070/RM9911
(Mi rm9911)
 

This article is cited in 4 scientific papers (total in 4 papers)

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
References:
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.
Received: 01.12.2018
Russian version:
Uspekhi Matematicheskikh Nauk, 2019, Volume 74, Issue 5(449), Pages 145–162
DOI: https://doi.org/10.4213/rm9911
Bibliographic databases:
Document Type: Article
UDC: 511.338
MSC: 11P21, 35P30, 58J51
Language: English
Original paper language: Russian
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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\pages 145--162
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Linking options:
  • https://www.mathnet.ru/eng/rm9911
  • https://doi.org/10.1070/RM9911
  • https://www.mathnet.ru/eng/rm/v74/i5/p145
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Успехи математических наук Russian Mathematical Surveys
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    Abstract page:367
    Russian version PDF:93
    English version PDF:46
    References:46
    First page:26
     
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