Abstract
In this paper, we study some shifted convolution sums for higher rank groups. In particular, we establish an asymptotic formula for a
where
Funding source: China Postdoctoral Science Foundation
Award Identifier / Grant number: 2017M620285
Funding source: Natural Science Foundation of Shandong Province
Award Identifier / Grant number: ZR2018QA004
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11801318
Award Identifier / Grant number: 11771252
Award Identifier / Grant number: 11531008
Funding source: Changjiang Scholar Program of Chinese Ministry of Education
Award Identifier / Grant number: IRT16R43
Funding statement: Jiang is supported by the China Postdoctoral Science Foundation (no. 2017M620285), the Natural Science Foundation of Shandong Province (no. ZR2018QA004) and NSFC (no. 11801318), and Lü is supported in part by NSFC (nos. 11771252, 11531008), IRT16R43 and Taishan Scholars.
Acknowledgements
The authors wish to thank the referee for valuable comments.
References
[1] F. V. Atkinson, The mean value of the zeta-function on the critical line, Proc. Lond. Math. Soc. (2) 47 (1941), 174–200. 10.1112/plms/s2-47.1.174Suche in Google Scholar
[2]
L. Barthel and D. Ramakrishnan,
A nonvanishing result for twists of L-functions of
[3] P. T. Bateman, On the representations of a number as the sum of three squares, Trans. Amer. Math. Soc. 71 (1951), 70–101. 10.1090/S0002-9947-1951-0042438-4Suche in Google Scholar
[4] J. Bourgain, On the Fourier–Walsh spectrum of the Moebius function, Israel J. Math. 197 (2013), no. 1, 215–235. 10.1007/s11856-013-0002-2Suche in Google Scholar
[5] P. Deligne, La conjecture de Weil. I, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–307. 10.1007/BF02684373Suche in Google Scholar
[6] J.-M. Deshouillers and H. Iwaniec, An additive divisor problem, J. Lond. Math. Soc. (2) 26 (1982), no. 1, 1–14. 10.1112/jlms/s2-26.1.1Suche in Google Scholar
[7] T. Estermann, Über die Darstellungen einer Zahl als Differenz von zwei Produkten, J. Reine Angew. Math. 164 (1931), 173–182. 10.1515/crll.1931.164.173Suche in Google Scholar
[8]
D. Goldfeld,
Automorphic Forms and L-functions for the Group
[9]
D. Goldfeld and X. Li,
The Voronoi formula for
[10] G. H. Hardy, On the representation of a number as the sum of any number of squares, and in particular of five, Trans. Amer. Math. Soc. 21 (1920), no. 3, 255–284. 10.1090/S0002-9947-1920-1501144-7Suche in Google Scholar
[11] D. R. Heath-Brown, The fourth power moment of the Riemann zeta function, Proc. Lond. Math. Soc. (3) 38 (1979), no. 3, 385–422. 10.1112/plms/s3-38.3.385Suche in Google Scholar
[12] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence, 2004. 10.1090/coll/053Suche in Google Scholar
[13] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499–558. 10.2307/2374103Suche in Google Scholar
[14]
H. H. Kim,
Functoriality for the exterior square of
[15] E. M. Kıral and F. Zhou, The Voronoi formula and double Dirichlet series, Algebra Number Theory 10 (2016), no. 10, 2267–2286. 10.2140/ant.2016.10.2267Suche in Google Scholar
[16] G. Lü, J. Wu and W. Zhai, Shifted convolution of cusp-forms with θ-series, Ramanujan J. 40 (2016), no. 1, 115–133. 10.1007/s11139-015-9678-8Suche in Google Scholar
[17] W. Luo, Shifted convolution of cusp-forms with θ-series, Abh. Math. Semin. Univ. Hambg. 81 (2011), no. 1, 45–53. 10.1007/s12188-010-0046-8Suche in Google Scholar
[18]
W. Luo, Z. Rudnick and P. Sarnak,
On the generalized Ramanujan conjecture for
[19]
S. D. Miller and W. Schmid,
Automorphic distributions, L-functions, and Voronoi summation for
[20]
S. D. Miller and W. Schmid,
A general Voronoi summation formula for
[21]
R. Munshi,
Shifted convolution sums for
[22]
N. J. E. Pitt,
On shifted convolutions of
[23]
D. Ramakrishnan,
Modularity of the Rankin–Selberg L-series, and multiplicity one for
[24]
X. Ren and Y. Ye,
Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for
[25] A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math. 8 (1965), 1–15. 10.1090/pspum/008/0182610Suche in Google Scholar
[26] R. A. Smith, The average order of a class of arithmetic functions over arithmetic progressions with applications to quadratic forms, J. Reine Angew. Math. 317 (1980), 74–87. 10.1515/crll.1980.317.74Suche in Google Scholar
[27]
Q. Sun,
Shifted convolution sums of
[28] R. C. Vaughan, The Hardy–Littlewood Method, 2nd ed., Cambridge Tracts in Math. 125, Cambridge University, Cambridge, 1997. 10.1017/CBO9780511470929Suche in Google Scholar
[29] L. Weinstein, The hyper-Kloosterman sum, Enseign. Math. (2) 27 (1981), no. 1–2, 29–40. Suche in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
- Infinite families of equivariantly formal toric orbifolds
- Bounds for GL3L-functions in depth aspect
- Nonlinear Dirichlet problems with unilateral growth on the reaction
- K-invariant cusp forms for reductive symmetric spaces of split rank one
- Cellularity in quotient spaces of topological groups
- Shifted convolution sums for higher rank groups
- Dimension quotients, Fox subgroups and limits of functors
- Large sums of Hecke eigenvalues of holomorphic cusp forms
- K-theory classification of graded ultramatricial algebras with involution
- Regularity of symbolic powers and arboricity of matroids
- On some problems concerning symmetrization operators
- An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
- Rankin–Selberg gamma factors of level zero representations of GLn
- Sobolev’s inequality for double phase functionals with variable exponents
- Independence of Artin L-functions
- Oscillatory singular integral operators with Hölder class kernels on Hardy spaces
Artikel in diesem Heft
- Frontmatter
- Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
- Infinite families of equivariantly formal toric orbifolds
- Bounds for GL3L-functions in depth aspect
- Nonlinear Dirichlet problems with unilateral growth on the reaction
- K-invariant cusp forms for reductive symmetric spaces of split rank one
- Cellularity in quotient spaces of topological groups
- Shifted convolution sums for higher rank groups
- Dimension quotients, Fox subgroups and limits of functors
- Large sums of Hecke eigenvalues of holomorphic cusp forms
- K-theory classification of graded ultramatricial algebras with involution
- Regularity of symbolic powers and arboricity of matroids
- On some problems concerning symmetrization operators
- An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
- Rankin–Selberg gamma factors of level zero representations of GLn
- Sobolev’s inequality for double phase functionals with variable exponents
- Independence of Artin L-functions
- Oscillatory singular integral operators with Hölder class kernels on Hardy spaces