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Asymptotic formula for shifted convolution sums involving the coefficients of an L-function

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 September 2025

Naveen K. Godara Open the ORCID record for Naveen K. Godara [Opens in a new window] ,
Anuj Jakhar  and
Kotyada Srinivas
Naveen K. Godara*
Affiliation:
Department of Mathematics, Indian Institute of Technology (IIT) Madras , Chennai 600036, Tamil Nadu, India e-mail: anujjakhar@iitm.ac.in
Anuj Jakhar
Affiliation:
Department of Mathematics, Indian Institute of Technology (IIT) Madras , Chennai 600036, Tamil Nadu, India e-mail: anujjakhar@iitm.ac.in
Kotyada Srinivas
Affiliation:
Department of Mathematics, Indian Institute of Science Education and Research Tirupati , Tirupati 517619, Andhra Pradesh, India e-mail: srini@imsc.res.in
*
e-mail: naveen.iiserb@gmail.com
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Abstract

Let $f $ be a normalized Hecke eigenform of even weight $k \geq 2$ for $SL_2(\mathbb {Z})$. In this article, we establish an asymptotic formula for the shifted convolution sum of a general divisor function, where the sum involves the Fourier coefficients of a multi-folded L-function weighted with a kernel function.

Keywords

Eigenformsautomorphic L-functionsshifted sum

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight 11F30: Fourier coefficients of automorphic forms 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations

