Abstract
Under a previously studied condition on the argument of the Selberg zeta function on the critical line, we reach the critical exponent \(\frac{1}{2}\) in the error term of the prime geodesic theorem for the modular group PSL(2,\( \mathbb {Z}\)) outside a set of finite logarithmic measure. We also prove a conditional prime geodesic theorem of Hejhal’s type in this setting without the latter exclusion.
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References
Avdispahić, M.: Gallagherian \(PGT\) on PSL(2, \(\mathbb{Z}\)). Funct. Approx. Comment. Math. 58, 207–213 (2018). https://doi.org/10.7169/facm/1686
Avdispahić, M.: Prime geodesic theorem for the modular surface. Hacet. J. Math. Stat. 49(2), 505–509 (2020). https://doi.org/10.15672/hujms.568323
Avdispahić, M., Šabanac, Z.: Gallagherian prime geodesic theorem in higher dimensions. Bull. Malays. Math. Sci. Soc. 43(4), 3019–3026 (2020). https://doi.org/10.1007/s40840-019-00849-y
Avdispahić, M., Gušić, Dž: On the error term in the prime geodesic theorem. Bull. Korean Math. Soc. 49(2), 367–372 (2012)
Balkanova, O., Frolenkov, D.: Bounds for a spectral exponential sum. J. Lond. Math. Soc. (2) 99(2), 249–272 (2019). https://doi.org/10.1112/jlms.12174
Balkanova, O., Frolenkov, D., Risager, M.S.: Prime geodesics and averages of the Zagier \(L\)-series. arXiv:1912.05277
Balog, A., Biró, A., Harcos, G., Maga, P.: The prime geodesic theorem in square mean. J. Number Theory 198, 239–249 (2019). https://doi.org/10.1016/j.jnt.2018.10.012
Cherubini, G., Guerreiro, J.: Mean square in the prime geodesic theorem. Algebra Number Theory 12(3), 571–597 (2018). https://doi.org/10.2140/ant.2018.12.571
Conrey, J.B., Iwaniec, H.: The cubic moment of central values of automorphic L-functions. Ann. Math. (2) 151(3), 1175–1216 (2000). https://doi.org/10.2307/121132
Gallagher, P.X.: Some consequences of the Riemann hypothesis. Acta Arith. 37, 339–343 (1980). http://eudml.org/doc/205704
Hejhal, D.A.: The Selberg trace formula for PSL(2,\({\mathbb{R}} \)). Vol 1, Lecture Notes in Mathematics 548, Springer, Berlin, (1976)
Hejhal, D.A.: The Selberg trace formula for PSL(2,\({\mathbb{R}} \)). Vol 2, Lecture Notes in Mathematics 1001, Springer, Berlin, (1983)
Ingham, A.E.: The Distribution of Prime Numbers. Cambridge University Press, Cambridge (1932)
Iwaniec, H.: Prime geodesic theorem. J. Reine Angew. Math. 349, 136–159 (1984). http://eudml.org/doc/152626
Iwaniec, H.: Spectral Methods of Automorphic Forms, 2nd edn. Amer. Math. Soc, Providence, Rhode Island (2002)
Koyama, S.: Prime geodesic theorem for arithmetic compact surfaces, Internat. Math. Res. Notices (1998), no. 8, 383–388. https://doi.org/10.1155/S1073792898000257
Park, J.: Ruelle zeta function and prime geodesic theorem for hyperbolic manifolds with cusps. In: van Dijk, G., Wakayama, M. (eds.) Casimir Force, Casimir Operators and the Riemann Hypothesis, 9–13 November 2009. Kyushu University, Fukuoka, Japan, Walter de Gruyter (2010)
Randol, B.: On the asymptotic distribution of closed geodesics on compact Riemann surfaces. Trans. Am. Math. Soc. 233, 241–247 (1977). https://doi.org/10.2307/1997834
Soundararajan, K., Young, M.P.: The prime geodesic theorem. J. Reine Angew. Math. 676, 105–120 (2013). https://doi.org/10.1515/crelle.2012.002
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Communicated by Rosihan M. Ali.
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Avdispahić, M. On von Koch Theorem for PSL(2,\(\mathbb {Z}\)). Bull. Malays. Math. Sci. Soc. 44, 2139–2150 (2021). https://doi.org/10.1007/s40840-020-01053-z
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DOI: https://doi.org/10.1007/s40840-020-01053-z