Vinogradov’s Three Prime Theorem
Ryan Alweiss, Evan Chen, Sammy Luo
18.099 Final Paper
Spring 2016
Abstract
We prove that every sufficiently large odd integer can be written as the sum
of three primes, conditioned on a strong form of the prime number theorem.
Contents
1 Introduction
2
2 Synopsis
2
3 Prime number theorem type bounds
3.1 Dirichlet characters . . . . . . . . . . . . . . . . . .
3.2 Prime number theorem for arithmetic progressions
3.3 Gauss sums . . . . . . . . . . . . . . . . . . . . . .
3.4 Dirichlet approximation . . . . . . . . . . . . . . .
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4 Bounds on S(x, α)
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5 Estimation of the arcs
5.1 Setting up the arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Estimate of the minor arcs . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Estimate on the major arcs . . . . . . . . . . . . . . . . . . . . . . .
8
9
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6 Completing the proof
13
1
1
Introduction
In this paper, we prove the following result:
Theorem 1.1 (Vinogradov). Every sufficiently large odd integer N is the sum of
three prime numbers.
In fact, the following result is also true, called the “weak Goldbach conjecture”.
Theorem 1.2 (Weak Goldbach conjecture). Every odd integer N ≥ 7 is the sum
of three prime numbers.
The proof of Vinogradov’s theorem becomes significantly simpler if one assumes
the generalized Riemann hypothesis; this allows one to use a strong form of the
prime number theorem (Theorem 3.4). This conditional proof was given by Hardy
and Littlewood in the 1923’s. In 1997, Deshouillers, Effinger, te Riele and Zinoviev
showed that the generalized Riemann hypothesis in fact also implies the weak Goldbach conjecture by improving the bound to 1020 and then exhausting the remaining
cases via a computer search. See [4] for details.
As for unconditional proofs, Vinogradov was able to eliminate the dependency
on the generalized Riemann hypothesis in 1937, which is why the Theorem 1.1
bears his name. However, Vinogradov’s bound used the ineffective Siegel-Walfisz
15
theorem; his student K. Borozdin showed that 33 is large enough. Over the years
the bound was improved, until recently in 2013 when Harald Helfgott claimed the
first unconditional proof of Theorem 1.2 in [3].
In this exposition we follow Hardy and Littlewood’s approach, i.e. we prove
Theorem 1.1 assuming the generalized Riemann hypothesis, following the exposition
of [1]. An exposition of the unconditional proof by Vinogradov can be found in [2].
2
Synopsis
We are going to prove that
X
a+b+c=N
where
1
Λ(a)Λ(b)Λ(c) N 2 G(N )
2
Y
G(N ) =
1−
def
p|N
1
(p − 1)2
Y
p-N
1
1+
(p − 1)3
(1)
and Λ is the von Mangoldt function defined as usual. Then so long as 2 - N , the
quantity G(N ) will be bounded away from zero; thus (1) will imply that in fact
there are many ways to write N as the sum of three distinct prime numbers.
The sum (1) is estimated using Fourier analysis. Let us define the following.
2
Definition 2.1. Let T = R/Z denote the circle group, and let e : T → C be the
exponential function θ 7→ exp(2πiθ). For α ∈ T, kαk denotes the minimal distance
from α to an integer.
Note that |e(θ) − 1| = Θ(kθk).
Definition 2.2. For α ∈ T and x > 0 we define
X
Λ(n)e(nα).
S(x, α) =
n≤x
Then we can rewrite (1) using S as a “Fourier coefficient”:
Proposition 2.3. We have
X
Z
Λ(a)Λ(b)Λ(c) =
S(N, α)3 e(−N α) dα.
(2)
α∈T
a+b+c=N
Proof. We have
S(N, α)3 =
X
Λ(a)Λ(b)Λ(c)e((a + b + c)α),
a,b,c≤N
so
Z
Z
3
X
S(N, α) e(−N α) dα =
α∈T
Λ(a)Λ(b)Λ(c)e((a + b + c)α)e(−N α) dα
α∈T a,b,c≤N
=
X
Z
=
X
e((a + b + c − N )α) dα
Λ(a)Λ(b)Λ(c)
α∈T
a,b,c≤N
Λ(a)Λ(b)Λ(c)I(a + b + c = N )
a,b,c≤N
=
X
Λ(a)Λ(b)Λ(c),
a+b+c=N
as claimed.
In order to estimate the integral in Proposition 2.3, we divide T into the so-called
“major” and “minor” arcs. Roughly,
• The “major arcs” are subintervals of T centered at a rational number with
small denominator.
• The “minor arcs” are the remaining intervals.
3
These will be made more precise later. This general method is called the HardyLittlewood circle method, because of the integral over the circle group T.
The rest of the paper is structured as follows. In Section 3, we define the
Dirichlet character and other number-theoretic objects, and state some estimates
for the partial sums of these objects conditioned on the Riemann hypothesis. These
