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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The number of Goldbach representations of an integer


Authors: Alessandro Languasco and Alessandro Zaccagnini
Journal: Proc. Amer. Math. Soc. 140 (2012), 795-804
MSC (2010): Primary 11P32; Secondary 11P55
Published electronically: July 20, 2011
MathSciNet review: 2869064
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Abstract: Let $ \Lambda$ be the von Mangoldt function and $ R(n) = \sum_{h+k=n} \Lambda(h)\Lambda(k) $ be the counting function for the Goldbach numbers. Let $ N \geq 2$ and assume that the Riemann Hypothesis holds. We prove that

$\displaystyle \sum_{n=1}^{N} R(n) = \frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + \mathcal{O}(N \log^{3}N), $

where $ \rho=1/2+i\gamma$ runs over the non-trivial zeros of the Riemann zeta-function $ \zeta(s)$. This improves a recent result by Bhowmik and Schlage-Puchta.


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Additional Information

Alessandro Languasco
Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email: languasco@math.unipd.it

Alessandro Zaccagnini
Affiliation: Dipartimento di Matematica, Università di Parma, Parco Area delle Scienze 53/a, Campus Universitario, 43124 Parma, Italy
Email: alessandro.zaccagnini@unipr.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10957-2
PII: S 0002-9939(2011)10957-2
Keywords: Goldbach-type theorems, Hardy-Littlewood method
Received by editor(s): November 11, 2010
Received by editor(s) in revised form: December 16, 2010
Published electronically: July 20, 2011
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.