Spectral decomposition formula and moments of symmetric square $L$-functions
- Authors: Balkanova O.G.1
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Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Issue: Vol 87, No 4 (2023)
- Pages: 3-46
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/133914
- DOI: https://doi.org/10.4213/im9330
- ID: 133914
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Abstract
We prove a spectral decomposition formula for averages of Zagier $L$-series in terms of moments of symmetric square $L$-functions associated to Maass and holomorphic cusp forms of levels 4, 16, 64.
Keywords
About the authors
Olga Germanovna Balkanova
Steklov Mathematical Institute of Russian Academy of SciencesPhD, no status
References
- D. Zagier, “Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields”, Modular functions of one variable, VI (Univ. Bonn, Bonn, 1976), Lecture Notes in Math., 627, Springer, Berlin, 1977, 105–169
- O. Balkanova, D. Frolenkov, “Convolution formula for the sums of generalized Dirichlet $L$-functions”, Rev. Mat. Iberoam., 35:7 (2019), 1973–1995
- O. Balkanova, D. Frolenkov, M. S. Risager, “Prime geodesics and averages of the Zagier $L$-series”, Math. Proc. Cambridge Philos. Soc., 172:3 (2022), 705–728
- A. Balog, A. Biro, G. Cherubini, N. Laaksonen, “Bykovskii-type theorem for the Picard manifold”, Int. Math. Res. Not. IMRN, 2022:3 (2022), 1893–1921
- В. А. Быковский, “Плотностные теоремы и среднее значение арифметических функций на коротких интервалах”, Аналитическая теория чисел и теория функций. 12, Зап. науч. сем. ПОМИ, 212, Наука, СПб., 1994, 56–70
- K. Soundararajan, M. P. Young, “The prime geodesic theorem”, J. Reine Angew. Math., 2013:676 (2013), 105–120
- G. Cherubini, Han Wu, G. Zabradi, “On Kuznetsov–Bykovskii's formula of counting prime geodesics”, Math. Z., 300:1 (2022), 881–928
- O. Balkanova, D. Frolenkov, “The mean value of symmetric square $L$-functions”, Algebra Number Theory, 12:1 (2018), 35–59
- O. Balkanova, “The first moment of Maass form symmetric square $L$-functions”, Ramanujan J., 55:2 (2021), 761–781
- E. M. Kiral, M. P. Young, “Kloosterman sums and Fourier coefficients of Eisenstein series”, Ramanujan J., 49:2 (2019), 391–409
- И. С. Градштейн, И. М. Рыжик, Таблицы интегралов, рядов и произведений, 7-е изд., БХВ-Петербург, СПб., 2011, 1176 с.
- G. Shimura, “On the holomorphy of certain Dirichlet series”, Proc. London Math. Soc. (3), 31:1 (1975), 79–98
- J.-M. Deshouillers, H. Iwaniec, “Kloosterman sums and Fourier coefficients of cusp forms”, Invent. Math., 70:2 (1982), 219–288
- E. M. Kiral, M. P. Young, “The fifth moment of modular $L$-functions”, J. Eur. Math. Soc. (JEMS), 23:1 (2021), 237–314
- S. Drappeau, “Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method”, Proc. Lond. Math. Soc. (3), 114:4 (2017), 684–732
- NIST handbook of mathematical functions, eds. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge Univ. Press, Cambridge, 2010, xvi+951 pp.
- T. Miyake, Modular forms, Transl. from the 1976 Japanese original, Springer Monogr. Math., Reprint of the 1st ed., Springer-Verlag, Berlin, 2006, x+335 pp.
- А. В. Малышев, “О представлении целых чисел положительными квадратичными формами”, Тр. МИАН СССР, 65 (1962), 3–212, Изд-во АН СССР, М.
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