Higher moment of coefficients of Dedekind zeta function

Abstract

Let K₃ be a non-normal cubic extension over \mathbb{Q}.We study the higher moment of the coefficients aK₃ (n) of Dedekind zeta function over sum of two squares {\displaystyle {\sum }_{n_{\mathrm{1}}^{\mathrm{2}}+n_{\mathrm{2}}^{\mathrm{2}}\le x}{a}_{K_{\mathrm{3}}}^l(n_{\mathrm{1}}^{\mathrm{2}}+n_{\mathrm{2}}^{\mathrm{2}})},where 2≤l≤8 and n1,n2,l∈\mathbb{Z}.

Keywords

Non-normal cubic field / Dedekind zeta function

References

[1] Bourgain J. Decoupling, exponential sums and the Riemann zeta function. J Amer Math Soc, 2017, 30(1): 205–224

[2] Chandrasekharan K, Good A. On the number of integral ideals in Galois extensions. Monatsh Math, 1983, 95(2): 99–109

[3] Chandrasekharan K, Narasimhan R. The approximate functional equation for a class of zeta-functions. Math Ann, 1963, 152: 30–64

[4] Fomenko O M. Mean values associated with the Dedekind zeta function. J Math Sci (N Y), 2008, 150(3): 2115–2122

[5] Gelbart S, Jacquet H. A relation between automorphic representations of GL(2) and GL(3).Ann Sci Éc Norm Supér (4), 1978, 11: 471–542

[6] Good A. The square mean of Dirichlet series associated with cusp forms. Mathematika 1982, 29(2): 278–295

[7] Heath-Brown D R. The growth rate of the Dedekind zeta-function on the critical line. Acta Arith, 1988, 49(4): 323–339

[8] Iwaniec H, Kowalski E. Analytic Number Theory. Amer Math Soc Colloq Publ, Vol 53. Providence: Amer Math Soc, 2004

[9] Jacquet H, Shalika J A. On Euler products and the classification of automorphic representations I. Amer J Math, 1981, 103(3): 499–558

[10] Jacquet H, Shalika J A. On Euler products and the classification of automorphic forms II. Amer J Math, 1981, 103(4): 777–815

[11] Jutila M. Lectures on a Method in the Theory of Exponential Sums. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 80. Berlin: Springer- Verlag, 1987

[12] Kim H H. Functoriality for the exterior square of GL₄ and the symmetric fourth of GL₂ (with Appendix 1 by Dinakar Ramakrishnan and Appendix 2 by Kim and Peter Sarnak). J Amer Math Soc, 2003, 16(1): 139–183

[13] Kim H H, Shahidi F. Functorial products for GL₂ × GL₃ and the symmetric cube for GL₂ (with an appendix by C. J. Bushnell and G. Henniart). Ann of Math, 2002, 155(2): 837–893

[14] Kim H H, Shahidi F. Cuspidality of symmetric powers with applications. Duke Math J, 2002, 112: 177–197

[15] Landau, E. Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale. New York: Chelsea Publishing Company, 1949

[16] Li X Q. Bounds for GL(3) × GL(2) L-functions and GL(3) L-functions. Ann of Math, 2011, 173(1): 301–336

[17] Lü G S. Mean values connected with the Dedekind zeta function of a non-normal cubic field. CEJOR Cent Eur J Oper Res, 2013, 11(2): 274–282

[18] Lü G S, Wang Y H. Note on the number of integral ideals in Galois extensions. Sci China Math, 2010, 53(9): 2417–2424

[19] Rudnick Z, Sarnak P. Zeros of principal L-functions and random matrix theory. Duke Math J, 1996, 81: 269–322

[20] Shahidi F. On certain L-functions. Amer J Math, 1981, 103(2): 297–355

[21] Shahidi F. Third symmetric power L-functions for GL(2).Compos Math, 1989, 70(3): 245–273

[22] Tenenbaum G. Introduction to Analytic and Probabilistic Number Theory. Cambridge Stud Adv Math, 46. Cambridge: Cambridge Univ Press, 1995

[23] Yang Z S. Ideal counting function in cubic fields. Front Math China, 2017, 12(4): 981–992

RIGHTS & PERMISSIONS

2020 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature