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Unconditional explicit Mertens’ theorems for number fields and Dedekind zeta residue bounds. (English) Zbl 1497.11278

Summary: We obtain unconditional, effective number-field analogues of the three Mertens’ theorems, all with explicit constants and valid for \(x\ge 2\). Our error terms are explicitly bounded in terms of the degree and discriminant of the number field. To this end, we provide unconditional bounds, with explicit constants, for the residue of the corresponding Dedekind zeta function at \(s=1\).

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11R29 Class numbers, class groups, discriminants
11N05 Distribution of primes
11R11 Quadratic extensions
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[1] Bardestani, M., Freiberg, T.: Mertens’s theorem for splitting primes and more. arXiv:1309.7482
[2] Bateman, PT; Grosswald, E., Imaginary quadratic fields with unique factorization, Ill. J. Math., 6, 187-192 (1962) · Zbl 0100.03103
[3] Borevich, A.I., Shafarevich, I.R.: Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London (1966) · Zbl 0145.04902
[4] Broadbent, S., Kadiri, H., Lumley, A., Ng, N., Wilk, K.: Sharper bounds for the Chebyshev function \(\theta (x)\). arXiv:2002.11068 · Zbl 1478.11108
[5] de la Vallée Poussin, C. J., La fonction \(\zeta (s)\) de Riemann et les nombres premiers en général, Ann. Soc. Sci. Bruxelles Sér. I, 20, 183-256 (1896)
[6] Diamond, H.G., Halberstam, H.: A higher-dimensional sieve method, Cambridge Tracts in Mathematics, vol. 177, Cambridge University Press, Cambridge, 2008, With an appendix (“Procedures for computing sieve functions”) by William F. Galway · Zbl 1207.11099
[7] Diamond, HG; Pintz, J., Oscillation of Mertens’ product formula, J. Théor. Nombres Bordeaux, 21, 3, 523-533 (2009) · Zbl 1214.11102 · doi:10.5802/jtnb.687
[8] Friedman, E., Analytic formulas for the regulator of a number field, Invent. Math., 98, 3, 599-622 (1989) · Zbl 0694.12006 · doi:10.1007/BF01393839
[9] Garcia, S.R., Lee, E.S.: Explicit Mertens’ theorems for number fields and Dedekind zeta residue bounds assuming GRH. arXiv:2006.03337
[10] Grenié, L.; Molteni, G., An explicit Chebotarev density theorem under GRH, J. Number Theory, 200, 441-485 (2019) · Zbl 1443.11236 · doi:10.1016/j.jnt.2018.12.005
[11] Hadamard, J., Sur la distribution des zéros de la fonction \(zeta (s)\) et ses conséquences arithmétiques, Bulletin de la Societé mathematique de France, 24, 199-220 (1896) · JFM 27.0154.01 · doi:10.24033/bsmf.545
[12] Hardy, GH, Note on a Theorem of Mertens, J. Lond. Math. Soc., 2, 2, 70-72 (1927) · JFM 53.0152.01 · doi:10.1112/jlms/s1-2.2.70
[13] Hardy, GH, Second note on a Theorem of Mertens, J. Lond. Math. Soc., 10, 2, 91-94 (1935) · Zbl 0011.29403 · doi:10.1112/jlms/s1-10.1.91
[14] Heilbronn, H.: On real simple zeros of Dedekind \(\zeta \)-functions. In: Proceedings of the Number Theory Conference (University of Colorado, Boulder, 1972), 1972, pp. 108-110 · Zbl 0341.12007
[15] Ingham, A.E.: The distribution of prime numbers, Cambridge Mathematical Library, Cambridge University Press, Cambridge: Reprint of the 1932 original. With a foreword by R. C, Vaughan (1990) · Zbl 0715.11045
[16] Lang, S., Algebraic Number Theory. Graduate Texts in Mathematics (1994), New York: Springer, New York · Zbl 0811.11001 · doi:10.1007/978-1-4612-0853-2
[17] Lebacque, P., Generalised Mertens and Brauer-Siegel theorems, Acta Arith., 130, 4, 333-350 (2007) · Zbl 1155.11032 · doi:10.4064/aa130-4-3
[18] Legendre, A.-M.: Essai sur la théorie des nombres, Cambridge Library Collection, Cambridge University Press, Cambridge, 2009, Reprint of the second (1808) edition
[19] Louboutin, S., Explicit bounds for residues of Dedekind zeta functions, values of \(L\)-functions at \(s=1\), and relative class numbers, J. Number Theory, 85, 2, 263-282 (2000) · Zbl 0967.11049 · doi:10.1006/jnth.2000.2545
[20] Mertens, F., Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math., 78, 46-62 (1874) · JFM 06.0116.01
[21] Montgomery, H. L., Vaughan, R. C.: Multiplicative number theory I: classical theory, reprint ed., Cambridge Studies in Advanced Mathematics (Book 97) (2012) · Zbl 1245.11002
[22] Pintz, J., Elementary methods in the theory of \(L\)-functions. II. On the greatest real zero of a real \(L\)-function, Acta Arith., 31, 3, 273-289 (1976) · Zbl 0307.10041 · doi:10.4064/aa-31-3-273-289
[23] Rosen, M., A generalization of Mertens’ theorem, J. Ramanujan Math. Soc., 14, 1, 1-19 (1999) · Zbl 1133.11317
[24] Rosser, JB; Schoenfeld, L., Approximate formulas for some functions of prime numbers, Ill. J. Math., 6, 64-94 (1962) · Zbl 0122.05001
[25] Stark, HM, A complete determination of the complex quadratic fields of class-number one, Mich. Math. J., 14, 1-27 (1967) · Zbl 0148.27802 · doi:10.1307/mmj/1028999653
[26] Stark, HM, Some effective cases of the Brauer-Siegel theorem, Invent. Math., 23, 135-152 (1974) · Zbl 0278.12005 · doi:10.1007/BF01405166
[27] Stewart, I.; Tall, D., Algebraic Number Theory and Fermat’s Last Theorem (2016), Boca Raton: CRC Press, Boca Raton · Zbl 1332.11001
[28] Sunley, J.E.S.: On the class numbers of totally imaginary quadratic extensions of totally real fields, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.). University of Maryland, College Park (1971) · Zbl 0231.12009
[29] Sunley, JES, On the class numbers of totally imaginary quadratic extensions of totally real fields, Bull. Am. Math. Soc., 78, 74-76 (1972) · Zbl 0231.12009 · doi:10.1090/S0002-9904-1972-12859-3
[30] Sunley, JES, Class numbers of totally imaginary quadratic extensions of totally real fields, Trans. Am. Math. Soc., 175, 209-232 (1973) · Zbl 0289.12010 · doi:10.1090/S0002-9947-1973-0311622-9
[31] Tenenbaum, G.: Generalized Mertens sums, Analytic number theory, modular forms and \(q\)-hypergeometric series, Springer Proc. Math. Stat., vol. 221, Springer, Cham, pp. 733-736 (2017) · Zbl 1426.11098
[32] Zimmert, R., Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung, Invent. Math., 62, 3, 367-380 (1981) · Zbl 0456.12003 · doi:10.1007/BF01394249
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