THE SELBERG SIEVE, APPLIED TO TWIN PRIMES The theory of ...

THE SELBERG SIEVE, APPLIED TO TWIN PRIMES
PART III PRIME NUMBERS, MICHAELMAS 2004
The theory of “small” sieves is large and rather bewildering for the beginner. In this
chapter we present one example of a small sieve in action, using the so-called Selberg
sieve to prove the following result:
Theorem 0.1. The number of twin primes less than N (that is, primes p such that
p + 2 is also prime) is at most CN/ log 2 N.
Remarks. Thus most primes are not twin primes. An amusing consequence of this
result, which is often quoted, is that the sum of the reciprocals of the twin primes
converges (to what is known as Brun’s constant). It is a famous unsolved problem to
decide whether there are infinitely many twin primes.
One of the biggest difficulties for the would-be sieve theorist is the bewildering array
of notation in the subject. By studying one specific problem we can avoid a lot of
that, whilst hardly losing any of the ideas, but it would be remiss not to make some
remarks relevant to the more general context. To place an upper bound on the number
of twin primes less than N, one studies the sequence A = (an), where an = n(n + 2). If
√ N � n � N and n is a twin prime, then an does not have any prime divisors smaller
than √ N. Thus, for any z � √ N we have the upper bound
number of twin primes p ∈ [ √ N, N] � |S(A, P, z)|,
where S(A, P, z) is the collection of all a ∈ A which are not divisible by any p ∈ P with
p � z. In our example we will take P = all primes, though there are problems where
we might wish to use just a subset of the primes.
What information do we know about A? Well, it is not very difficult to estimate the
number of a ∈ A which are divisible by 2, 3, 4, . . . quite accurately. Write
Ad := {a ∈ A : d|a}.
Then |Ad| can be studied using the function ω, defined as follows. If p is prime, then
ω(p) is the number of residues x(mod p) such that x(x + 2) ≡ 0(mod p). Thus ω(2) = 1
and ω(p) = 2 for p � 3. Extend ω to a function on all of N by insisting that it be
completely 1 multiplicative, that is to say ω(mm ′ ) = ω(m)ω(m ′ ) for all m, m ′ .
As regards |Ad|, one easily sees that if d is squarefree then ω(d) is the number of solutions
to x(x + 2) ≡ 0(mod d). Thus in this case (which will be the only one that interests us)
where |Rd| � ω(d).
|Ad| = ω(d)
d N + Rd, (0.1)
1 as opposed to just multiplicative, which means that ω(mm ′ ) = ω(m)ω(m ′ ) when (m, m ′ ) = 1
1

2 PART III PRIME NUMBERS, MICHAELMAS 2004
All of the above features are more-or-less typical in sieve theory. Once one has an
understanding of Ad one might hope to estimate S(A, P, z) by using inclusion-exclusion:
|S(A, P, z)| = N − �
|Ap1| + �
p1�z
p1

THE SELBERG SIEVE, APPLIED TO TWIN PRIMES 3
2. selberg’s observation
Let λ : {1, . . . , z} → R be any function whatsoever with λ1 = 1. Then if a ∈ S(A, P, z),
we have
� � �2 λd � 1,
d|a
d�z
since the sum collapses to just the one term with d = 1. It follows that
|S(A, P, z)| � � � � �2 λd =
� �
λd1λd2 = �
|A[d1,d2]|λd1λd2
n�N
d|an
d�z
d1,d2�z n�N
d1,d2|n
d1,d2�z
Let us assume furthermore that λd = 0 if d is not squarefree. Then we may use (0.1) to
write this in the form
|S(A, P, z)| � N �
d1,d2�z
µ(di)�=0
ω([d1, d2])
[d1, d2] λd1λd2 + O( �
d1,d2�z
µ(di)=0
ω([d1, d2])λd1λd2). (2.1)
We have ω(d) = Oɛ(d ɛ ), since by multiplicativity ω(d) is no more than 2 ϖ(d) , where
ϖ(d) is the number of prime factors of d (this is an exercise on the second example
sheet – we also used it in [PN9]). It will turn out later on that whenever we apply (2.1)
we will have the bound
λd = Oɛ(d ɛ ) (2.2)
In this case the error term in (2.1) is O(z 2 N ɛ ), which will be dominated by the main
term if z = N 1/2−δ for some δ > 0.
Write
Q := �
d1,d2�z
µ(di)�=0
ω([d1, d2])
[d1, d2] λd1λd2
for the main term in (2.1). Thus under the assumption (2.2) we have
|S(A, P, z)| � NQ + Oɛ(z 2 N ɛ ). (2.3)
Q is a quadratic form, and we wish to choose values of λd so that it is as small as
possible. We do this by diagonalising the form. To begin with, we rewrite Q in the form
Q = � ω(d1)ω(d2) (d1, d2)
. (2.4)
d1,d2�z
µ(di)�=0
d1d2
λd1λd2
ω((d1, d2))
To get a handle on this, we use a fairly standard trick. Write g(k) = k/ω(k), and
observe that by Möbius inversion we have
g(k) = �
f(δ)
where
f(k) = �
δ|k
δ|k
µ( k
δ )g(δ).

