Difference between revisions of "Brun's constant"
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==Proof of convergence== | ==Proof of convergence== | ||
− | Everywhere below, <math>p</math> will stand for an odd [[prime number]]. Let | + | Everywhere below, <math>p</math> will stand for an [[odd integer | odd]] [[prime number]]. Let |
<math>\pi_2(x)=\#\{p\le x:p+2\mathrm{\ is\ also\ prime\,}\}</math>. We shall prove that <math>\pi_2(x)\le C\frac{x}{(\ln x)^2}(\ln\ln x)^2</math> for large <math>x</math> with some absolute constant <math>C<+\infty</math>. | <math>\pi_2(x)=\#\{p\le x:p+2\mathrm{\ is\ also\ prime\,}\}</math>. We shall prove that <math>\pi_2(x)\le C\frac{x}{(\ln x)^2}(\ln\ln x)^2</math> for large <math>x</math> with some absolute constant <math>C<+\infty</math>. | ||
− | The technique used in the proof is a version of the [[ | + | The technique used in the proof is a version of the [[Principle of Inclusion-Exclusion]] and is known nowadays as '''Brun's simple pure sieve'''. |
− | + | ===Lemma=== | |
Let <math>a_1,\dots,a_n\in[0,1]</math>. | Let <math>a_1,\dots,a_n\in[0,1]</math>. | ||
Let <math>\sigma_k=\sum_{1\le i_1<\dots<i_k\le n}a_{i_1}\dots a_{i_k}</math> be the <math>k</math>-th [[symmetric sum]] of the numbers <math>a_1,\dots, a_n</math>. Then | Let <math>\sigma_k=\sum_{1\le i_1<\dots<i_k\le n}a_{i_1}\dots a_{i_k}</math> be the <math>k</math>-th [[symmetric sum]] of the numbers <math>a_1,\dots, a_n</math>. Then | ||
− | <math>1-\sigma_1+\sigma_2-\dots-\sigma_k\le \prod_{j=1}^n(1-a_j)\le 1-\sigma_1+\sigma_2-\dots+\sigma_\ell</math> for every odd <math>k</math> and even <math>\ell</math>. | + | <math>1-\sigma_1+\sigma_2-\dots-\sigma_k\le \prod_{j=1}^n(1-a_j)\le 1-\sigma_1+\sigma_2-\dots+\sigma_\ell</math> for every odd <math>k</math> and [[even integer | even]] <math>\ell</math>. |
====Proof of Lemma==== | ====Proof of Lemma==== | ||
Induction on <math>n</math>. | Induction on <math>n</math>. |
Latest revision as of 17:48, 12 October 2006
Definition
Brun's constant is the (possibly infinite) sum of reciprocals of the twin primes . It turns out that this sum is actually convergent. Brun's constant is equal to approximately .
Proof of convergence
Everywhere below, will stand for an odd prime number. Let . We shall prove that for large with some absolute constant . The technique used in the proof is a version of the Principle of Inclusion-Exclusion and is known nowadays as Brun's simple pure sieve.
Lemma
Let . Let be the -th symmetric sum of the numbers . Then for every odd and even .
Proof of Lemma
Induction on .
Now, take a very big and fix some to be chosen later. For each odd prime , let
.
Clearly, if , and for some , then either or is not prime. Thus, the number of primes such that is also prime does not exceed .
Let now be an even number. By the inclusion-exclusion principle,
Let us now estimate . Note that the condition depends only on the remainder of modulo and that, by the Chinese Remainder Theorem, there are exactly remainders that satisfy this condition (for each , we must have or and the remainders for different can be chosen independently). Therefore
where . It follows that
where is the -th symmetric sum of the set . Indeed, we have not more than terms in the inclusion-exclusion formula above and each term is estimated with an error not greater than .
Now notice that by the lemma. The product does not exceed (see the prime number article), so it remains to estimate . But we have
.
This estimate yields the final inequality
.
It remains to minimize the right hand side over all possible choices of and . We shall choose and . With this choice, every term on the right does not exceed and we are done.