International Mathematics Research Notices

International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm156, 41 pages, doi:10.1093/imrn/rnm156 published on February 6, 2008

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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Bounded gaps between products of primes with applications to ideal class groups and elliptic curves

Frank Thorne

Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA

Correspondence: Correspondence to be sent to: thorne{at}math.wisc.edu

In their recent papers, Goldston, Graham, Pintz, and Yldrm [12, 13] use a variant of the Selberg sieve to prove the existence of small gaps between E₂ numbers, that is, square-free numbers with exactly two prime factors. We apply their techniques to prove similar bounds for E_r numbers for any r 2, where these numbers are required to have all of their prime factors in a set of primes . Our result holds for any of positive density that satisfies a Siegel–Walfisz condition regarding distribution in arithmetic progressions. We also prove a stronger result in the case that satisfies a Bombieri–Vinogradov condition. We were motivated to prove these generalizations because of recent results of Ono [22] and Soundararajan [25]. These generalizations yield applications to divisibility of class numbers, nonvanishing of critical values of L-functions, and triviality of ranks of elliptic curves.

References

1. Balog A., Ono K. Elements of class groups and Shafarevich-Tate groups of elliptic curves. Duke Mathematical Journal (2003) 120:35–63.[CrossRef]
2. Bombieri E. Le grand crible dans la théorie analytique des nombres (1987) 2nd ed. Paris: Société Mathématique de France. Astérisque 18.
3. Bombieri E., Friedlander J. B., Iwaniec H. Primes in progression to large moduli. Acta Mathematica (1986) 156:203–51.[CrossRef]
4. Booker A. The nth prime page. http://primes.utm.edu/nthprime/index.php#nth.
5. Chen J.-R. On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Scientia Sinica (1973) 16:157–76.
6. Cojocaru A. C., Murty M. R. An Introduction to Sieve Methods and their Applications. (2005) Cambridge, MA: Cambridge University Press.
7. Davenport H. Multiplicative Number Theory. (2000) New York: Springer. Graduate Texts in Mathematics 74.
8. Deligne P., Serre J.-P. Formes modulaires de poids 1. Annales Scientifiques de l'École Normale Supèrieure (1974) 4(no. 7):507–30.
9. Engelsma T. J. k-tuple permissible patterns. http://www.opertech.com/primes/k-tuples. html.
10. Goldfeld D. Conjectures on Elliptic Curves over Quadratic Fields (1979) New York: Springer. 108–18. Springer Lecture Notes 751.
11. Goldston D. A., Pintz J., Yldrm C. Y. Primes in tuples 1. (2005) preprint arXiv:math/0508185.
12. Goldston D. A., Graham S. W., Pintz J., Yldrm C. Y. Small gaps between primes or almost primes. (2005) preprint arXiv:math.NT/0506067.
13. Goldston D. A., Graham S. W., Pintz J., Yldrm C. Y. Small gaps between products of two primes. (2006) preprint arXiv:math/0609615.
14. Iwaniec H., Kowalski E. Analytic Number Theory. (2005) Providence, RI: American Mathematical Society.
15. Jiménez Urroz J. Bounded gaps between almost prime numbers representable as sum of two squares. Preprint.
16. Knapp A. Elliptic Curves. (1992) Princeton, MA: Princeton University Press.
17. Koblitz N. Introduction to Elliptic Curves and Modular Forms. (1993) New York: Springer. Graduate Texts in Mathematics 97.
18. Kolyvagin V. Finiteness of and for a subclass of Weil curves" [in Russian]. Izvestiya Rossijskoj Akademii Nauk. Seriya Matematicheskaya. Rossijskaya Akademiya Nauk, USSR (1988) 52:522–40.
19. Murty M. R., Murty V. K. A variant of the Bombieri-Vinogradov theorem. Canadian Mathematical Society Conference Proceedings (1987) 7:243–72.
20. Motohashi Y. An induction principle for the generalization of Bombieri's prime number theorem. Proceedings of the Japan Academy (1976) 52:273–5.
21. Ono K. Twists of elliptic curves. Compositio Mathematica (1997) 106:349–60.[CrossRef]
22. Ono K. Nonvanishing of quadratic twists of modular L-functions and applications to elliptic curves. Journal für die reine und angewandte Mathematik (2001) 533:81–97.
23. Ono K., Skinner C. Nonvanishing of quadratic twists of modular L-functions. Inventiones Mathematicae (1998) 134:651–60.[CrossRef]
24. Silverman J. The Arithmetic of Elliptic Curves. (1986) New York: Springer. Graduate Texts in Mathematics 106.
25. Soundararajan K. Divisibility of class numbers of imaginary quadratic fields. Journal of the London Mathematical Society (2000) 61:681–90.[Abstract/Free Full Text]

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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions How to cite this article Google Scholar Articles by Thorne, F. Search for Related Content Social Bookmarking          What's this?