# On the Euler-Kronecker constants of global fields and primes with small norms

• Yasutaka Ihara
Chapter
Part of the Progress in Mathematics book series (PM, volume 253)

## Abstract

Let K be a global field, i.e., either an algebraic number field of finite degree (abbreviated NF), or an algebraic function field of one variable over a finite field (FF). Let ζK(s) be the Dedekind zeta function of K, with the Laurent expansion at s = 1:
$$\zeta _K \left( s \right) = c_{ - 1} \left( {s - 1} \right)^{ - 1} + c_0 + c_1 \left( {s - 1} \right) + \cdots \left( {c_{ - 1} \ne 0} \right)$$
(0.1)
In this paper, we shall present a systematic study of the real number
$$\gamma _K = {{c_0 } \mathord{\left/ {\vphantom {{c_0 } {c_{ - 1} }}} \right. \kern-\nulldelimiterspace} {c_{ - 1} }}$$
(0.2)
attached to each K, which we call the Euler-Kronecker constant (or invariant) of K. When K = ℚ (the rational number field), it is nothing but the Euler-Mascheroni constant
$$\gamma _\mathbb{Q} = \mathop {\lim }\limits_{n \to \infty } \left( {1 + \frac{1} {2} + \cdots + \frac{1} {n} - \log n} \right) = 0.57721566...,$$
and when K is imaginary quadratic, the well-known Kronecker limit formula expresses γ K in terms of special values of the Dedekind η function. This constant γ K appears here and there in several articles in analytic number theory, but as far as the author knows, it has not played a main role nor has it been systematically studied. We shall consider γ K more as an invariant of K.

## Keywords

Explicit Formula Rational Point Finite Field Main Lemma Small Norm
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [D-V]
V. G. Drinfeld and S. G. Vlăduţ, On the number of points of algebraic curves, Functional Anal., 17 (1983), 68–69 (inRussian).
2. [E]
N. Elkies, E. Howe, A. Kresch, B. Poonen, J. Wetherell, and M. Zieve, Curves of every genus with many points II: Asymptotically good families, Duke Math. J., 122 (2004), 399–422.
3. [G]
K. Győry, Discriminant form and index form equations, in F. Halter-Koch and R. F. Tichy, eds., Algebraic Number Theory and Diophantine Analysis, Walter De Gruyter, Berlin, New York, 2000, 191–214.Google Scholar
4. [G-S]
A. Granville and H. M. Stark, ABC implies no “Siegel zeros” for L-functions of characters with negative discriminant, Invent. Math., 139 (2000), 509–523.
5. [H]
E. Hecke, Über die Kroneckersche Grenzformel für reelle Quadratische Körper und die Klassenzahl relativ-abelscher Körper, Verh. Naturf. Ges. Basel, 28 (1917), 363–372.; also in Mathematische Werke, 2nd ed., Vandenhoeck and Ruprecht, Göttingen, the Netherlands, 1970, 198–207.Google Scholar
6. [HIKW]
Y. Hashimoto, Y. Iijima, N. Kurokawa, and M. Wakayama, Euler’s constants for the Selberg and the Dedekind zeta functions, Bull. Belgian Math. Soc., 11-4 (2004), 493–516.
7. [I1]_Y. Ihara, The congruence monodromy problems, J. Math. Soc. Japan, 20 (1968), 107–121.
8. [I2]_Y. Ihara, Shimura curves over finite fields and their rational points, Contemp. Math., 245 (1999), 15–23.
9. [L-O]
J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, in A. Fröhlich, ed., Algebraic Number Fields: Proceedings of the 1975 Durham Symposium, Academic Press, London, New York, 1977, 409–464.Google Scholar
10. [La]
11. [Le]
H.W. Lenstra, Jr., Miller’s primality test, Inform. Process. Lett., 8 (1979), 86–88.
12. [Li]
J. E. Littlewood, On the class-number of the corpus $$P\left( {\sqrt { - k} } \right)$$, Proc. London Math. Soc. Ser. 2, 27 (1928}), 358–372; also in Collected Papers, Vol. II, Oxford University Press, Oxford, UK, 920–
13. [Sc]
I. Schur, Über die Verteilung derWurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z., 1 (1918), 377–402; also in Gesammelte Abhandlungen, Vol. II, Springer-Verlag, Berlin, 1973, number 32, 213–238.
14. [St]
H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math., 23 (1974), 135–152.
15. [Ts1]_M. A. Tsfasman, Some remarks on the asymptotic number of points, in Coding Theory and Algebraic Geometry (Luminy, 1991), Lecture Notes in Mathematics, Vol. 1518, Springer-Verlag, Berlin, 1992, 178–192.
16. [Ts2]_M. A. Tsfasman, Asymptotic behavior of the Euler-Kronecker constant, in V. Ginzburg, ed., Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday, Progress in Mathematics, Vol. 850, Birkhäuser Boston, Cambridge, MA, 2006, 453–458 (this volume).Google Scholar
17. [T-V]
M. A. Tsfasman and S. G. Vlăduţ, Infinite global fields and the generalized Brauer-Siegel theorem, Moscow Math. J., 2 (2002), 329–402.
18. [TVZ]
M. A. Tsfasman, S. G. Vlăduţ, and Th. Zink, Modular curves, Shimura curves, and Goppa codes better than the Varshamov-Gilbert bound, Math. Nachr., 109 (1982), 21–28.
19. [W1]_A. Weil, Sur les “formules explicites” de la théorie des nombres premiers, Comm. Sém. Math. Lund, Suppl. vol. (1952), 252–265; also in Collected Works, Vol. 2, 48–62.Google Scholar
20. [W2]_A. Weil, Sur les formules explicites de la théorie des nombres, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 3–18; also in Collected Works, Vol. 3, 249–264 (in French). 