A Bombieri-Vinogradov theorem for all number fields
HTML articles powered by AMS MathViewer
- by M. Ram Murty and Kathleen L. Petersen PDF
- Trans. Amer. Math. Soc. 365 (2013), 4987-5032 Request permission
Abstract:
The classical theorem of Bombieri and Vinogradov is generalized to a non-abelian, non-Galois setting. This leads to a prime number theorem of “mixed-type” for arithmetic progressions “twisted” by splitting conditions in number fields. One can view this as an extension of earlier work of M. R. Murty and V. K. Murty on a variant of the Bombieri-Vinogradov theorem. We develop this theory with a view to applications in the study of the Euclidean algorithm in number fields and arithmetic orbifolds.References
- Hideo Aramata, Über die Teilbarkeit der Dedekindschen Zetafunktionen, Proc. Imp. Acad. Tokyo 9 (1933), no. 2, 31–34 (German). MR 1568340
- Paul T. Bateman and Harold G. Diamond, Analytic number theory, Monographs in Number Theory, vol. 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004. An introductory course. MR 2111739, DOI 10.1142/5605
- E. Bombieri, On the large sieve, Mathematika 12 (1965), 201–225. MR 197425, DOI 10.1112/S0025579300005313
- Enrico Bombieri, Le grand crible dans la théorie analytique des nombres, Astérisque 18 (1987), 103 (French, with English summary). MR 891718
- Richard Brauer, On the zeta-functions of algebraic number fields, Amer. J. Math. 69 (1947), 243–250. MR 20597, DOI 10.2307/2371849
- J.W. Cogdell, On Artin $L$-functions, www.math.ohio-state.edu/$\sim$cogdell.
- Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931, DOI 10.1007/978-1-4757-5927-3
- Max Deuring, Über den Tschebotareffschen Dichtigkeitssatz, Math. Ann. 110 (1935), no. 1, 414–415 (German). MR 1512947, DOI 10.1007/BF01448036
- Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636
- P. X. Gallagher, Bombieri’s mean value theorem, Mathematika 15 (1968), 1–6. MR 237442, DOI 10.1112/S002557930000231X
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
- Malcolm Harper and M. Ram Murty, Euclidean rings of algebraic integers, Canad. J. Math. 56 (2004), no. 1, 71–76. MR 2031123, DOI 10.4153/CJM-2004-004-5
- J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 409–464. MR 0447191
- C. R. MacCluer, A reduction of the Čebotarev density theorem to the cyclic case, Acta Arith. 15 (1968), 45–47. MR 233796, DOI 10.4064/aa-15-1-45-47
- J. Martinet, Character theory and Artin $L$-functions, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 1–87. MR 0447187
- M. Ram Murty, Problems in analytic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 206, Springer, New York, 2008. Readings in Mathematics. MR 2376618
- M. Ram Murty and V. Kumar Murty, A variant of the Bombieri-Vinogradov theorem, Number theory (Montreal, Que., 1985) CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 243–272. MR 894326
- M. Ram Murty and Kathleen L. Petersen, The generalized Artin conjecture and arithmetic orbifolds, Groups and symmetries, CRM Proc. Lecture Notes, vol. 47, Amer. Math. Soc., Providence, RI, 2009, pp. 259–265. MR 2500566, DOI 10.1090/crmp/047/17
- Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, Monografie Matematyczne, Tom 57, PWN—Polish Scientific Publishers, Warsaw, 1974. MR 0347767
- Hans Rademacher, On the Phragmén-Lindelöf theorem and some applications, Math. Z 72 (1959/1960), 192–204. MR 0117200, DOI 10.1007/BF01162949
- K. Ramachandra, A simple proof of the mean fourth power estimate for $\zeta (1/2+it)$ and $L(1/2+it,\,X)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1 (1974), 81–97 (1975). MR 376562
- H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 342472, DOI 10.1007/BF01405166
- N. Tschebotareff, Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören, Math. Ann. 95 (1926), no. 1, 191–228 (German). MR 1512273, DOI 10.1007/BF01206606
- A. I. Vinogradov, The density hypothesis for Dirichet $L$-series, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 903–934 (Russian). MR 0197414
Additional Information
- M. Ram Murty
- Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6
- MR Author ID: 128555
- Email: murty@mast.queensu.ca
- Kathleen L. Petersen
- Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306
- MR Author ID: 811372
- Email: petersen@math.fsu.edu
- Received by editor(s): July 26, 2010
- Received by editor(s) in revised form: December 17, 2011, and February 7, 2012
- Published electronically: December 13, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 4987-5032
- MSC (2010): Primary 11M26; Secondary 11M06, 11N36
- DOI: https://doi.org/10.1090/S0002-9947-2012-05805-3
- MathSciNet review: 3066777