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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Sums of Three or More Primes

Authors: J. B. Friedlander and D. A. Goldston
Journal: Trans. Amer. Math. Soc. 349 (1997), 287-310
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 $\sum _{p \le x} \log p - x$ in the Prime Number Theorem, such bounds being within a factor of $(\log x)^{2}$ 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 $k$ primes for $k \ge 4$, and, in a mean square sense, for $k \ge 3$. 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 ``Quasi-Riemann Hypothesis''. We incidentally give a slight sharpening to a well-known exponential sum estimate of Vinogradov-Vaughan.

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Additional Information

J. B. Friedlander
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada

D. A. Goldston
Affiliation: Department of Mathematics and Computer Science, San Jose State University, San Jose, California 95192

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