Geodesic
Spectral decomposition formula and moments of symmetric square $L$-functions
Izvestiya. Mathematics , Tome 87 (2023) no. 4, pp. 641-682.

Voir la notice de l'article provenant de la source Math-Net.Ru

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: $L$-functions, Eisenstein series, Kuznetsov trace formula.
Mots-clés : Gauss sums 
@article{IM2_2023_87_4_a0,
     author = {O. G. Balkanova},
     title = {Spectral decomposition formula and moments of symmetric square $L$-functions},
     journal = {Izvestiya. Mathematics },
     pages = {641--682},
     publisher = {mathdoc},
     volume = {87},
     number = {4},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a0/}
}
TY  - JOUR
AU  - O. G. Balkanova
TI  - Spectral decomposition formula and moments of symmetric square $L$-functions
JO  - Izvestiya. Mathematics 
PY  - 2023
SP  - 641
EP  - 682
VL  - 87
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a0/
LA  - en
ID  - IM2_2023_87_4_a0
ER  - 
%0 Journal Article
%A O. G. Balkanova
%T Spectral decomposition formula and moments of symmetric square $L$-functions
%J Izvestiya. Mathematics 
%D 2023
%P 641-682
%V 87
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a0/
%G en
%F IM2_2023_87_4_a0
O. G. Balkanova. Spectral decomposition formula and moments of symmetric square $L$-functions. Izvestiya. Mathematics , Tome 87 (2023) no. 4, pp. 641-682. http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a0/

[1] 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 | DOI | MR | Zbl

[2] O. Balkanova and D. Frolenkov, “Convolution formula for the sums of generalized Dirichlet $L$-functions”, Rev. Mat. Iberoam., 35:7 (2019), 1973–1995 | DOI | MR | Zbl

[3] O. Balkanova, D. Frolenkov, and M. S. Risager, “Prime geodesics and averages of the Zagier $L$-series”, Math. Proc. Cambridge Philos. Soc., 172:3 (2022), 705–728 | DOI | MR | Zbl

[4] A. Balog, A. Biro, G. Cherubini, and N. Laaksonen, “Bykovskii-type theorem for the Picard manifold”, Int. Math. Res. Not. IMRN, 2022:3 (2022), 1893–1921 | DOI | MR | Zbl

[5] V. A. Bykovskii, “Density theorems and the mean value of arithmetical functions in short intervals”, Analytical theory of numbers and theory of functions. Part 12, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 212, Nauka, St. Petersburg, 1994, 56–70 ; English transl. J. Math. Sci. (N.Y.), 83:6 (1997), 720–730 | MR | Zbl | DOI

[6] K. Soundararajan and M. P. Young, “The prime geodesic theorem”, J. Reine Angew. Math., 2013:676 (2013), 105–120 | DOI | MR | Zbl

[7] G. Cherubini, Han Wu, and G. Zábrádi, “On Kuznetsov–Bykovskii's formula of counting prime geodesics”, Math. Z., 300:1 (2022), 881–928 | DOI | MR | Zbl

[8] O. Balkanova and D. Frolenkov, “The mean value of symmetric square $L$-functions”, Algebra Number Theory, 12:1 (2018), 35–59 | DOI | MR | Zbl

[9] O. Balkanova, “The first moment of Maaß form symmetric square $L$-functions”, Ramanujan J., 55:2 (2021), 761–781 | DOI | MR | Zbl

[10] E. M. Kiral and M. P. Young, “Kloosterman sums and Fourier coefficients of Eisenstein series”, Ramanujan J., 49:2 (2019), 391–409 | DOI | MR | Zbl

[11] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., BKhV-Petersburg, St. Petersburg, 2011; English transl. 7th ed., Elsevier/Academic Press, Amsterdam, 2007 | MR | Zbl

[12] G. Shimura, “On the holomorphy of certain Dirichlet series”, Proc. London Math. Soc. (3), 31:1 (1975), 79–98 | DOI | MR | Zbl

[13] J.-M. Deshouillers and H. Iwaniec, “Kloosterman sums and Fourier coefficients of cusp forms”, Invent. Math., 70:2 (1982), 219–288 | DOI | MR | Zbl

[14] E. M. Kiral and M. P. Young, “The fifth moment of modular $L$-functions”, J. Eur. Math. Soc. (JEMS), 23:1 (2021), 237–314 | DOI | MR | Zbl

[15] 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 | DOI | MR | Zbl

[16] NIST handbook of mathematical functions, eds. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge Univ. Press, Cambridge, 2010 | MR | Zbl

[17] T. Miyake, Modular forms, Transl. from the 1976 Japanese original, Springer Monogr. Math., Reprint of the 1st ed., Springer-Verlag, Berlin, 2006 | DOI | MR | Zbl

[18] A. V. Malyshev, “On the representation of integers by positive quadratic forms”, Trudy Mat. Inst. Steklov., 65, Acad. Sci. USSR, Moscow, 1962, 3–212 | MR | Zbl