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Mathematics of Computation

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Irregularities in the distribution of primes and twin primes

Author: Richard P. Brent
Journal: Math. Comp. 29 (1975), 43-56
MSC: Primary 10H15; Secondary 10-04
Corrigendum: Math. Comp. 30 (1976), 198.
MathSciNet review: 0369287
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Abstract: The maxima and minima of $ \langle L(x)\rangle - \pi (x),\langle R(x)\rangle - \pi (x)$, and $ \langle {L_2}(x)\rangle - {\pi _2}(x)$ in various intervals up to $ x = 8 \times {10^{10}}$ are tabulated. Here $ \pi (x)$ and $ {\pi _2}(x)$ are respectively the number of primes and twin primes not exceeding $ x,L(x)$ is the logarithmic integral, $ R(x)$ is Riemann's approximation to $ \pi (x)$, and $ {L_2}(x)$ is the Hardy-Littlewood approximation to $ {\pi _2}(x)$. The computation of the sum of inverses of twin primes less than $ 8 \times {10^{10}}$ gives a probable value $ 1.9021604 \pm 5 \times {10^{ - 7}}$ for Brun's constant.

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Keywords: Prime, twin prime, Riemann's approximation, error bounds, Hardy-Littlewood conjecture, Brun's constant, logarithmic integral
Article copyright: © Copyright 1975 American Mathematical Society

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