bounds are then used in Section 4 to provide corresponding estimates on S(x, α). In
Section 5 we then define the major and minor arcs rigorously and use the previous
estimates to given an upper bound for the integral over both areas. Finally, we
complete the proof in Section 6.
3
Prime number theorem type bounds
In this section, we collect the necessary number-theoretic results that we will need.
It is in this section only that we will require the generalized Riemann hypothesis.
As a reminder, the notation f (x) g(x), where f is a complex function and g
a nonnegative real one, means f (x) = O(g(x)), a statement about the magnitude of
f . Likewise, f (x) = g(x) + O(h(x)) simply means that for some C, |f (x) − g(x)| ≤
C|h(x)| for all sufficiently large x.
3.1
Dirichlet characters
In what follows, q denotes a positive integer.
Definition 3.1. A Dirichlet character modulo q χ is a homomorphism χ :
(Z/q)× → C× . It is said to be trivial if χ = 1; we denote this character by χ0 .
By slight abuse of notation, we will also consider χ as a function Z → C∗ by
setting χ(n) = χ(n (mod q)) for gcd(n, q) = 1 and χ(n) = 0 for gcd(n, q) > 1.
Remark 3.2. The Dirichlet characters form a multiplicative group of order φ(q)
under multiplication, with inverse given by complex conjugation. Note that χ(m)
is a primitive φ(q)th root of unity for any m ∈ (Z/q)× , thus χ takes values in the
unit circle.
Moreover, the Dirichlet characters satisfy an orthogonality relation
Experts may recognize that the Dirichlet characters are just the elements of the
Pontryagin dual of (Z/q)× . In particular, they satisfy an orthogonality relationship
(
X
1 n = a (mod q)
1
χ(n)χ(a) =
(3)
φ(q)
0 otherwise
χ mod q
and thus form an orthonormal basis for functions (Z/q)× → C.
4
3.2
Prime number theorem for arithmetic progressions
Definition 3.3. The generalized Chebyshev function is defined by
X
Λ(n)χ(n).
ψ(x, χ) =
n≤x
The Chebyshev function is studied extensively in analytic number theory, as it
is the most convenient way to phrase the major results of analytic number theory.
For example, the prime number theorem is equivalent to the assertion that
X
Λ(n) x
ψ(x, χ0 ) =
n≤x
where q = 1 (thus χ0 is the constant function 1). Similarly, Dirichlet’s theorem
actually asserts that any q ≥ 1,
(
x + oq (x) χ = χ0 trivial
ψ(x, χ) =
oq (x)
χ 6= χ0 nontrivial.
However, the error term in these estimates is quite poor (more than x1−ε for every ε).
However, by assuming the Riemann Hypothesis for a certain “L-function” attached
to χ, we can improve the error terms substantially.
Theorem 3.4 (Prime number theorem for arithmetic progressions). Let χ be a
Dirichlet character modulo q, and assume the Riemann hypothesis for the L-function
attached to χ.
(a) If χ is nontrivial, then
ψ(x, χ) √
x(log qx)2 .
(b) If χ = χ0 is trivial, then
ψ(x, χ0 ) = x + O
√
x(log x)2 + log q log x .
Theorem 3.4 is the strong estimate that we will require when putting good estimates on S(x, α), and is the only place in which the generalized Riemann Hypothesis
is actually required.
3.3
Gauss sums
Definition 3.5. For χ a Dirichlet character modulo q, the Gauss sum τ (χ) is
defined by
q−1
X
τ (χ) =
χ(a)e(a/q).
a=0
5
We will need the following fact about Gauss sums.
Lemma 3.6. Consider Dirichlet characters modulo q. Then:
(a) We have τ (χ0 ) = µ(q).
(b) For any χ modulo q, |τ (χ)| ≤
3.4
√
q.
Dirichlet approximation
We finally require Dirichlet approximation theorem in the following form.
Theorem 3.7 (Dirichlet approximation). Let α ∈ R be arbitrary, and M a fixed
integer. Then there exists integers a and q = q(α), with 1 ≤ q ≤ M and gcd(a, q) =
1, satisfying
α − a ≤ 1 .
q qM
4
Bounds on S(x, α)
In this section, we use our number-theoretic results to bound S(x, α).
First, we provide a bound for S(x, α) if α is a rational number with “small”
denominator q.
Lemma 4.1. Let gcd(a, q) = 1. Assuming Theorem 3.4, we have
S(x, a/q) =
µ(q)
√
x + O qx(log qx)2
φ(q)
where µ denotes the Möbius function.
Proof. Write the sum as
S(x, a/q) =
X
Λ(n)e(na/q).
n≤x
First we claim that the terms gcd(n, q) > 1 (and Λ(n) 6= 0) contribute a negligibly
small log q log x. To see this, note that
• The number q has log q distinct prime factors, and
• If pe | q, then Λ(p) + · · · + Λ(pe ) = e log p = log(pe ) < log x.
So consider only terms with gcd(n, q) = 1. To bound the sum, notice that
X
e(n · a/q) =
e(b/q) · 1(b ≡ an)
b mod q

