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On the prime number theorem for a compact Riemann surface. (English) Zbl 1279.11086

Summary: We improve the estimate of the error term in Selberg’s and Huber’s formula for the distribution of the eigenvalues of the Laplace-Beltrami operator on a compact Riemann surface.

MSC:

11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11N45 Asymptotic results on counting functions for algebraic and topological structures
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References:

[1] M. Avdispahić and L. Smajlović, An explicit formula and its application to the Selberg trace formula , Monatsh. Math. 147 (2006), 183-198. · Zbl 1092.11026 · doi:10.1007/s00605-005-0317-0
[2] D.A. Hejhal, The Selberg trace formula for \(PSL(2,\r)\) , Vol. I, Lecture Notes Math. 548 , Springer Verlag, Berlin, 1976. · Zbl 0347.10018 · doi:10.1007/BFb0079608
[3] H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen II, Math. Ann. 142 (1961), 385-398. · Zbl 0094.05703 · doi:10.1007/BF01451031
[4] ——–, Nachtrag zu [3], Math. Ann. 143 (1961), 463-464. · Zbl 0101.05702 · doi:10.1007/BF01470758
[5] H. Iwaniec and E. Kowalski, Analytic number theory , Amer. Math. Soc. Colloq. Publ. 53 (2004). · Zbl 1059.11001
[6] B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse , Mon. Not. Berlin Akad. (1859), 671-680.
[7] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series , J. Ind. Math. Soc. 20 (1956), 47-87. · Zbl 0072.08201
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