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Irregularities in the Distribution of Primes and Twin Primes

Richard P. Brent
Mathematics of Computation
Vol. 29, No. 129 (Jan., 1975), pp. 43-56
DOI: 10.2307/2005460
Stable URL: http://www.jstor.org/stable/2005460
Page Count: 14
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Irregularities in the Distribution of Primes and Twin Primes
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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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