Summary
This appendix to the beautiful paper [1] of Ihara puts it in the context of infinite global fields of our papers [2] and [3]. We study the behaviour of Euler-Kronecker constant γK when the discriminant (genus in the function field case) tends to infinity. Results of [2] easily give us good lower bounds on the ratio \( {{\gamma _K } \mathord{\left/ {\vphantom {{\gamma _K } {\log \sqrt {\left| {d_K } \right|} }}} \right. \kern-\nulldelimiterspace} {\log \sqrt {\left| {d_K } \right|} }} \). In particular, for number fields, under the generalized Riemann hypothesis we prove
Then we produce examples of class-field towers, showing that
To Volodya Drinfeld with friendship and admiration.
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References
Y. Ihara, On the Euler-Kronecker constants of global fields and primes with small norms, 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, 407–452 (this volume).
M. A. Tsfasman and S. G. Vlăduţ, Infinite global fields and the generalized Brauer-Siegel theorem, Moscow Math. J., 2-2 (2002), 329–402.
M. A. Tsfasman and S. G. Vlăduţ, Asymptotic properties of zeta-functions, J. Math. Sci. (N.Y.), 84-5 (1997), 1445–1467.
F. Hajir and C. Maire, Tamely ramified towers and discriminant bounds for number fields II, J. Symbol. Comput., 33-4 (2002), 415–423.
A. Zykin, Private communication.
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Tsfasman, M.A. (2006). Asymptotic behaviour of the Euler-Kronecker constant. In: Ginzburg, V. (eds) Algebraic Geometry and Number Theory. Progress in Mathematics, vol 253. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4532-8_6
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DOI: https://doi.org/10.1007/978-0-8176-4532-8_6
Publisher Name: Birkhäuser Boston
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Online ISBN: 978-0-8176-4532-8
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