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
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.
circle problem, Voronoi's formula, short intervals, quantum chaos, universality conjecture.
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Russian Mathematical Surveys, 2019, 74:5, 909–925
MSC: 11P21, 35P30, 58J51
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
\paper Circle problem and the spectrum of the Laplace operator on closed 2-manifolds
\jour Uspekhi Mat. Nauk
\jour Russian Math. Surveys
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