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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Irregularities in the distribution of primes and twin primes
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by Richard P. Brent PDF
Math. Comp. 29 (1975), 43-56 Request permission

Corrigendum: Math. Comp. 30 (1976), 198.


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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Math. Comp. 29 (1975), 43-56
  • MSC: Primary 10H15; Secondary 10-04
  • DOI:
  • MathSciNet review: 0369287