The Ramanujan Journal

, Volume 47, Issue 3, pp 685–700 | Cite as

Higher-power moments of Fourier coefficients of holomorphic cusp forms for the congruence subgroup \(\varGamma _0(N)\)

  • Deyu Zhang
  • Yingnan WangEmail author


Let \(S_k(N)\) be the space of all holomorphic cusp forms of even integral weight k for the congruence group \(\varGamma _0(N).\) For any \(f\in S_k(N)\) with \(\Vert f\Vert _2=1,\) we study the higher-power moments of \(\sum _{n\le x}a_f(n),\) where \(a_f(n)\) is the nth normalized Fourier coefficient of f. Furthermore, as an application, we investigate the higher-power moments of Fourier coefficients in arithmetic progressions.


Fourier coefficients Holomorphic cusp forms Arithmetic progressions 

Mathematics Subject Classification

11F30 11F11 



The authors would like to thank the referee for his or her very valuable inputs and guidance. The authors would also like to thank Dr. Yuk-Kam Lau for his helpful suggestions.


  1. 1.
    Andrianov, A.N., Fomenko, O.M.: On square means of progressions of Fourier coefficients of parabolic forms. Trudy Mat. Inst. Steklova 80, 5–15 (1965)MathSciNetGoogle Scholar
  2. 2.
    Blomer, V.: The average value of divisor sums in arithmetic progressions. Q. J. Math. 59, 275–286 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blomer, V., Milićević, D.: The second moments of twisted modular $L$-functions. Geom. Funct. Anal. 25, 453–516 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cai, Y.C.: On the third and fourth power moments of Fourier coefficients of cusp forms. Acta Math. Sin. N. Ser. 13, 443–452 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chandrasekharan, K., Narasimhan, R.: The approximate functional equation for a class of zeta-functions. Math. Ann. 152, 30–64 (1963)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Davenport, H.: On certain exponential sums. J. Reine Angew. Math. 169, 158–176 (1932)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Deligne, P.: La conjecture de Weil. I. Publ. Math. IHES 43, 273–307 (1974)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fouvry, É., Ganguly, S., Kowalski, E., Michel, P.: Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions. Comment. Math. Helv. 89, 979–1014 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hafner, J.L., Ivić, A.: On sums of Fourier coefficients of cusp forms. Enseign. Math. 35, 373–382 (1989)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hecke, E.: Theorie der Eisensteinsche Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik. Abh. Math. Semin. Univ. Hambg 5, 199–224 (1927)CrossRefGoogle Scholar
  11. 11.
    Ivić, A.: The Riemann Zeta Function Theory and Applications. Wiley, New York (1985)zbMATHGoogle Scholar
  12. 12.
    Ivić, A.: Large values of certain number-theoretic error terms. Acta Arith. 56, 135–159 (1990)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Iwaniec, H.: Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  14. 14.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory, American Mathematical Society Colloquium Publication, vol. 53. American Mathematical Society, Providence (2004)Google Scholar
  15. 15.
    Jiang, Y., Lau, Y.-K., Lü, G., Royer, E., Wu, J.: Sign changes of Fourier coefficients of modular forms of half integral weight. arxiv:1602.08922v1 Google Scholar
  16. 16.
    Jutia, M.: A Method in the Theory of Exponential Sums. Tata Institute, Bombay Lectures, vol. 80. Springer, Berlin (1987)Google Scholar
  17. 17.
    Kloosterman, H.D.: Asymptotische Formeln für die Fourier-koeffizienten ganzer Modulformen. Abh. Math. Semin. Univ. Hambg 5, 337–353 (1927)CrossRefGoogle Scholar
  18. 18.
    Lau, Y.-K., Wu, J.: A density theorem on automorphic L-functions and some applications. Trans. Am. Math. Soc. 358, 441–472 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lau, Y.-K., Zhao, L.: On the variance of Hecke eigenvalues in arithmetic progressions. J. Number Theory 132, 869–887 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lü, G.: The average value of Fourier coefficients of cusp forms in arithmetic progressions. J. Number Theory 129, 488–494 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ng, M.H.: The basis for space of cusp forms and Petersson trace formula. MPhil Thesis, The University of Hong Kong (2012)Google Scholar
  22. 22.
    Rankin, R.A.: Sums of cusp form coefficients. In: Automorphic Forms and Analytic Number Theory (Montreal, PQ, 1989), pp. 115–121. University of Montreal, Montreal (1990)Google Scholar
  23. 23.
    Salié, H.: Zur Abschätzung der Fourierkoeffizienten ganzer Modulformen. Math. Z. 36, 263–278 (1933)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tenenbaum, G.: Introduction to Analytic and Probabilistic Number Theory. Translated from the second French edition (1995) by Thomas, C.B. Cambridge Studies in Advanced Mathematics, vol. 46. Cambridge University Press, Cambridge (1995)Google Scholar
  25. 25.
    Walfisz, A.: Über die Koeffizientensummen einiger Moduformen. Math. Ann. 108, 75–90 (1933)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Weil, A.: On some exponential sums. Proc. Natl Acad. Sci. USA 34, 204–207 (1948)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wilton, J.R.: A note on Ramanujans arithmetical function $\tau (n)$. Proc. Camb. Philos. Soc. 25, 121–129 (1928)CrossRefGoogle Scholar
  28. 28.
    Wu, J.: Power sums of Hecke eigenvalues and applications. Acta Arith. 137, 333–344 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Yao, W.: A Barban–Davenport–Halberstam theorem for integers with fixed number of prime divisors. Sci. China A 57, 2103–2110 (2014)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhai, W.G.: On higher-power moments of $\varDelta (x)$. Acta Arith. 112, 367–396 (2004)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zhai, W.G.: On higher-power moments of $\varDelta (x)$ (IV). Acta Math. Sin. (Chin. Ser.) 49, 639–646 (2006)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Zhai, W.G.: On the error term in Weyl’s law for Heisenberg manifolds. Acta Arith. 134, 219–257 (2008)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhang, D.Y., Lau, Y.-K., Wang, Y.: Remark on the paper “On products of Fourier coefficients of cusp forms”. Arch. Math. 108, 263–269 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShandong Normal UniversityJinanChina
  2. 2.College of Mathematics and StatisticsShenzhen UniversityShenzhenChina

Personalised recommendations