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Extreme residues of Dedekind zeta functions

  • PETER J. CHO (a1) and HENRY H. KIM (a2)

In a family of S d+1-fields (d = 2, 3, 4), we obtain the conjectured upper and lower bounds of the residues of Dedekind zeta functions except for a density zero set. For S 5-fields, we need to assume the strong Artin conjecture. We also show that there exists an infinite family of number fields with the upper and lower bounds, resp.

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[1] Cho, P.J. and Kim, H.H. Probabilistic properties of number fields. J. Number Theory, 133 (2013), 41754187.
[2] Cho, P.J. and Kim, H.H. Central limit theorem for Artin L-functions. Int. J. Number Theory, 13 (2017), no. 1, 114.
[3] Chowla, S. Improvement of a theorem of Linnik and Walfisz. Proc. London Math. Soc., 50 (1949), 423429.
[4] Daileda, R.C. Non-abelian number fields with very large class numbers. Acta Arith., 125 (2006), 215255.
[5] Granville, A. and Soundararajan, K. The Distribution of Values of L(1, χ d ). Geom. Funct. Anal., 13 (2003), no. 5, 9921028.
[6] Large character sums. J. Amer. Math. Soc., 14 (2000), no. 2, 365397.
[7] Klingen, N. Arithmetical Similarities. Oxford Math. Monogr., Oxford Sci. Publ. (The Clarendon Press, Oxford University Press, New York 1998).
[8] Kowalski, E. and Michel, P. Zeros of families of automorphic L-functions close to 1. Pacific J. Math., 207 (2002), No. 2, 411431.
[9] Lamzouri, Y. Extreme values of class numbers of real quadratic fields. Int. Math. Res. Not. 2015, no. 2, 632653.
[10] Lamzouri, Y. Large values of L(1,χ) for k-th order characters χ and applications to character sums. Mathematika, 63 (2016), 5371
[11] Littlewood, J.E. On the class number of corpus . Proc. London Math. Soc., 27, no.1 (1928): 358372.
[12] Montgomery, H.L. and Vaughan, R.C. Extreme values of Dirichlet L-functions at 1. Number Theory in Progress, Vol. 2 (Zakopane-Kościelisko, 1997) (de Gruyter, Berlin, 1999), 10391052.
[13] Murty, M.R. Problems in analytic number theory, Second edition. Graduate Texts in Math. 206 (Springer, New York, 2008).
[14] Taniguchi, T. and Thorne, F. Secondary terms in counting functions for cubic fields. Duke Math. J., 162 (2013), 24512508.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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