Research article

On triple correlation sums of Fourier coefficients of cusp forms

  • Received: 17 June 2022 Revised: 02 August 2022 Accepted: 12 August 2022 Published: 01 September 2022
  • MSC : 11F67, 11F66

  • Let $ p $ be a prime. In this paper, we study the sum

    $ \sum\limits_{m\ge 1} \sum\limits_{n\ge 1} a_n \lambda_g(m)\lambda_{f}(m+pn) \,U{ \left( \frac{m}{X} \right) }V{ \left ( \frac{n}{H} \right)} $

    for any newforms $ g\in \mathcal{B}_k(1) $ (or $ \mathcal{B}_\lambda^\ast(1) $) and $ f\in \mathcal{B}_k(p) $ (or $ \mathcal{B}_\lambda^\ast(p) $), with the aim of determining the explicit dependence on the level, where $ {\bf{a}} = \{a_n\in\mathbb{C}\} $ is an arbitrary complex sequence. As a result, we prove a uniform bound with respect to the level parameter $ p $, and present that this type of sum is non-trivial for any given $ H, X\ge 2 $.

    Citation: Fei Hou, Bin Chen. On triple correlation sums of Fourier coefficients of cusp forms[J]. AIMS Mathematics, 2022, 7(10): 19359-19371. doi: 10.3934/math.20221063

    Related Papers:

  • Let $ p $ be a prime. In this paper, we study the sum

    $ \sum\limits_{m\ge 1} \sum\limits_{n\ge 1} a_n \lambda_g(m)\lambda_{f}(m+pn) \,U{ \left( \frac{m}{X} \right) }V{ \left ( \frac{n}{H} \right)} $

    for any newforms $ g\in \mathcal{B}_k(1) $ (or $ \mathcal{B}_\lambda^\ast(1) $) and $ f\in \mathcal{B}_k(p) $ (or $ \mathcal{B}_\lambda^\ast(p) $), with the aim of determining the explicit dependence on the level, where $ {\bf{a}} = \{a_n\in\mathbb{C}\} $ is an arbitrary complex sequence. As a result, we prove a uniform bound with respect to the level parameter $ p $, and present that this type of sum is non-trivial for any given $ H, X\ge 2 $.



