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Some problems involving Hecke eigenvalues

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Abstract

We study a square mean value problem and two general divisor problems related to Hecke eigenvalues of classical holomorphic cusp forms and classical Maass cusp forms, respectively. We improve previous results.

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References

  1. Barthel, L., Ramakrishnan, D.: A nonvanishing result for twists of \(L\)-functions of \(GL(n)\). Duke Math. J. 74, 681–700 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bourgain, J.: Decoupling, exponential sums and the Riemann zeta function. Amer. Math. Soc. 30, 205–224 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chandrasekharan, K., Narasimhan, R.: Functional equations with multiple gamma factors and the average order of arithmetical functions. Ann. of Math. 76, 93–136 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  4. Deligne, P.: La conjecture de Weil. Inst. Hautes Études Sci. Publ. Math. 43, 29–39 (1974)

    Article  MathSciNet  Google Scholar 

  5. Gelbart, S., Jacquet, H.: A relation between automorphic representations of \(GL(2)\) and \(GL(3)\). Ann. Sci. École Norm. Sup. 11, 471–542 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Heath-Brown, D.R.: The twelfth power moment of the Riemann zeta-function. Quart. J. Math. 29, 443–462 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ivić, A.: Exponent pairs and the zeta function of Riemann. Stud. Sci. Math. Hungar. 15, 157–181 (1980)

    MathSciNet  MATH  Google Scholar 

  8. Iwaniec, H.: Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17. American Mathematical Society (Providence, RI (1997)

    MATH  Google Scholar 

  9. H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS Coll. Publ.  53, American Mathematical Society (Providence, RI, 2004)

  10. Kanemitsu, S., Sankaranarayanan, A., Tanigawa, Y.: A mean value theorem for Dirichlet series and a general divisor problem. Monatsh. Math. 136, 17–34 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kim, H.: Functoriality for the exterior square of \(GL_4\) and symmetric fourth of \(GL_2\). J. Amer. Math. Soc. 16, 139–183 (2003)

    Article  MathSciNet  Google Scholar 

  12. Kim, H.H., Shahidi, F.: Functorial products for \(GL_2 \times GL_3\) and the symmetric cube for \(GL_2\). Ann. Math. 155, 837–893 (2002)

    Article  MathSciNet  Google Scholar 

  13. Kim, H.H., Shahidi, F.: Cuspidality of symmetric powers with applications. Duke Math. J. 112, 177–197 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim, H., Sarnak, P.: Functoriality for the exterior square of \(GL_4\) and the symmetric fourth of \(GL_2\) (by Kim); with appendix, Refined estimates towards the Ramanujan and Selberg conjectures (by Kim and Sarnak). J. Amer. Math. Soc. 16, 139–183 (2003)

    Article  MathSciNet  Google Scholar 

  15. Liu, H.F.: Mean value estimates of the coefficients of product \(L\)-functions. Acta. Math. Hungar. 156, 102–111 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lü, G.S.: On general divisor problems involving Hecke eigenvalues. Acta. Math. Hungar. 135, 148–159 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. R. M. Nunes, On the subconveity estimate for self-dual \(GL(3)\) \(L\)-functions in the \(t\)-aspect, arXiv:1703.04424v1 (2017)

  18. Perelli, A.: General \(L\)-functions. Ann. Mat. Pura Appl. 130, 287–306 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. R. A. Rankin, Contributions to the theory of Ramanujan's function \(\tau (n)\) and similar arithemtical functions. II, The order of the Fourier coefficients of the integral modular forms, Proc. Cambridge Phil. Soc., 35 (1939), 357–372

  20. Selberg, A.: Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist. Arch. Math. Naturvid. 43, 47–50 (1940)

    MathSciNet  MATH  Google Scholar 

  21. Wang, D.: On general divisor problems involving Hecke eigenvalues. Acta. Math. Hungar. 153, 509–523 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

The authors thank the referee for comments.

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Correspondence to R. Zhang.

Additional information

This work is supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2018BA006) and the National Natural Science Foundation of China (Grant No. 11801328).

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Liu, H.F., Zhang, R. Some problems involving Hecke eigenvalues. Acta Math. Hungar. 159, 287–298 (2019). https://doi.org/10.1007/s10474-019-00913-w

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  • DOI: https://doi.org/10.1007/s10474-019-00913-w

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