Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T07:00:28.219Z Has data issue: false hasContentIssue false
  • English
  • Français

A pair correlation hypothesis and the exceptional set in Goldbach's problem

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

A. Languasco  and
A. Perelli
A. Languasco
Affiliation:
Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy. e-mail: [email protected]
A. Perelli
Affiliation:
Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy, e-mail: [email protected]
Get access

Extract

In 1973, Montgomery [12] introduced, in order to study the vertical distribution of the zeros of the Riemann zeta function, the pair correlation function

where w(u) = 4/(4 + u2) and γjj = 1, 2, run over the imaginary part of the nontrivial zeros of ζ(s). It is easy to see that, for T → ∞,

uniformly in X, and Montgomery [12], see also Goldston-Montgomery [7], proved that under the Riemann Hypothesis (RH)

uniformly for X ≤ T ≤ XA, for any fixed A > 1. He also conjectured, under RH, that (1) holds uniformly for Xε ≤ T ≤ X, for every fixed ε > 0. We denote by MC the above conjecture.

MSC classification

Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes
Type
Research Article
Information
Mathematika , Volume 43 , Issue 2 , December 1996 , pp. 349 - 361
Copyright
Copyright © University College London 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Baker, R. C. and Harman, G.. Diophantine approximation by prime numbers. J. London Math. Soc. (2), 25 (1982), 201–215.CrossRefGoogle Scholar
2.Davenport, H.. Multiplicative Number Theory, 2nd ed. (Springer Verlag, 1980).CrossRefGoogle Scholar
3.Gallagher, P. X. and Mueller, J. H.. Primes and zeros in short intervals. J. reine angew. Math., 303/304 (1978), 205–220.Google Scholar
4.Goldston, D. A.. Linnik's theorem on Goldbach numbers in short intervals. Glasgow Math. J., 32 (1990), 285–297.CrossRefGoogle Scholar
5.Goldston, D. A.. On Hardy and Littlewood's contribution to the Goldbach conjecture. Proc. Amalfi Conf. Analytic Number Theory, Bombieri, E.et al., editors (Universita di Salerno, 1992), 115–155.Google Scholar
6.Goldston, D. A.. An exponential sum over primes. Number Theory with an Emphasis on the Markoff Spectrum, Pollington, A. D. and Moran, W., editors (Marcel Dekker Inc., 1993), 101–106.Google Scholar
7.Goldston, D. A. and Montgomery, H. L.. Pair correlation of zeros and primes in short intervals. Analytic Number Theory and Dioph. Probl., Adolphson, A. C.et al., editors (Birkhäuser Verlag, 1987), 183–203.Google Scholar
8.Hardy, G. H. and Littlewood, J. E.. Some problems of “Partitio Numerorum”, V: a further contribution to the study of Goldbach's problem. Proc. London Math. Soc. (2), 22 (1924), 46–56.CrossRefGoogle Scholar
9.Heath-Brown, D. R.. Gaps between primes, and the pair correlation of zeros of the zetafunction. Acta Arith., 41 (1982), 85–99.CrossRefGoogle Scholar
10.Kaczorowski, J.Perelli, A. and Pintz, J.. A note on the exceptional set for Goldbach's problem in short intervals. Mh. Math., 116 (1993), 275–282; corrigendum, 119 (1995), 215–216.CrossRefGoogle Scholar
11.Languasco, A. and Perelli, A.. On Linnik's theorem on Goldbach numbers in short intervals and related problems. Ann. Inst. Fourier, 44 (1994), 307–322.CrossRefGoogle Scholar
12.Montgomery, H. L.. The pair correlation of zeros of the zeta function. Proc. A.M.S. Symp. Pure Math., 24 (1973), 181–193.CrossRefGoogle Scholar
13.Montgomery, H. L.. Topics in Multiplicative Number Theory. Springer Lecture Notes, 227 (Springer, 1971).CrossRefGoogle Scholar
14.Perelli, A. and Pintz, J.. On the exceptional set for Goldbach's problem in short intervals. J. London Math. Soc. (2), 47 (1993), 41–49.CrossRefGoogle Scholar
15. B. Saffari and R. C. Vaughan. On the fractional parts of x/n and related sequences II. Ann. Inst. Fourier, 27 (1977), 1–30.CrossRefGoogle Scholar