Shifted convolution sums for \({{\varvec{SL}}}(m)\)

Abstract

In this paper, we study the shifted convolution sums of the Fourier coefficients \(\lambda _\pi (1,\ldots ,1,n)\) and \(r_{s,k}(n)\) with \(k\ge 3\), where \(r_{s,k}(n)\) denotes the number of representations of the positive integer n as sums of s kth powers. We are able to generalize or improve previous results.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Bourgain, J., Demeter, C., Guth, L.: Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. Math. 184(2), 633–682 (2016)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Goldfeld, D., Li, X.Q.: The Voronoi formula for \(GL(n,R)\). Int. Math. Res. Not. IMRN, No. 2, Art. ID rnm144 (2008)

  3. 3.

    Ingham, A.E.: Some asymptotic formulae in the theory of numbers. J. Lond. Math. Soc. 2(3), 202–208 (1927)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Iwaniec, H., Kowalski, E.: Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)

    Google Scholar 

  5. 5.

    Jiang, Y.J., Lü, G.S.: Shifted convolution sums for higher rank groups. Forum Math. 31(2), 361–383 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Kim, H.: Functoriality for the exterior square of \(GL_4\) and the symmetric fourth of \(GL_2\). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. J. Am. Math. Soc. 16(1), 139–183 (2003)

    Article  Google Scholar 

  7. 7.

    Lü, G.S., Wu, J., Zhai, W.G.: Shifted convolution of cusp-forms with \(\theta \)-series. Ramanujan J. 40(1), 115–133 (2016)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Luo, W.Z.: Shifted convolution of cusp-forms with \(\theta \)-series. Abh. Math. Semin. Univ. Hambg. 81(1), 45–53 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Luo, W.Z., Rudnick, Z., Sarnak, P.: On the generalized Ramanujan conjecture for \({ GL}(n)\). Automorphic forms, automorphic representations, and arithmetic. Proc. Sympos. Pure Math. 66, 301–310 (1996)

    Google Scholar 

  10. 10.

    Miller, S.D.: Cancellation in additively twisted sums on \(GL(n)\). Am. J. Math. 128(3), 699–729 (2006)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Munshi, R.: Shifted convolution sums for \({GL}(3)\times {GL}(2)\). Duke Math. J. 162(13), 2345–2362 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Pitt, N.J.E.: On shifted convolutions of \(\zeta _3(s)\) with automorphic \(L\)-functions. Duke Math. J. 77(2), 383–406 (1995)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Vaughan, R.: The Hardy–Littlewood Method, Cambridge Tracts in Mathematics, 2nd ed., (Vol. 125), Cambridge University, Cambridge, (1997)

  14. 14.

    Wooley, Trevor D.: The asymptotic formula in Waring’s problem. Int. Math. Res. Not. IMRN 7, 1485–1504 (2012)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the referee for some extremely helpful remarks.

Funding

This work is supported in part by NSFC (Nos. 11771252, 11531008), IRT16R43, and the Taishan Scholar Project.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Guangwei Hu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hu, G., Lü, G. Shifted convolution sums for \({{\varvec{SL}}}(m)\). manuscripta math. 163, 375–394 (2020). https://doi.org/10.1007/s00229-019-01166-1

Download citation

Mathematics Subject Classification (2010)

  • 11E76
  • 11F30
  • 11P55