# Best provable and unconditional lower and upper bounds for Brun's constant

Question: What are the currently known best provable and unconditional lower and upper bounds for Brun's constant $$B$$, corresponding to the sum of the reciprocals of the twin primes?

Remark. According to Dominic Klyve's thesis "Explicit bounds on twin primes and Brun's Constant" (2007, p.23) the best provable unconditional bounds known at the time were given by

$$1,830484424658 < B < 2,347$$

the lower bound being obtained by computation of the sum up to $$10^{16}$$. Hence, apparently,

not even the first digit was known at the time i.e. the question $$B<2$$ was open.

Was there any significant progress since then?

Note that the finner estimates that one usually sees, as that of Nicely

$$B=1.90216 05823 +/- 0.00000 00008$$

are in fact conjectured but not rigorously proved (95% "confidence interval" according to its author).

## 2 Answers

In March 7, 2018 (several months after you asked this question!), Dave Platt and Tim Trudgian published a paper called Improved bounds on Brun’s constant, where they show the following improved bounds:

$$1.840503 < B < 2.288513$$

The best known bounds seem to be due to Nicely (A new error analysis for Brun's constant, Virginia J. Sci. 52 (2001), no. 1, 45–55), who showed that Brun's constant is $$1.9021605823 \pm 0.0000000008$$ To do so he computed all prime twins up to $3\cdot 10^{15}$. Since the known upper bound for prime twins differs from the expected number by a factor of 4 (better bounds are known, but making them explicit would probably be extremely difficult), it is likely that the only possible way to improve on this would be to extend the range of computation.

Edit. The article by Nicely can be obtained here: http://www.trnicely.net/twins/twins4.html . As nautilus already remarked, these computations do not give a proven bound, but a heuristic estimate, and the computations by Klyve are superior anyway. I took my information from MathSciNet, which is not always correct.

• please see the edited question. – Nautilus Jul 27 '17 at 15:08