Mean values connected with the Dedekind zeta-function of a non-normal cubic field

• Autorzy: Guangshi Lü
• Języki publikacji: EN

Abstrakty

EN

After Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K 3/ℚ, i.e. $$ S_{l,K_3 } (x) = \sum\nolimits_{m \leqslant x} {M^l (m)} $$, where M(m) denotes the number of integral ideals of the field K 3 of norm m and l ∈ ℕ. We improve the previous results for $$ S_{2,K_3 } (x) $$ and $$ S_{3,K_3 } (x) $$.

Słowa kluczowe

EN

Cusp form; Number field; Dedekind zeta function

Kategorie tematyczne

• 11N37: Asymptotic results on arithmetic functions
• 11R42: Zeta functions and L -functions of number fields
• 11F30: Fourier coefficients of automorphic forms
• 11F66: Langlands L -functions; one variable Dirichlet series and functional equations

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 2
• Strony: 274-282

Daty

wydano

2013-02-01

online

2012-11-21

Twórcy

autor

Guangshi Lü

• Shandong University

Bibliografia

• [1] Cassels J.W.S., Fröhlich A. (Eds.), Algebraic Number Theory, Brighton, September 1–17, 1965, Academic Press/Thompson Book, London/Washington, 1967
• [2] Chandrasekharan K., Good A., On the number of integral ideals in Galois extensions, Monatsh. Math., 1983, 95(2), 99–109 http://dx.doi.org/10.1007/BF01323653
• [3] Chandrasekharan K., Narasimhan R., The approximate functional equation for a class of zeta-functions, Math. Ann., 1963, 152, 30–64 http://dx.doi.org/10.1007/BF01343729
• [4] Deligne P., Serre J.-P., Formes modulaires de poids 1, Ann. Sci. École Norm. Sup., 1974, 7, 507–530
• [5] Fomenko O.M., Mean values associated with the Dedekind zeta function, J. Math. Sci. (N.Y.), 2008, 150(3), 2115–2122 http://dx.doi.org/10.1007/s10958-008-0126-9
• [6] Gelbart S., Jacquet H., A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. École Norm. Sup., 1978, 11(4), 471–542
• [7] Good A., The square mean of Dirichlet series associated with cusp forms, Mathematika, 1982, 29(2), 278–295 http://dx.doi.org/10.1112/S0025579300012377
• [8] Huxley M.N., Watt N., The number of ideals in a quadratic field II, Israel J. Math., 2000, 120(A), 125–153
• [9] Ivic A., Exponent pairs and the zeta function of Riemann, Studia Sci. Math. Hungar., 1980, 15(1–3), 157–181
• [10] Iwaniec H., Kowalski E., Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., 53, American Mathematical Society, Providence, 2004
• [11] Jutila M., Lectures on a Method in the Theory of Exponential Sums, Tata Inst. Fund. Res. Lectures on Math. and Phys., 80, Springer, Berlin, 1987
• [12] Kim H.H., Functoriality for the exterior square of GL4 and symmetric fourth of GL2, J. Amer. Math. Soc., 2003, 16(1), 139–183 http://dx.doi.org/10.1090/S0894-0347-02-00410-1
• [13] Kim H.H., An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type, Invent. Math., 2004, 156(3), 495–502 http://dx.doi.org/10.1007/s00222-003-0340-5
• [14] Kim H.H., Functoriality and number of solutions of congruences, Acta Arith., 2007, 128(3), 235–243 http://dx.doi.org/10.4064/aa128-3-4
• [15] Kim H.H., Shahidi F., Symmetric cube L-functions for GL2 are entire, Ann. of Math., 1999, 150(2), 645–662 http://dx.doi.org/10.2307/121091
• [16] Kim H.H., Shahidi F., Cuspidality of symmetric power with applications, Duke Math. J., 2002, 112(1), 177–197 http://dx.doi.org/10.1215/S0012-9074-02-11215-0
• [17] Kim H.H., Shahidi F., Functorial products for GL2×GL3 and the symmetric cube for GL2, Ann. of Math., 2002, 155(3), 837–893 http://dx.doi.org/10.2307/3062134
• [18] Landau E., Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, Chelsea, New York, 1949
• [19] Li X., Bounds for GL(3)×GL(2) L-functions and GL(3) L-functions, Ann. of Math., 2011, 173(1), 301–336 http://dx.doi.org/10.4007/annals.2011.173.1.8
• [20] Müller W., On the distribution of ideals in cubic number fields, Monatsh. Math., 1988, 106(3), 211–219 http://dx.doi.org/10.1007/BF01318682
• [21] Nowak W.G., On the distribution of integer ideals in algebraic number fields, Math. Nachr., 1993, 161, 59–74 http://dx.doi.org/10.1002/mana.19931610107
• [22] Pan C.D., Pan C.B., Fundamentals of Analytic Number Theory, Science Press, Beijing, 1991 (in Chinese)

Identyfikatory

DOI

10.2478/s11533-012-0133-4