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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