4 PART III PRIME NUMBERS, MICHAELMAS 2004
Substituting into (2.4) and swapping the order of summation yields
Q = �
�
�
f(δ)
ω(d)
d λd
�2 . (2.5)
δ�z
δ|d,d�z
µ(d)�=0
This is indeed a diagonal quadratic form, in the variables
uδ := � ω(d)
d λd.
δ|d,d�z
µ(d)�=0
To minimise Q we need to express the constraint λ1 = 1 in terms of these new variables
uδ. This can be achieved by applying Lemma 1.2. One obtains
ω(d)
d λd = �
d|δ,δ�z
µ(δ)�=0
µ( δ
d )uδ, (2.6)
and so that constraint becomes simply
�
µ(δ)uδ = 1. (2.7)
δ�z
The minimisation of Q, as given in (2.5), subject to the constraint (2.7), is a simple
matter of completing the square. The minimum value is Q0 = 1/D, where
D = D(z) := �
d�z
µ 2 (d)
f(d) .
The optimal choice for uδ is uδ = µ(δ)/Df(δ), which corresponds in view of (2.6) to
λd =
d
ω(d)D
� µ(δ/d)µ(δ)
.
f(δ)
d|δ,δ�z
We may now verify the claimed bound (2.2) for our choice of λd. We start with the
observation that since g (as defined above) is multiplicative, then so is f = g ∗ µ.
Furthermore it is easy to check that f(p) = p/ω(p) − 1. Thus (note that f � 0)
|ω(d)λd| = | d
D
�
d|δ,δ�z
µ(δ/d)µ(δ)
| �
f(δ)
dµ2 (d)
Df(d)
�
δ�z
µ 2 (δ)
f(δ) � dµ2 (d)
f(d) .
Remember that µ 2 (n) is simply either 1 or 0 according as n is or is not squarefree, thus
the above is not as intimidating as it looks. Now if d is squarefree we have, further,
d
f(d)
This establishes (2.2).
= �
p|d
p
f(p)
= �
p|d
1
ω(p)
1
− 1
p
� 6 ϖ(d) = Oɛ(d ɛ ).
Remark. There are times when it is helpful to know more about the coefficients λd.
In our particular problem they behave rather like µ(d)
� log(z/d)
log z
� 2
, the 2 here being a
consequence of the fact that ω(p) = 2 for almost all primes. They are bounded by 1,
rather than just by Oɛ(d ɛ ).

THE SELBERG SIEVE, APPLIED TO TWIN PRIMES 5
Now that (2.2) is established, we may consider (2.3) to have been confirmed, with
Q = D:
|S(A, P, z)| � N
D + Oɛ(z 2 N ɛ ). (2.8)
When z = N 1/3 , the error term here is small. In the next section we will obtain the
bound D(N 1/3 ) ≫ log 2 N, which will conclude the proof of Theorem 0.1.
3. twin primes
Our aim here is to finish the proof of Theorem 0.1. In view of (2.8) it is enough to
prove the bound D(N 1/3 ) ≫ log 2 N, where
Thus we have
D = �
To bound � ω(m)
m�z m
d�z
µ 2 (d)
f(d)
D(z) = �
= �
d�z
µ(d)=0
d�z
d�z
p|d
�
p|d
µ 2 (d)
f(d) .
ω(p)/p
1 − ω(p)
p
= � � � ω(p)
1 +
p + ω(p2 )
p2 �
+ . . .
� �
m�z
ω(m)
m .
below one can observe that ω(m) � d(m), the number of divisors
of m, for all odd m. This follows by writing m = p α1
1 . . . p αk
k , so that ω(m) = 2α1 . . . 2 αk
and d(m) = (α1 + 1) . . . (αk + 1). From this observation one can deduce that
� ω(m) � d(m)
�
m m �
�
�
�2 1
≫ (log z)
m
2 .
m�z
m odd
m�z
m odd
m� √ z
m odd
This concludes our establishment of a lower bound for D(z), and hence the proof of
Theorem 0.1.