Brun’s pure sieve

In the first quarter of the twentieth century Viggo Brun developed an extension of the sieve of Eratosthenes that yielded good estimates on the number of elements of a set $𝒜$ that are not divisible (http://planetmath.org/Divisibility) by any of the primes $p_1,\mathrm{\dots },p_k$ provided only that $𝒜$ is “sufficiently regularly distributed” modulo these primes. That allowed him to prove that the sum of reciprocals of twin primes converges (the limit is now known as Brun’s constant), and that every sufficiently large even number is a sum of two numbers each having at most $9$ prime factors. In what follows we describe the simplest form of Brun’s sieve, known as Brun’s pure sieve.

The sieve of Eratosthenes (http://planetmath.org/SieveOfEratosthenes2) is based on the principle of inclusion-exclusion in the form

${\displaystyle \sum _{n\in 𝒜}}{\displaystyle \sum _{\begin{array}{c}{p}_1\nmid n\\ \mathrm{\dots }\\ p_k\nmid n\end{array}}}1$ $={\displaystyle \sum _{n\in 𝒜}}{\displaystyle \sum _{(n,p_1\mathrm{\cdots }{p}_k)=1}}1={\displaystyle \sum _{n\in 𝒜}}{\displaystyle \sum _{d\mid (n,p_1\mathrm{\cdots }{p}_k)}}\mu (d)$

$={\displaystyle \sum _{d\mid p_1\mathrm{\cdots }{p}_k}}\mu (d){\displaystyle \sum _{\begin{array}{c}n\mid 𝒜\\ d\mid n\end{array}}}1={\displaystyle \sum _{d\mid p_1\mathrm{\cdots }{p}_k}}\mu (d)A_d$

$=A_1-{\displaystyle \sum _{p\mid p_1\mathrm{\cdots }{p}_k}}{A}_p+{\displaystyle \sum _{pq\mid p_1\mathrm{\cdots }{p}_k}}{A}_{pq}+\mathrm{\cdots }.$ (1)

Brun’s sieve is based on the observation that the partial sums of (1) alternatively overcount and undercount the number of elements of $𝒜$ counted by the left side, that is,

$$\sum _{\begin{array}{c}d\mid p_1\mathrm{\cdots }{p}_k\\ \nu (d)\le 2h-1\end{array}}\mu (d)A_d\le \sum _{n\in 𝒜}\sum _{\begin{array}{c}{p}_1\nmid n\\ \mathrm{\dots }\\ p_k\nmid n\end{array}}1\le \sum _{\begin{array}{c}d\mid p_1\mathrm{\cdots }{p}_k\\ \nu (d)\le 2h\end{array}}\mu (d)A_d$$ (2)

where $\nu (d)$ denotes the number of prime (http://planetmath.org/Prime) factors of $d$. In applications $A_d$ can usually be approximated by a multiple of a multiplicative function, that is,

$$A_d=Xw(d)/d+R_d$$

where $w(d)$ is some multiplicative function of $d$, and $R_d$ is small compared to $Xw(d)/d$. Then the estimate in (2) takes the form

${\displaystyle \sum _{\begin{array}{c}d\mid p_1\mathrm{\cdots }{p}_k\\ \nu (d)\le t\end{array}}}\mu (d)A_d$ $={\displaystyle \sum _{\begin{array}{c}d\mid p_1\mathrm{\cdots }{p}_k\\ \nu (d)\le t\end{array}}}\mu (d)(Xw(d)/d+R_d)$

$=X{\displaystyle \sum _{\begin{array}{c}d\mid p_1\mathrm{\cdots }{p}_k\\ \nu (d)\le t\end{array}}}\mu (d){\displaystyle \frac{w(d)}{d}}+O\left({\displaystyle \sum _{\begin{array}{c}d\mid p_1\mathrm{\cdots }{p}_k\\ \nu (d)\le t\end{array}}}{R}_d\right)$ (3)

If the truncated sum is a good approximation to the full sum, then this formula is an improvement over the sieve of Eratosthenes (http://planetmath.org/SieveOfEratosthenes2) because the sum over error terms $R_d$ is shorter.

Since $u^{\nu (d)-t}$ is greater than $1$ for every $u>1$ and every $d$ such that $\nu (d)>t$, we have that

${\displaystyle \sum _{\begin{array}{c}d\mid p_1\mathrm{\cdots }{p}_k\\ \nu (d)\le t\end{array}}}\mu (d){\displaystyle \frac{w(d)}{d}}$ $={\displaystyle \sum _{d\mid p_1\mathrm{\cdots }{p}_k}}\mu (d){\displaystyle \frac{w(d)}{d}}+O\left({\displaystyle \sum _{\begin{array}{c}d\mid p_1\mathrm{\cdots }{p}_k\\ \nu (d)>t\end{array}}}{\displaystyle \frac{w(d)}{d}}\right)$

$={\displaystyle \sum _{d\mid p_1\mathrm{\cdots }{p}_k}}\mu (d){\displaystyle \frac{w(d)}{d}}+O\left({\displaystyle \sum _{d\mid p_1\mathrm{\cdots }{p}_k}}{\displaystyle \frac{w(d)}{d}}{u}^{\nu (d)-t}\right)$

$={\displaystyle \sum _{d\mid p_1\mathrm{\cdots }{p}_k}}\mu (d){\displaystyle \frac{w(d)}{d}}+O\left(u^{-t}{\displaystyle \sum _{d\mid p_1\mathrm{\cdots }{p}_k}}{\displaystyle \frac{w(d)}{d}}{u}^{\nu (d)}\right)$

and since the sum of a multiplicative function can be written as an Euler product, we get

$={\displaystyle \prod _{p\in 𝒫}}\left(1-{\displaystyle \frac{w(p)}{p}}\right)+O\left(u^{-t}{\displaystyle \prod _{p\in 𝒫}}\left(1+u{\displaystyle \frac{w(p)}{p}}\right)\right)$

$={\displaystyle \prod _{p\in 𝒫}}\left(1-{\displaystyle \frac{w(p)}{p}}\right)+O\left(u^{-t}{\displaystyle \prod _{p\in 𝒫}}{\left(1+{\displaystyle \frac{w(p)}{p}}\right)}^u\right)$

to minimize the error term we choose $u=t/{\sum }_{p\in 𝒫}\log (1+\frac{w(p)}{p})$, and get

$={\displaystyle \prod _{p\in 𝒫}}\left(1-{\displaystyle \frac{w(p)}{p}}\right)+O{\left({\displaystyle \frac{1}{t}}{\displaystyle \sum _{p\in 𝒫}}{\displaystyle \frac{w(p)}{p}}\right)}^t$

Combining this with (3) and (2) we obtain

$$\sum _{n\in 𝒜}\sum _{\begin{array}{c}{p}_1\nmid n\\ \mathrm{\dots }\\ p_k\nmid n\end{array}}1=X\prod _{p\in 𝒫}\left(1-\frac{w(p)}{p}\right)+X\cdot O{\left(\frac{1}{t}\sum _{p\in 𝒫}\frac{w(p)}{p}\right)}^t+O\left(\sum _{\begin{array}{c}d\mid p_1\mathrm{\cdots }{p}_k\\ \nu (d)\le t\end{array}}{R}_d\right)$$ (4)

provided that $u=t/{\sum }_{p\in 𝒫}\log (1+\frac{w(p)}{p})\ge 1$.

Example. (Upper bound on the number of primes.) If we set $𝒫$ to be all the primes less $y$, and $𝒜=\{1,2,\mathrm{\dots },x\}$, then the right hand side of (4) provides an upper bound for the number of primes between $y$ and $x$.

We have that $w(d)=1$ and $R_d\le 1$. The second error term in (4) has at most $y^t$ summands, and the first error term is bounded by ${(\frac{c}{t}\log \log y)}^t$ by Merten’s theorem. Thus, the estimate (4) becomes

$$\pi (x)-\pi (y)\le x\frac{e^{-\gamma }+o(1)}{\log y}+x\cdot O{\left(\frac{1}{t}\log \log y\right)}^t+O(y^t)$$

In order to squeeze out the best upper bound on the number of primes not exceeding $x$ we have to minimize the right side of the inequality above. The nearly optimal choice is $t=c\log \log x$, and $y=x^{1/2c\log \log x}$ for a sufficiently large constant $c$. With this choice we obtain

$$\pi (x)\le O\left(\frac{x\log \log x}{\log x}\right).$$

This inequality is stronger than the one obtained by an application of the sieve of Eratosthenes (http://planetmath.org/SieveOfEratosthenes2).

Example. (Upper bound on the number of twin primes.) If and $p\mid n(n+2)$, then the integers $n$ and $n+2$ cannot both be primes. Let $𝒫$ be as above, and set $𝒜=\{n(n+1):n\in [y..x]\}$. Then $w(2)=1$, and $w(p)=2$ if $p>2$. The remainder $R_d$ does not exceed $2^{\nu (d)}$. Like above we obtain

$${\pi }_2(x)-{\pi }_2(y)\le x\frac{c}{{(\log y)}^2}+x\cdot O{\left(\frac{1}{t}\log \log y\right)}^t+O({(2y)}^t)$$

where ${\pi }_2(x)$ denotes the number of twin primes not exceeding $x$. Upon setting $t=c\log \log x$, and $y=x^{1/2c\log \log x}$ we obtain

$${\pi }_2(x)\le O\left(\frac{x{(\log \log x)}^2}{{(\log x)}^2}\right).$$

This is the original result for which Viggo Brun developed the sieve now bearing his name. This result can be put in following striking form:

Theorem.

The sum

$$\sum _{p,p+2\mathit{\text{ primes}}}\left(\frac{1}{p}+\frac{1}{p+2}\right)$$

converges.

General combinatorial sieve

The inequality (2) was based on the observation that

$$\sum _{d\mid n}\mu (d){\chi }_{-}(d)\le \sum _{d\mid n}\mu (d)\le \sum _{d\mid n}\mu (d){\chi }_{+}(d)$$ (5)

where ${\chi }_{-}$ is the characteristic function of the set of integers with no more than $2h-1$ prime factors, and ${\chi }_{+}$ is the characteristic function of the set of integers with no more than $2h$ prime factors. It is possible to choose different functions ${\chi }_{-}$ and ${\chi }_{+}$ that satisfy the inequality (5) to obtain bounds similar to (4). The problem of optimal choice of these functions in general is very hard. For more detailed information on Brun’s sieve and sieves in general one should consult the monographs [1, 2, 3].

Title	Brun’s pure sieve
Canonical name	BrunsPureSieve
Date of creation	2013-03-22 14:10:54
Last modified on	2013-03-22 14:10:54
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	11
Author	bbukh (348)
Entry type	Topic
Classification	msc 11N36
Classification	msc 11N35
Related topic	PrincipleOfInclusionExclusion
Related topic	SieveOfEratosthenes2
Related topic	BrunsConstant
Related topic	BonferroniInequalities
Defines	combinatorial sieve