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Estimating

Explicit bounds for residues of Dedekind zeta functions, values of {

J. Number Theory, 85(2), 263--282.] and in [Louboutin, 2001 †Louboutin, S. 2001

Explicit upper bounds for residues of Dedekind zeta functions and values of {

Canad. J. Math., 53(6), 1194--1222.] with additional properties of log-convexity of some functions related to

Approximate Formulae for

Acta Arith., 100, 245--266.], we get for instance

Approximate Formulae for

Acta Arith., 100, 245--266.] implies that the constant may be replaced by 0, so that

Algorithmic algebraic number theory

Encyclopedia of Mathematics and its Applications, vol. 30. Cambridge: Cambridge University Press.]. In this case, one rather uses explicit bounds for the Piltz-Dirichlet divisor functions

Explicit upper bounds for the average order of {

JIPAM. J. Inequal. Pure Appl. Math., 3(3), Article 38, 15 pp. (electronic).] and [Bordellès, 2006 †Bordellès, O. 2006

An inequality for the class number

JIPAM. J. Inequal. Pure Appl. Math., 7(3), Article 87, 8 pp. (electronic).]) and get

On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes

J. Théor. Nombres Bordeaux, 17(2), 559--573.] how the behavior of certain small primes may subtantially improve on the previous bounds. To make things more significant, define, for a rational prime

On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes

J. Théor. Nombres Bordeaux, 17(2), 559--573.], we have among other things

Explicit bounds for residues of Dedekind zeta functions, values of {

J. Number Theory, 85(2), 263--282.]:

Theorem (2000)

From [Louboutin, 2003 †Louboutin, S. 2003We haveh−K≤2QKwK(dKdK+)1/2(elog(dK/dK+)4πn)n.

Explicit lower bounds for residues at {

Trans. Amer. Math. Soc., 355(8), 3079--3098 (electronic).]:

Theorem (2003)

Again from [Louboutin, 2003 †Louboutin, S. 2003Assume that(ζK/ζK+)(σ)≥0 whenever0<σ<1 . Then we haveh−K≥QKwKπelogdK(dKdK+)1/2(n−1πelogdK)n−1.

Explicit lower bounds for residues at {

Trans. Amer. Math. Soc., 355(8), 3079--3098 (electronic).]:

Theorem (2003)

And a third result from [Louboutin, 2003 †Louboutin, S. 2003Letc=6−42√=0.3431⋯ . Assume thatdK≥2800n and that eitherK does not contain any imaginary quadratic subfield, or that the real zeros in the range1−clogdN≤σ<1 of the Dedekind zeta-functions of the imaginary quadratic subfields ofK are nor zeros ofζK(s) , whereN is the normal closure ofK . Then we haveh−K≥cQKwK4nec/2[N:Q](dKdK+)1/2(nπelogdK)n.

Explicit lower bounds for residues at {

Trans. Amer. Math. Soc., 355(8), 3079--3098 (electronic).]:

Theorem (2003)

Assumen>2 ,dK>2800n and thatK contains an imaginary quadratic subfieldF such thatζF(β)=ζK(β)=0 for someβ satisfying1−2logdK≤β<1 . Then we haveh−K≥6(πe)2(dKdK+)1/2−1/n(nπelogdK)n−1.

Last updated on August 23rd, 2012, by Olivier Bordellès