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    [1] V. Blomer, Shifted convolution sums and subconvexity bounds for automorphic $L$-functions, IMRN, 2004 (2004), 3905–3926. https://doi.org/10.1155/S1073792804142505 doi: 10.1155/S1073792804142505
    [2] V. Blomer, On triple correlations of divisor functions, Bull. Lond. Math. Soc., 49 (2017), 10–22. https://doi.org/10.1112/blms.12004 doi: 10.1112/blms.12004
    [3] V. Blomer, G. Harcos, P. Michel, A Burgess-like subconvex bound for twisted $L$-functions (with Appendix 2 by Z. Mao), Forum Math., 19 (2007), 61–105. https://doi.org/10.1515/FORUM.2007.003 doi: 10.1515/FORUM.2007.003
    [4] T. D. Browning, The divisor problem for binary cubic form, J. Théor. Nombr. Bordx., 23 (2011), 579–602.
    [5] W. Duke, J. B. Friedlander, H. Iwaniec, Bounds for automorphic $L$-functions, Invent. Math., 112 (1993), 1–8. https://doi.org/10.1007/BF01232422 doi: 10.1007/BF01232422
    [6] D. Goldfeld, Automorphic forms and $L$-functions for the group $GL(n, \mathbb{R} )$, Cambridge Studies in Advanced Mathematics, Vol. 99, Cambridge University Press, 2009. https://doi.org/10.1017/CBO9780511542923
    [7] A. Good, Beitrage zur theorie der Dirichletreihen, die spitzenformen zugeordnet sind, J. Number Theory, 13 (1981), 18–65. https://doi.org/10.1016/0022-314X(81)90028-7 doi: 10.1016/0022-314X(81)90028-7
    [8] A. Good, Cusp forms and eigenfunctions of the Laplacian, Math. Ann., 255 (1981), 523–548. https://doi.org/10.1007/BF01451932 doi: 10.1007/BF01451932
    [9] G. Harcos, P. Michel, The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points, Ⅱ, Invent. Math., 163 (2006), 581–655. https://doi.org/10.1007/s00222-005-0468-6 doi: 10.1007/s00222-005-0468-6
    [10] G. Harcos, N. Templier, On the sup-norm of Maaß cusp forms of large level, Ⅲ, Math. Ann., 356 (2013), 209–216. https://doi.org/10.1007/s00208-012-0844-7 doi: 10.1007/s00208-012-0844-7
    [11] D. R. Heath-Brown, The fourth power moment of the Riemann zeta function, Proc. Lond. Math. Soc., S3-38 (1979), 385–422. https://doi.org/10.1112/plms/s3-38.3.385 doi: 10.1112/plms/s3-38.3.385
    [12] R. Holowinsky, K. Soundararajan, Mass equidistribution for Hecke eigenforms, Ann. Math., 172 (2010), 1517–1528. http://doi.org/10.4007/annals.2010.172.1517 doi: 10.4007/annals.2010.172.1517
    [13] T. A. Hulse, C. I. Kuan, D. Lowry-Duda, A. Walker, Second moments in the generalized Gauss circle problem, Forum Math. Sigma, 6 (2018), 1–49. https://doi.org/10.1017/fms.2018.26 doi: 10.1017/fms.2018.26
    [14] T. A. Hulse, C. I. Kuan, D. Lowry-Duda, A. Walker, Triple correlation sums of coefficients of cusp forms, J. Number Theory, 220 (2021), 1–18. https://doi.org/10.1016/j.jnt.2020.08.007 doi: 10.1016/j.jnt.2020.08.007
    [15] A. Ivić, A note on the Laplace transform of the square in the circle problem, Studia Sci. Math. Hung., 37 (2001), 391–399.
    [16] H. Iwaniec, E. Kowalski, Analytic number theory, Vol. 53, American Mathematical Society Colloquium Publications, 2004.
    [17] M. Jutila, Sums of the additive divisor problem type and the inner product method, J. Math. Sci., 137 (2006), 4755–4761. https://doi.org/10.1007/s10958-006-0271-y doi: 10.1007/s10958-006-0271-y
    [18] E. Kowalski, P. Michel, J. VanderKam, Rankin-Selberg $L$-functions in the level aspect, Duke Math. J., 114 (2002), 123–191. https://doi.org/10.1215/S0012-7094-02-11416-1 doi: 10.1215/S0012-7094-02-11416-1
    [19] Y. K. Lau, J. Liu, Y. Ye, Shifted convolution sums of Fourier coefficients of cusp forms, In: Number theory: Sailing on the sea of number theory, 2007,108–135. https://doi.org/10.1142/9789812770134_0005
    [20] Y. Lin, Triple correlations of Fourier coefficients of cusp forms, Ramanujan J., 45 (2018), 841–858. https://doi.org/10.1007/s11139-016-9874-1 doi: 10.1007/s11139-016-9874-1
    [21] G. Lü, P. Xi, On triple correlations of Fourier coefficients of cusp forms, J. Number Theory, 183 (2018), 485–492. https://doi.org/10.1016/j.jnt.2017.08.028 doi: 10.1016/j.jnt.2017.08.028
    [22] G. Lü, P. Xi, On triple correlations of Fourier coefficients of cusp forms. Ⅱ, Int. J. Number Theory, 15 (2019), 713–722. https://doi.org/10.1142/S1793042119500374 doi: 10.1142/S1793042119500374
    [23] R. Munshi, The circle method and bounds for $L$-functions-Ⅱ. Subconvexity for twists of $GL(3)$ $L$-functions, Amer. J. Math., 137 (2012), 791–812. https://doi.org/10.1353/AJM.2015.0018 doi: 10.1353/AJM.2015.0018
    [24] R. Munshi, The circle method and bounds for $L$-functions-Ⅲ. $t$-aspect subconvexity for $GL(3)$ $L$-functions, J. Amer. Math. Soc., 28 (2015), 913–938.
    [25] R. Munshi, On some recent applications of circle method, Math. Students, 84 (2015), 23–38.
    [26] R. Munshi, On a shifted convolution sum problem, J. Number Theory, 230 (2022), 225–232. https://doi.org/10.1016/j.jnt.2020.12.011 doi: 10.1016/j.jnt.2020.12.011
    [27] A. Saha, Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, Math. Ann., 376 (2020), 609–644. https://doi.org/10.1007/s00208-019-01923-3 doi: 10.1007/s00208-019-01923-3
    [28] S. K. Singh, On double shifted convolution sum of $SL(2, \mathbb{Z})$ Hecke eigenforms, J. Number Theory, 191 (2018), 258–272. https://doi.org/10.1016/j.jnt.2018.03.008 doi: 10.1016/j.jnt.2018.03.008
    [29] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.
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