Sums of Three or More Primes
Authors:
J. B. Friedlander and D. A. Goldston
Journal:
Trans. Amer. Math. Soc. 349 (1997), 287310
MSC (1991):
Primary 11P32
MathSciNet review:
1357393
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Abstract: It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error in the Prime Number Theorem, such bounds being within a factor of of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided ``Riemann Hypothesis'' is replaced by ``Generalized Riemann Hypothesis'', results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of primes for , and, in a mean square sense, for . We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a ``QuasiRiemann Hypothesis''. We incidentally give a slight sharpening to a wellknown exponential sum estimate of VinogradovVaughan.
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 R. C. Baker and G. Harman, Diophantine approximation by prime numbers, J. London Math. Soc. 25 (1982), 201215. MR 84g:10067
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 H. Davenport, Multiplicative Number Theory, 2nd ed., revised by H. L. Montgomery, SpringerVerlag (Berlin), 1980. MR 82m:10001
 [Go]
 D. A. Goldston, On Hardy and Littlewood's contribution to the Goldbach conjecture, Proc. Amalfi Conf. on Analytic Number Theory (Sept. 1989), Univ. Salerno, 1992. MR 94m:11122.
 [HL]
 G. H. Hardy and J. E. Littlewood, Some problems of `Partitio Numerorum': III On the expression of a number as a sum of primes, Acta Math. 44 (1923), 170.
 [HB]
 D. R. HeathBrown, Zero density estimates for the Riemann zetafunction and Dirichlet Lfunctions, J. London Math. Soc. 19 (1979), 221232. MR 80i:10055
 [Hu]
 M. N. Huxley, Large values of Dirichlet polynomials, III, Acta Arith. 26 (1975), 435444. MR 52:10620
 [HJ]
 M. N. Huxley and M. Jutila, Large values of Dirichlet Polynomials, IV, Acta Arith. 32 (1977), 297312. MR 58:5550
 [In]
 A. E. Ingham, Some asymptotic formulae in the theory of numbers, J. London Math. Soc. 2 (1927), 202208.
 [Li]
 Yu. V. Linnik, A new proof of the GoldbachVinogradov theorem, Mat. Sb. 61 (1946), 38.
 [Mo]
 H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, vol. 227, SpringerVerlag (Berlin), 1971. MR 49:2616
 [MV]
 H. L. Montgomery and R. C. Vaughan, Error terms in additive prime number theory, Quart. J. Math. Oxford (2) 24 (1973), 207216. MR 49:2624
 [Oz]
 A. E. Özlük , Pair correlation of zeros of Dirichlet functions Thesis, University of Michigan, 1982.
 [Ti]
 E. C. Titchmarsh, The Theory of the Riemann ZetaFunction, 2nd ed. revised by D. R. HeathBrown, Clarendon (Oxford), 1986. MR 88c:11049
 [Va1]
 R. C. Vaughan, Sommes trigonométriques sur les nombres premiers, C. R. Acad. Sci. Paris Ser. A 258 (1977), 981983. MR 58:16555
 [Va2]
 R. C. Vaughan, The HardyLittlewood Method, Cambridge Tracts in Math., vol. 80, (Cambridge), 1981. MR 84b:10002
 [Vi]
 I. M. Vinogradov, Representation of an odd number as a sum of three primes, Dokl. Akad. Nauk SSSR 15 (1937), 67.
 [Yi]
 C. Y. Yildirim, The pair correlation of zeros of Dirichlet Lfunctions and primes in arithmetic progressions, Manuscripta Math., vol. 72, 1991, pp. 325334. MR 93b:11111
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Additional Information
J. B. Friedlander
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada
Email:
frdlndr@math.toronto.edu
D. A. Goldston
Affiliation:
Department of Mathematics and Computer Science, San Jose State University, San Jose, California 95192
Email:
goldston@sjsumcs.sjsu.edu
DOI:
http://dx.doi.org/10.1090/S0002994797016528
PII:
S 00029947(97)016528
Received by editor(s):
April 29, 1994
Received by editor(s) in revised form:
September 22, 1995
Additional Notes:
Research of the first author supported in part by NSERC Grant A5123 and NSF Grant DMS8505550.
Research of the second author supported in part by NSF Grant DMS9205533.
Article copyright:
© Copyright 1997
American Mathematical Society
