Download MR3004004 (Review) 11R18 (11R42) Fouvry, ´Etienne (F

Citations
Previous
Up
Next
From References: 0
From Reviews: 0
Article
MR3004004 (Review) 11R18 (11R42)
Fouvry, Étienne (F-PARIS11-M)
Sum of Euler-Kronecker constants over consecutive cyclotomic fields. (English summary)
J. Number Theory 133 (2013), no. 4, 1346–1361.
Let K be a number field and ζK (s) be its Dedekind zeta-function. The Euler-Kronecker constant
γK associated to K is defined by
0
ζK
(s)
1
−
=
− γK + O(s − 1).
ζK (s) s − 1
For q a positive integer, let γq denote the Euler-Kronecker constant of the cyclotomic field Q(ζq ).
The aim of the paper is to prove that uniformly for Q ≥ 3, one has
1 X
γq = log Q + O(log log Q).
Q
Q<q≤2Q
In the present setting, this result is more precise than V. K. Murty’s in [Ann. Sci. Math. Québec 35
(2011), no. 2, 239–247; MR2917834], which in a different setting asserts that
X
1
|γp | log Q,
π ∗ (Q)
Q<p≤2Q
where p ranges over the π ∗ (Q) prime numbers in (Q, 2Q]. Let us take this opportunity to mention
that this constant γK is involved in some explicit bounds on residues at s = 1 of Dedekind zeta
functions and on values at s = 1 of L-functions, as in [S. R. Louboutin, Canad. J. Math. 53 (2001),
no. 6, 1194–1222; MR1863848 (2003d:11167)], since the constant µK is this article is equal to
n
1
µK = log dK − (γ + log π) − (r1 + r2 ) log 2 + 1 + γK ,
2
2
where K is a number field of degree n = r1 + 2r2 .
Reviewed by Stéphane R. Louboutin
References
1. A.I. Badzyan, The Euler-Kronecker constant, Math. Notes 87 (1–2) (2010) 31–42. MR2730381
2. E. Bombieri, J.B. Friedlander, H. Iwaniec, Primes in arithmetic progressions to large moduli,
Acta Math. 156 (1986) 203–251. MR0834613 (88b:11058)
3. Z.I. Borevitch, I.R. Chafarevitch, Théorie des Nombres, Gauthier-Villars, Paris, 1967.
MR0205908 (34 #5734)
4. H. Davenport, Multiplicative Number Theory, second edition, Grad. Texts in Math., vol. 74,
Springer Verlag, 1980. MR0606931 (82m:10001)
5. K. Ford, F. Luca, P. Moree, Values of the Euler ϕ-function not divisible by a given odd prime,
and the distribution of Euler-Kronecker constants for cyclotomic fields, arXiv:1108.3805v2
6.
7.
8.
9.
10.
11.
12.
13.
14.
[math.NT], 2 January 2012.
E. Fouvry, Sur le problème des diviseurs de Titchmarsh, J. Reine Angew. Math. 357 (1985)
51–76. MR0783533 (87b:11090)
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: Progr. Math., vol. 253,
Birkhäuser, 2006, pp. 407–451. MR2263195 (2007h:11127)
Y. Ihara, The Euler-Kronecker invariants in various families of global fields, in: Arithmetic,
Geometry and Coding Theory (AGCT 2005), in: Séminaires et Congrès, vol. 21, Société
Mathématique de France, Paris, 2010, pp. 79–102. MR2856562 (2012k:11182)
Y. Ihara, V. Kumar Murty, M. Shimura, On the logarithmic derivatives of Dirichlet L-functions
at s = 1 Acta Arith. 137 (2009) 253–276. MR2496464 (2009m:11135)
H. Iwaniec, E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., vol. 53,
Amer. Math. Soc., Providence, RI, 2004. MR2061214 (2005h:11005)
V. Kumar Murty, The Euler-Kronecker constant of a cyclotomic field, Ann. Sci. Math. Québec
35 (2) (2012) 239–247. MR2917834
H.L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math., vol. 227,
Springer Verlag, 1971. MR0337847 (49 #2616)
W. Narkiewicz, Elementary and Analytic Theory of Numbers, second edition, PWN-Polish
Scientific Publishers, Warsaw, 1990. MR1055830 (91h:11107)
M.A. Tsfasman, Asymptotic behaviour of the Euler-Kronecker constant, in: Algebraic Geometry and Number Theory, in: Progr. Math., vol. 253, Birkhaüser Boston, Boston, MA, 2006,
pp. 453–458. MR2263196 (2007h:11129)
Note: This list reflects references listed in the original paper as accurately as possible with no
attempt to correct errors.
c Copyright American Mathematical Society 2013