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Average prime-pair counting formula

Authors: Jaap Korevaar and Herman te Riele
Journal: Math. Comp. 79 (2010), 1209-1229
MSC (2000): Primary 11P32; Secondary 65-05
Published electronically: September 25, 2009
MathSciNet review: 2600563
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Abstract: Taking $ r>0$, let $ \pi_{2r}(x)$ denote the number of prime pairs $ (p, p+2r)$ with $ p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $ \pi_{2r}(x)\sim 2C_{2r} {li}_2(x)$ with an explicit constant $ C_{2r}>0$. There seems to be no good conjecture for the remainders $ \omega_{2r}(x)=\pi_{2r}(x)- 2C_{2r} {li}_2(x)$ that corresponds to Riemann's formula for $ \pi(x)-{li}(x)$. However, there is a heuristic approximate formula for averages of the remainders $ \omega_{2r}(x)$ which is supported by numerical results.

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Additional Information

Jaap Korevaar
Affiliation: KdV Institute of Mathematics, University of Amsterdam, Science Park 904, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands

Herman te Riele
Affiliation: CWI: Centrum Wiskunde en Informatica, Science Park 123, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

Keywords: Hardy--Littlewood conjecture, prime-pair functions, representation by repeated complex integral, zeta's complex zeros
Received by editor(s): February 25, 2009
Received by editor(s) in revised form: June 5, 2009
Published electronically: September 25, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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