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Type
Article
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Canadian Mathematical Bulletin , First View , pp. 1 - 20
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Acharya, R., A twist of the Gauss circle problem by holomorphic cusp forms . Res. Number Theory. 8(2022), no. 1, Article no. 5, 14 pp.10.1007/s40993-021-00299-1CrossRefGoogle Scholar
Banerjee, S. and Pandey, M. K., Signs of Fourier coefficients of cusp form at sum of two squares . Proc. Indian Acad. Sci. Math. Sci. 130(2020), no. 1, Article no. 2, 9 pp.10.1007/s12044-019-0534-4CrossRefGoogle Scholar
Blomer, V., Sums of Hecke eigenvalues over values of quadratic polynomials . Int. Math. Res. Not. (2008), no. 16, Article no. rnn059, 29 pp.Google Scholar
Bourgain, J., Decoupling, exponential sums and the Riemann zeta function . J. Am. Math. Soc. 30(2017), no. 1, 205–224.10.1090/jams/860CrossRefGoogle Scholar
Clozel, L. and Thorne, J. A., Level-raising and symmetric power functoriality, I . Compos. Math. 150(2014), no. 5, 729–748.10.1112/S0010437X13007653CrossRefGoogle Scholar
Clozel, L. and Thorne, J. A., Level raising and symmetric power functoriality, II . Ann. Math. (2) 181(2015), no. 1, 303–359.10.4007/annals.2015.181.1.5CrossRefGoogle Scholar
Clozel, L. and Thorne, J. A., Level-raising and symmetric power functoriality, III . Duke Math. J. 166(2017), no. 2, 325–402.10.1215/00127094-3714971CrossRefGoogle Scholar
Cogdell, J. and Michel, P., On the complex moments of symmetric power $L$ -functions at $s=1$ . Int. Math. Res. Not. (2004), no. 31, 1561–1617.10.1155/S1073792804132455CrossRefGoogle Scholar
Deligne, P., La conjecture de Weil. I . Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273–307.CrossRefGoogle Scholar
Erdös, P. and Ivić, A., The distribution of values of a certain class of arithmetic functions at consecutive integers . In: Number theory Vol. I (Budapest, 1987), volume 51 of Colloquia Mathematica Societatis János Bolyai, North-Holland, Amsterdam, 1990, pp. 45–91.Google Scholar
Garrett, P. B. and Harris, M., Special values of triple product $L$ -functions . Am. J. Math. 115(1993), no. 1, 161–240.CrossRefGoogle Scholar
Gelbart, S. and Jacquet, H., A relation between automorphic representations of and GL(2) and GL(3) . Ann. Sci. École Norm. Sup. (4). 11(1978), no. 4, 471–542.10.24033/asens.1355CrossRefGoogle Scholar
Good, A., The square mean of Dirichlet series associated with cusp forms . Mathematika. 29(1982), no. 2, 278–295.10.1112/S0025579300012377CrossRefGoogle Scholar
Heath-Brown, D. R., Hybrid bounds for Dirichlet $L$ -functions. Invent. Math. 47(1978), no. 2, 149–170.CrossRefGoogle Scholar
Hecke, E., Theorie der Eisensteinschen Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik . Abh. Math. Sem. Univ. Hamburg. 5(1927), no. 1, 199–224.10.1007/BF02952521CrossRefGoogle Scholar
Hua, G., On the Fourier coefficients for general product $L$ -functions . Math. Pannon (N. S.). 30(2024), no. 1, 41–60.10.1556/314.2024.00003CrossRefGoogle Scholar
Hua, G., On the asymptotics of coefficients associated to $l$ -fold product $L$ -functions and its applications . Ramanujan J. 66(2025), no. 2, Article no. 31, 20 p.10.1007/s11139-024-01003-4CrossRefGoogle Scholar
Huang, B., On the Rankin-Selberg problem, II . Q. J. Math. 75(2024), no. 1, 1–10.10.1093/qmath/haad037CrossRefGoogle Scholar
Ivić, A., Exponent pairs and the zeta function of Riemann . Stud. Sci. Math. Hung. 15(1980), nos. 1–3, 157–181.Google Scholar
Ivić, A. and Tenenbaum, G., Local densities over integers free of large prime factors . Q. J. Math. 37(1986), no. 148, 401–417.CrossRefGoogle Scholar
Iwaniec, H., Topics in classical automorphic forms, volume 17 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1997.Google Scholar
Jiang, Y. and Lü, G., On the higher mean over arithmetic progressions of Fourier coefficients of cusp forms . Acta Arith. 166(2014), no. 3, 231–252.10.4064/aa166-3-2CrossRefGoogle Scholar
Jutila, M., Lectures on a method in the theory of exponential sums, volume 80 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1987.Google Scholar
Kim, H. H., Functoriality for the exterior square of $\mathrm{GL}_4$ and the symmetric fourth of $\mathrm{GL}_2$ . J. Am. Math. Soc. 16(2003), no. 1, 139–183 With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak.CrossRefGoogle Scholar
Kim, H. H. and Shahidi, F., Cuspidality of symmetric powers with applications . Duke Math. J. 112(2002), no. 1, 177–197.10.1215/S0012-9074-02-11215-0CrossRefGoogle Scholar
Kim, H. H. and Shahidi, F., Functorial products for $\mathrm{GL}_2 \times \mathrm{GL}_3$ and the symmetric cube for $\mathrm{GL}_2$ . Ann. Math. (2) 155(2002), no. 3, 837–893. With an appendix by Colin J. Bushnell and Guy Henniart.10.2307/3062134CrossRefGoogle Scholar
Lin, Y., Nunes, R., and Qi, Z., Strong subconvexity for self-dual $GL(3)$ $L$ -functions . Int. Math. Res. Not. (2023), no. 13, 11453–11470.10.1093/imrn/rnac153CrossRefGoogle Scholar
Lü, G. and Sankaranarayanan, A., On the coefficients of triple product $L$ -functions . Rocky Mountain J. Math. 47(2017), no. 2, 553–570.10.1216/RMJ-2017-47-2-553CrossRefGoogle Scholar
Matsumoto, K., The mean values and the universality of Rankin-Selberg $L$ -functions. In: M. Jutila and T. Metsänkylä. (eds.), Number theory (Turku, 1999), de Gruyter, Berlin, 2001, pp. 201–221.CrossRefGoogle Scholar
Murty, M. R., Problems in analytic number theory. 2nd ed., volume 206 of Graduate Texts in Mathematics, Readings in Mathematics, Springer, New York, NY, 2008.Google Scholar
Newton, J. and Thorne, J. A., Symmetric power functoriality for holomorphic modular forms . Publ. Math. Inst. Hautes Études Sci. 134(2021), 1–116.10.1007/s10240-021-00127-3CrossRefGoogle Scholar
Newton, J. and Thorne, J. A., Symmetric power functoriality for holomorphic modular forms, II . Publ. Math. Inst. Hautes Études Sci. 134(2021), 117–152.10.1007/s10240-021-00126-4CrossRefGoogle Scholar
Perelli, A., General $L$ -functions . Annali di Matematica (4) 130(1982), 287–306.10.1007/BF01761499CrossRefGoogle Scholar
Rankin, R. A., Contributions to the theory of Ramanujan’s function $\tau (n)$ and similar arithmetical functions. I. The zeros of the function $\sum_{n=1}^{\infty}\tau (n)/{n}^s$ on the line $\mathit{\operatorname{Re}}(s)=13/2$ . II. The order of the Fourier coefficients of integral modular forms . Proc. Camb. Philos. Soc. 35(1939), 351–372.10.1017/S0305004100021095CrossRefGoogle Scholar
Selberg, A., Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist . Arch. Math. Naturvid. 43(1940), 47–50.Google Scholar
Shahidi, F., Third symmetric power $L$ -functions for GL(2) . Compos. Math. 70(1989), no. 3, 245–273.Google Scholar
Tiwari, P. and Godara, N. K., On the power moments of hybrid arithmetic functions associated with the Hecke eigenvalues . Ramanujan J. 66(2025), no. 46.10.1007/s11139-024-01019-wCrossRefGoogle Scholar
Venkatasubbareddy, K. and Sankaranarayanan, A., On the average behavior of coefficients related to triple product $L$ -functions . Funct. Approx. Comment. Math. 68(2023), no. 2, 195–206.10.7169/facm/2046CrossRefGoogle Scholar
Venkatasubbareddy, K. and Sankaranarayanan, A., On the tetra, penta, hexa, hepta and octa product $L$ -functions . Eur. J. Math. 9(2023), no. 1, Article no. 17, 24 pp.10.1007/s40879-023-00612-5CrossRefGoogle Scholar
Venkatasubbareddy, K. and Sankaranarayanan, A., Corrigendum to “on certain kernel functions and shifted convolution sums ”. J. Number Theory 258(2024), 414–450]. J. Number Theory, 262:577–578, 2024.10.1016/j.jnt.2023.12.006CrossRefGoogle Scholar
Venkatasubbareddy, K. and Sankaranarayanan, A., On certain kernel functions and shifted convolution sums of the Fourier coefficients . J. Number Theory 258(2024), 414–450.10.1016/j.jnt.2023.12.006CrossRefGoogle Scholar
Young, M. P., Weyl-type hybrid subconvexity bounds for twisted $L$ -functions and Heegner points on shrinking sets . J. Eur. Math. Soc. 19(2017), no. 5, 1545–1576.10.4171/jems/699CrossRefGoogle Scholar