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.
\Bibitem{Pop19}
\by D.~A.~Popov
\paper Circle problem and the spectrum of the Laplace operator on closed 2-manifolds
\jour Russian Math. Surveys
\yr 2019
\vol 74
\issue 5
\pages 909--925
\mathnet{http://mi.mathnet.ru/eng/rm9911}
\crossref{https://doi.org/10.1070/RM9911}
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https://www.mathnet.ru/eng/rm9911
https://doi.org/10.1070/RM9911
https://www.mathnet.ru/eng/rm/v74/i5/p145
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