Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

Average prime-pair counting formula

Author(s): Jaap Korevaar; Herman te Riele.
Journal: Math. Comp. 79 (2010), 1209-1229.
MSC (2000): Primary 11P32; Secondary 65-05
Posted: September 25, 2009
MathSciNet review: 2600563
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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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