Skip to main content Accessibility help
×
Home
Hostname: page-component-5bf98f6d76-gckwl Total loading time: 3.103 Render date: 2021-04-21T06:17:10.780Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Mean value theorem connected with Fourier coefficients of Hecke-Maass forms for SL(m, ℤ)

Published online by Cambridge University Press:  25 April 2016

YUJIAO JIANG
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China. e-mails: yujiaoj@hotmail.com
GUANGSHI LÜ
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China. e-mails: gslv@sdu.edu.cn
XIAOFEI YAN
Affiliation:
School of Mathematics, Shandong Normal University, Jinan, Shandong 250014, China. e-mails: xfyan.sdu@gmail.com
Corresponding

Abstract

Let F(z) be a Hecke–Maass form for SL(m, ℤ) with m ⩽ 3, or be the symmetric power lift of a Hecke–Maass form for SL(2, ℤ) if m = 4, 5 and let AF (n, 1, . . ., 1) be the coefficients of L-function attached to F. We establish

$$\sum_{q\leq Q}\max_{(a,q)=1}\max_{y\leq x}\left|\sum_{n\leq y \atop n\equiv a\bmod q}A_F(n,1, \dots, 1)\Lambda(n)\right| \ll x\log^{-A}x,$$
where Q = x ϑ−ϵ with some ϑ > 0, the implied constant depends on F, A, ϵ.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below.

References

[1] Banks, W. D. Twisted symmetric-square L-functions and the nonexistence of siegel zeros on gl (3). Duke Math. J. 87 (2) (1997), 343354.CrossRefGoogle Scholar
[2] Bump, D. Automorphic Forms and Representations, volume 55 (Cambridge University Press, 1998).Google Scholar
[3] Chandrasekharan, K. and Narasimhan, R. Functional equations with multiple gamma factors and the average order of arithmetical functions. Ann. of Math. (2), 76 (1962), 93136.CrossRefGoogle Scholar
[4] Friedlander, J. B. and Iwaniec, H. Summation formulae for coefficients of L-functions. Canad. J. Math. 57 (3) (2005), 494505.CrossRefGoogle Scholar
[5] Goldfeld, D. Automorphic Forms and L-Functions for the Group GL(n, R) Cambridge Studies in Advanced Math. vol 99 (Cambridge University Press, Cambridge, 2006). With an appendix by Kevin A. Broughan.Google Scholar
[6] Hoffstein, J. and Ramakrishnan, D. Siegel zeros and cusp forms. Internat. Math. Res. Notices. (6) (1995), 279308.CrossRefGoogle Scholar
[7] Iwaniec, H. and Kowalski, E. Analytic Number Theory. Amer. Math. Soc. Colloq. Publ. vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[8] Kim, H. H. Functoriality for the exterior square of GL4 and the symmetric fourth of GL2 . J. Amer. Math. Soc. 16 (1) (2003), 139183. With appendix 1 by D. Ramakrishnan and appendix 2 by K. and P. Sarnak.CrossRefGoogle Scholar
[9] Kim, H. H. and Shahidi, F. Functorial products for gl2×gl3 and the symmetric cube for gl2. Annals of math. 155 (3) (2002), 837893.CrossRefGoogle Scholar
[10] Kim, H. H., Shahidi, F. Cuspidality of symmetric powers with applications. Duke Math. J. 112 (1) (2002), 177197.Google Scholar
[11] , G. On sums involving coefficients of automorphic L-functions. Proc. Amer. Math. Soc. 137 (9) (2009), 28792887.CrossRefGoogle Scholar
[12] , G. On averages of Fourier coefficients of Maass cusp forms. Arch. Math. (Basel) 100 (3) (2013), 255265.CrossRefGoogle Scholar
[13] Molteni, G. L-functions: Siegel-type theorems and structure theorems. PhD thesis. University of Milan (1999).Google Scholar
[14] Perelli, A. Exponential sums and mean-value theorems connected with Ramanujan's τ-function. In Seminar on Number Theory (Talence, 1983/1984), pages Exp. No. 25, 9. (Univ. Bordeaux I, Talence, 1984).Google Scholar
[15] Smith, R. A. The average order of a class of arithmetic functions over arithmetic progressions with applications to quadratic forms. J. Reine Angew. Math. 317 (1980), 7487.Google Scholar
[16] Vaughan, R. An elementary method in prime number theory. Acta Arithmetica 37 (1) (1980), 111115.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 128 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 21st April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Mean value theorem connected with Fourier coefficients of Hecke-Maass forms for SL(m, ℤ)
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Mean value theorem connected with Fourier coefficients of Hecke-Maass forms for SL(m, ℤ)
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Mean value theorem connected with Fourier coefficients of Hecke-Maass forms for SL(m, ℤ)
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *