Brun’s pure sieve

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

,

…

,

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

∑

n

∈

𝒜

∑

p

1

∤

n

…

p

k

∤

n

1

=

∑

n

∈

𝒜

∑

(

n

,

p

1

⁢

⋯

⁢

p

k

)

=

1

1

=

∑

n

∈

𝒜

∑

d

∣

(

n

,

p

1

⋯

p

k

)

μ

⁢

(

d

)

=

∑

d

∣

p

1

⋯

p

k

μ

⁢

(

d

)

⁢

∑

n

∣

𝒜

d

∣

n

1

=

∑

d

∣

p

1

⋯

p

k

μ

⁢

(

d

)

⁢

A

d

=

A

1

-

∑

p

∣

p

1

⋯

p

k

A

p

+

∑

p

q

∣

p

1

⋯

p

k

A

p

⁢

q

+

⋯

.

(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,

∑

d

∣

p

1

⋯

p

k

ν

⁢

(

d

)

≤

2

⁢

h

-

1

μ

⁢

(

d

)

⁢

A

d

≤

∑

n

∈

𝒜

∑

p

1

∤

n

…

p

k

∤

n

1

≤

∑

d

∣

p

1

⋯

p

k

ν

⁢

(

d

)

≤

2

⁢

h

μ

⁢

(

d

)

⁢

A

d

(2)

where

ν

⁢

(

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

=

X

⁢

w

⁢

(

d

)

/

d

+

R

d

where

w

⁢

(

d

)

is some multiplicative function of

d

, and

R

d

is
small compared to

X

⁢

w

⁢

(

d

)

/

d

. Then the estimate
in (

2

) takes the form

∑

d

∣

p

1

⋯

p

k

ν

⁢

(

d

)

≤

t

μ

⁢

(

d

)

⁢

A

d

=

∑

d

∣

p

1

⋯

p

k

ν

⁢

(

d

)

≤

t

μ

⁢

(

d

)

⁢

(

X

⁢

w

⁢

(

d

)

/

d

+

R

d

)

=

X

⁢

∑

d

∣

p

1

⋯

p

k

ν

⁢

(

d

)

≤

t

μ

⁢

(

d

)

⁢

w

⁢

(

d

)

d

+

O

⁢

(

∑

d

∣

p

1

⋯

p

k

ν

⁢

(

d

)

≤

t

R

d

)

(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

ν

⁢

(

d

)

-

t

is

greater

than

1

for every

u

>

1

and every

d

such that

ν

⁢

(

d

)

>

t

, we have that

∑

d

∣

p

1

⋯

p

k

ν

⁢

(

d

)

≤

t

μ

⁢

(

d

)

⁢

w

⁢

(

d

)

d

=

∑

d

∣

p

1

⋯

p

k

μ

⁢

(

d

)

⁢

w

⁢

(

d

)

d

+

O

⁢

(

∑

d

∣

p

1

⋯

p

k

ν

⁢

(

d

)

>

t

w

⁢

(

d

)

d

)

=

∑

d

∣

p

1

⋯

p

k

μ

⁢

(

d

)

⁢

w

⁢

(

d

)

d

+

O

⁢

(

∑

d

∣

p

1

⋯

p

k

w

⁢

(

d

)

d

⁢

u

ν

⁢

(

d

)

-

t

)

=

∑

d

∣

p

1

⋯

p

k

μ

⁢

(

d

)

⁢

w

⁢

(

d

)

d

+

O

⁢

(

u

-

t

⁢

∑

d

∣

p

1

⋯

p

k

w

⁢

(

d

)

d

⁢

u

ν

⁢

(

d

)

)

and since the sum of a multiplicative function can be
written as an

Euler product

, we get

=

∏

p

∈

𝒫

(

1

-

w

⁢

(

p

)

p

)

+

O

⁢

(

u

-

t

⁢

∏

p

∈

𝒫

(

1

+

u

⁢

w

⁢

(

p

)

p

)

)

=

∏

p

∈

𝒫

(

1

-

w

⁢

(

p

)

p

)

+

O

⁢

(

u

-

t

⁢

∏

p

∈

𝒫

(

1

+

w

⁢

(

p

)

p

)

u

)

to minimize the error term we choose

u

=

t

/

∑

p

∈

𝒫

log

⁡

(

1

+

w

⁢

(

p

)

p

)

, and get

=

∏

p

∈

𝒫

(

1

-

w

⁢

(

p

)

p

)

+

O

⁢

(

1

t

⁢

∑

p

∈

𝒫

w

⁢

(

p

)

p

)

t

Combining this with (

3

) and (

2

) we
obtain

∑

n

∈

𝒜

∑

p

1

∤

n

…

p

k

∤

n

1

=

X

⁢

∏

p

∈

𝒫

(

1

-

w

⁢

(

p

)

p

)

+

X

⋅

O

⁢

(

1

t

⁢

∑

p

∈

𝒫

w

⁢

(

p

)

p

)

t

+

O

⁢

(

∑

d

∣

p

1

⋯

p

k

ν

⁢

(

d

)

≤

t

R

d

)

(4)

provided that

u

=

t

/

∑

p

∈

𝒫

log

⁡

(

1

+

w

⁢

(

p

)

p

)

≥

1

.

Example. (

Upper bound

on the number of primes.)

If we set

𝒫

to be all the primes less

y

, and

𝒜

=

{

1

,

2

,

…

,

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

≤

1

. The second error term
in (

4

) has at most

y

t

summands, and the first
error term is bounded by

(

c

t

⁢

log

⁡

log

⁡

y

)

t

by Merten’s

theorem

. Thus, the estimate (

4

) becomes

π

⁢

(

x

)

-

π

⁢

(

y

)

≤

x

⁢

e

-

γ

+

o

⁢

(

1

)

log

⁡

y

+

x

⋅

O

⁢

(

1

t

⁢

log

⁡

log

⁡

y

)

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

/

2

⁢

c

⁢

log

⁡

log

⁡

x

for a sufficiently large constant

c

.
With this choice we obtain

π

⁢

(

x

)

≤

O

⁢

(

x

⁢

log

⁡

log

⁡

x

log

⁡

x

)

.

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

p

<

n

and

p

∣

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

∈

[

y

.

.

x

]

}

. Then

w

⁢

(

2

)

=

1

, and

w

⁢

(

p

)

=

2

if

p

>

2

. The

remainder

R

d

does not exceed

2

ν

⁢

(

d

)

. Like above we obtain

π

2

⁢

(

x

)

-

π

2

⁢

(

y

)

≤

x

⁢

c

(

log

⁡

y

)

2

+

x

⋅

O

⁢

(

1

t

⁢

log

⁡

log

⁡

y

)

t

+

O

⁢

(

(

2

⁢

y

)

t

)

where

π

2

⁢

(

x

)

denotes the number of twin primes not exceeding

x

. Upon setting

t

=

c

⁢

log

⁡

log

⁡

x

, and

y

=

x

1

/

2

⁢

c

⁢

log

⁡

log

⁡

x

we
obtain

π

2

⁢

(

x

)

≤

O

⁢

(

x

⁢

(

log

⁡

log

⁡

x

)

2

(

log

⁡

x

)

2

)

.

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

∑

p

,

p

+

2

⁢

primes

(

1

p

+

1

p

+

2

)

converges.

General combinatorial sieve

The inequality (

2

) was based on the observation
that

∑

d

∣

n

μ

⁢

(

d

)

⁢

χ

-

⁢

(

d

)

≤

∑

d

∣

n

μ

⁢

(

d

)

≤

∑

d

∣

n

μ

⁢

(

d

)

⁢

χ

+

⁢

(

d

)

(5)

where

χ

-

is the

characteristic function

of the set of
integers with no more than

2

⁢

h

-

1

prime factors, and

χ

+

is
the characteristic function of the set of integers with no more
than

2

⁢

h

prime factors. It is possible to choose different

functions

χ

-

and

χ

+

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

]

.

References

1

George Greaves.

Sieves in

number theory

.

Springer, 2001.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=1003.11044

Zbl 1003.11044.

2

H. Halberstam and H.-E. Richert.

Sieve methods

, volume 4 of

London Mathematical Society

Monographs

.

1974.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0298.10026

Zbl 0298.10026.

3

C. Hooley.

Application of sieve methods to the theory of numbers

,
volume 70 of

Cambridge Tracts in Mathematics

.

Cambridge Univ. Press, 1976.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0327.10044

Zbl 0327.10044.

4

Gérald Tenenbaum.

Introduction

to

Analytic

and Probabilistic Number Theory

,
volume 46 of

Cambridge Studies in Advanced Mathematics

.

Cambridge Univ. Press.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0831.11001

Zbl 0831.11001.

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

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