Mathematics of Computation

Author(s): Jaap Korevaar; Herman te Riele.
Journal: Math. Comp.
MSC (2000): Primary 11P32; Secondary 65-05
Posted: September 25, 2009
Retrieve article in: PDF

Abstract: Taking

, let $\pi_{2r}(x)$ denote the number of prime pairs

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

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

DOI: 10.1090/S0025-5718-09-02312-6
PII: S 0025-5718(09)02312-6
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
Posted: September 25, 2009
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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