## The signs of the Stieltjes constants associated with the Dedekind zeta function.(English)Zbl 1453.11148

Summary: The Stieltjes constants $$\gamma_{n}(K)$$ of a number field $$K$$ are the coefficients of the Laurent expansion of the Dedekind zeta function $$\zeta_{K}(s)$$ at its pole $$s=1$$. In this paper, we establish a similar expression of $$\gamma_{n}(K)$$ as Stieltjes obtained in 1885 for $$\gamma_{n}(\mathbb{Q})$$. We also study the signs of $$\gamma_{n}(K)$$.

### MSC:

 11R42 Zeta functions and $$L$$-functions of number fields 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
Full Text:

### References:

 [1] J. A. Adell and A. Lekuona, Fast computation of the Stieltjes constants, Math. Comp. 86 (2017), no. 307, 2479–2492. · Zbl 1378.11082 [2] J. A. Adell, Asymptotic estimates for Stieltjes constants: a probabilistic approach, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2128, 954–963. · Zbl 1219.11185 [3] W. E. Briggs, Some constants associated with the Riemann zeta-function, Michigan Math. J. 3 (1955–56), 117–121. · Zbl 0073.29303 [4] M. W. Coffey, Hypergeometric summation representations of the Stieltjes constants, Analysis (Munich) 33 (2013), no. 2, 121–142. · Zbl 1284.11118 [5] M. W. Coffey, Series representations for the Stieltjes constants, Rocky Mountain J. Math. 44 (2014), no. 2, 443–477. · Zbl 1320.11085 [6] Y. Hashimoto, Y. Iijima, N. Kurokawa and M. Wakayama, Euler’s constants for the Selberg and the Dedekind zeta functions, Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 4, 493–516. · Zbl 1080.11062 [7] Y. Ihara, On the Euler-Kronecker constants of global fields and primes with small norms, in Algebraic geometry and number theory, Progr. Math., 253, Birkhäuser Boston, Boston, MA, 2006, pp. 407–451. · Zbl 1185.11069 [8] M. I. Israilov, The Laurent expansion of the Riemann zeta function, Trudy Mat. Inst. Steklov. 158 (1981), 98–104. · Zbl 0477.10031 [9] C. Knessl and M. W. Coffey, An effective asymptotic formula for the Stieltjes constants, Math. Comp. 80 (2011), no. 273, 379–386. · Zbl 1208.41020 [10] E. Landau, Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, Chelsea Publishing Company, New York, NY, 1949. · Zbl 0045.32202 [11] Y. Matsuoka, Generalized Euler constants associated with the Riemann zeta function, in Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984), 279–295, World Sci. Publishing, Singapore, 1985. [12] D. Mitrović, The signs of some constants associated with the Riemann zeta-function, Michigan Math. J. 9 (1962), 395–397. [13] A. Reich, Zur Universalität und Hypertranszendenz der Dedekindschen Zetafunktion, Abh. Braunschweig. Wiss. Ges. 33 (1982), 197–203. · Zbl 0505.12018 [14] S. Saad Eddin, Explicit upper bounds for the Stieltjes constants, J. Number Theory 133 (2013), no. 3, 1027–1044. · Zbl 1282.11106 [15] S. Saad Eddin, Applications of the Laurent-Stieltjes constants for Dirichlet $$L$$-series, Proc. Japan Acad. Ser. A Math. Sci. 93 (2017), no. 10, 120–123. · Zbl 1430.11114 [16] M. A. Tsfasman, Asymptotic behaviour of the Euler-Kronecker constant, in Algebraic geometry and number theory, Progr. Math., 253, Birkhäuser Boston, Boston, MA, 2006, pp. 453–458. · Zbl 1185.11070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.