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Prime geodesics and averages of the Zagier L-series

Part of: Zeta and $L$-functions: analytic theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  11 May 2021

OLGA BALKANOVA ,
DMITRY FROLENKOV  and
MORTEN S. RISAGER
OLGA BALKANOVA
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences 8 Gubkina st., Moscow, 119991, Russia. e-mail: balkanova@mi-ras.ru
DMITRY FROLENKOV
Affiliation:
HSE University and Steklov Mathematical Institute of Russian Academy of Sciences 8 Gubkina st., Moscow, 119991, Russia. e-mail: frolenkov@mi-ras.ru
MORTEN S. RISAGER
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark. e-mail: risager@math.ku.dk
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Abstract

The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

MSC classification

Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 11F72: Spectral theory; Selberg trace formula 11M41: Other Dirichlet series and zeta functions
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 3 , May 2022 , pp. 705 - 728
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Balkanova, O.. The first moment of Maass form symmetric square L-functions. Ramanujan Journal, (2020).CrossRefGoogle Scholar
Balkanova, O. and Frolenkov, D.. Convolution formula for the sums of generalized Dirichlet L-series. Rev. Mat. Iberoam. (2019), no. 7, 1973–1995.Google Scholar
Balkanova, O. and Frolenkov, D.. The mean value of symmetric square L-functions. Algebra Number Theory, 12 (2018), no. 1, 35–59.CrossRefGoogle Scholar
Balkanova, O. and Frolenkov, D.. Bounds for a spectral exponential sum. J. London Math. Soc. 99 (2019), no. 2, 249–272.CrossRefGoogle Scholar
Balkanova, O. and Frolenkov, D.. Sums of Kloosterman sums in the prime geodesic theorem. The Quarterly Journal of Mathematics 70 (2019), no. 2, 649–674.CrossRefGoogle Scholar
Balog, A., Biró, A., Harcos, G. and Maga, P.. The prime geodesic theorem in square mean. J. Number Theory 198 (2019), 239–249.CrossRefGoogle Scholar
Blomer, V., Khan, R. and Young, M.. Distribution of mass of holomorphic cusp forms. Duke Math. J. 162 (2013), no. 14, 2609–2644.CrossRefGoogle Scholar
Bykovsk, V. A.. Density theorems and the mean value of arithmetic functions on short intervals. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 212 (1994), no. Anal. Teor. Chisel i Teor. Funktsii. 12, 56–70, 196.Google Scholar
Cai, Y.. Prime geodesic theorem. J. Théor. Nombres Bordeaux 14 (2002), no. 1, 59–72.CrossRefGoogle Scholar
Cherubini, G. and Guerreiro, J.. Mean square in the prime geodesic theorem. Algebra & Number Theory 12 (2018), no. 3, 571–597.CrossRefGoogle Scholar
Conrey, J. B. and Iwaniec, H.. The cubic moment of central values of automorphic L-functions. Ann. of Math. (2) 151 (2000), no. 3, 1175–1216.CrossRefGoogle Scholar
Frolenkov, D. A.. Uniform asymptotic formulas for a hypergeometric function, Dal’nevostochny Matematicheski Zhurnal. Far East. Math. J. 15 (2015), no. 2, 288–298 (Russian).Google Scholar
Good, A.. Beiträge zur Theorie der Dirichletreihen, die Spitzenformen zugeordnet sind. J. Number Theory 13 (1981), no. 1, 18–65.CrossRefGoogle Scholar
Hejhal, D. A.. The Selberg trace formula for PSL(2,R). Vol. 2, Lecture Notes in Math., vol. 1001 (Springer-Verlag, Berlin, 1983).CrossRefGoogle Scholar
Hoffstein, J. and Lockhart, P.. Coefficients of Maass forms and the Siegel zero. Ann. of Math. (2) 140 (1994), no. 1, 161–181, With an appendix by Dorian Goldfeld, Jeffrey Hoffstein and Daniel Lieman.CrossRefGoogle Scholar
Huber, H.. Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II. Math. Ann. 142 (1960), 385–398.CrossRefGoogle Scholar
Ivić, A.. On the error term for the fourth moment of the Riemann zeta-function. J. London Math. Soc. 60 (1999), no. 1, 21–32.CrossRefGoogle Scholar
Ivić, A. and Jutila, M.. On the moments of Hecke series at central points. II. Funct. Approx. Comment. Math. 31 (2003), 93–108.CrossRefGoogle Scholar
Ivić, A. and Motohashi, Y.. A note on the mean value of the zeta and L-functions. VII. Proc. Japan Acad. Ser. A Math. Sci. 66 (1990), no. 6, 150–152.CrossRefGoogle Scholar
Iwaniec, H.. Non-holomorphic modular forms and their applications. Modular Forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, (1984), pp. 157–196.Google Scholar
Iwaniec, H.. Prime geodesic theorem. J. Reine Angew. Math. 349 (1984), 136–159.Google Scholar
Iwaniec, H.. Spectral methods of automorphic forms, second ed., Graduate Studies in Math., vol. 53 (Amer. Math. Soc., Providence, RI, 2002).CrossRefGoogle Scholar
Kuznetsov, N. V.. The arithmetic form of the Selberg trace formula and the distribution of norms of primitive hyperbolic classes in the modular group. Preprint (Russian) 1978.Google Scholar
Luo, W. Z. and Sarnak, P.. Quantum ergodicity of eigenfunctions on PSL2(Z)\H2, Inst. Hautes Études Sci. Publ. Math. (1995), no. 81, 207–237.Google Scholar
McKee, M., Sun, H. and Ye, Y.. Weighted stationary phase of higher orders. Frontiers of Mathematics in China 12 (2017), no. 3, 675–702.CrossRefGoogle Scholar
Ng., M.-h. Moments of automorphic L-functions, PhD thesis. The University of Hong Kong (2016).Google Scholar
Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W.. NIST handbook of mathematical functions (Cambridge University Press, 2010).Google Scholar
Sarnak, P.. Class numbers of indefinite binary quadratic forms. J. Number Theory 15 (1982), no. 2, 229–247.CrossRefGoogle Scholar
Soundararajan, K. and Young, M. P.. The prime geodesic theorem, J. Reine Angew. Math. 676 (2013), 105–120.Google Scholar
Tang, H.. Central value of the symmetric square L-functions related to Maass forms, Sci Sin Math 42 (2012), no. 12, 1213–1224 (Chinese).CrossRefGoogle Scholar
Young, M. P.. Weyl-type hybrid subconvexity bounds for twisted L-functions and Heegner points on shrinking sets. J. Eur. Math. Soc. (JEMS) 19 (2017), no. 5, 1545–1576.CrossRefGoogle Scholar
Zagier, D.. Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, Modular Functions of One Variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) (Springer, Berlin, 1977), pp. 105–169. Lecture Notes in Math., Vol. 627.CrossRefGoogle Scholar