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2020 | OriginalPaper | Chapter

Improved Bounds on Brun’s Constant

Authors : Dave Platt, Tim Trudgian

Published in: From Analysis to Visualization

Publisher: Springer International Publishing

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Abstract

Brun’s constant is \(B=\sum _{p \in P_{2}} p^{-1} + (p+2)^{-1}\), where the summation is over all twin primes. We improve the unconditional bounds on Brun’s constant to \(1.840503< B < 2.288490\), which are about 13% tighter.

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previous chapter Metrical Theory for Small Linear Forms and Applications to Interference Alignment
next chapter Extending the PSLQ Algorithm to Algebraic Integer Relations
Footnotes
1
We cannot resist referencing an anecdote from Jon Borwein (and his co-authors). Nicely’s calculations on Brun’s constant are mentioned in [2, p. 40]. Nicely discovered a bug in an Intel Pentium chip, which, according to [2] ‘cost Intel about a billion dollars’ although the actual amount written off was a mere US$475 million. We believe Jon would have seen this as an excellent application of pure mathematics in the modern world.
 
2
We were reminded by the referee that Jon Borwein had worked on Hilbert’s inequality, although we do not believe his results to be applicable here.
 
3
We remark that Selberg conjectured that (8) holds with 1 in place of 3/2. It seems difficult to improve further on Preissmann’s work.
 
Literature
1.
Berkane, D., Bordellès, O., Ramaré, O.: Explicit upper bounds for the remainder term in the divisor problem. Math. Comput. 81(278), 1025–1051 (2012)MathSciNetCrossRef
2.
Borwein, J., et al.: Organic Mathematics: Proceedings of the Organic Mathematics Workshop, December 12–14, 1995, Simon Fraser University, Burnaby. British Columbia. AMS, Providence (1997)MATH
3.
Brent, R.P.: Tables concerning irregularities in the distribution of primes and twin primes up to \(10^{11}\). Math. Comput. 30(134), 379 (1976)
4.
Brun, V.: La série \(1/5 + 1/7 + 1/11 + 1/13 + 1/17\) [etc.] où les dénominateurs sont nombres premiers jumeaux est convergente ou finie. Bull. Sci. Math. 43, 124–128 (1919)
5.
Crandall., Pomerance.: Prime Numbers: A Computational Perspective, 2nd edn. Springer, New York (2005)
6.
Johansson, F.: Arb: efficient arbitrary-precision midpoint-radius interval arithmetic. IEEE Trans. Comput. 66, 1281–1292 (2017)MathSciNetCrossRef
7.
Klyve, D.: Explicit bounds on twin primes and Brun’s constant. Ph.D thesis, Dartmouth College (2007)
8.
Molin, P.: Intégration numérique et calculs de fonctions L. Ph.D thesis, Institut de Mathématiques de Bordeaux (2010)
9.
10.
11.
12.
13.
14.
15.
Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Illinos J. Math. 6, 64–94 (1962)MathSciNetCrossRef
16.
17.
18.
Wrench Jr., J.W.: Evaluation of Artin’s constant and the twin-prime constant. Math. Comput. 15(76), 396–398 (1961)MathSciNetMATH
Metadata
Title
Improved Bounds on Brun’s Constant
Authors
Dave Platt
Tim Trudgian
Copyright Year
2020
Book
From Analysis to Visualization

Print ISBN: 978-3-030-36567-7

Electronic ISBN: 978-3-030-36568-4

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-36568-4_25

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