=
X
b mod q

X
1
e(b/q) 
χ(b)χ(an)
φ(q)
χ mod q
6
by the orthogonality relations. Now we swap the order of summation to obtain a
Gauss sum:


X
X
1
e(n · a/q) =
χ(an) 
χ(b)e(b/q)
φ(q)
χ mod q
=
1
φ(q)
X
b mod q
χ(an)τ (χ).
χ mod q
Thus, we swap the order of summation to obtain that
X
S(x, α) =
Λ(n)e(n · a/q)
n≤x
gcd(n,q)=1
=
=
1
φ(q)
X
Λ(n)χ(an)τ (χ)
n≤x
χ mod q
gcd(n,q)=1
X
1
τ (χ)
φ(q)
χ mod q
=
X
X
X
1
χ(a)τ (χ)
φ(q)
χ mod q
Λ(n)χ(an)
n≤x
gcd(n,q)=1
X
Λ(n)χ(n)
n≤x
gcd(n,q)=1
X
1
χ(a)τ (χ)ψ(x, χ)
φ(q)
χ mod q

X
1 
τ (χ0 )ψ(x, χ0 ) +
=
φ(q)
=

χ(a)τ (χ)ψ(x, χ) .
16=χ mod q
Now applying both parts of Lemma 3.6 in conjunction with Theorem 3.4 gives
√
√
µ(q)
x + O x(log qx)2 + O x(log x)2
φ(q)
µ(q)
√
=
x + O qx(log qx)2
φ(q)
S(x, α) =
as desired.
We then provide a bound when α is “close to” such an a/q.
Lemma 4.2. Let gcd(a, q) = 1 and β ∈ T. Assuming Theorem 3.4, we have


X
µ(q) 
√
e(βn) + O (1 + kβkx) qx(log qx)2 .
S(x, a/q + β) =
φ(q)
n≤x
7
Proof. For convenience let us assume x ∈ Z. Let α = a/q + β. Let us denote
√
Err(x, α) = S(x, α) − µ(q)
qx(log x)2 . We
φ(q) x, so by Lemma 4.1 we have Err(x, α) have
X
Λ(n)e(na/q)e(nβ)
S(x, α) =
n≤x
=
X
e(nβ) (S(n, a/q) − S(n − 1, a/q))
n≤x
µ(q)
e(nβ)
=
+ Err(n, α) − Err(n − 1, α)
φ(q)
n≤x


X
X
µ(q) 
=
e(nβ) +
(e((m + 1)β) − e(mβ)) Err(m, α)
φ(q)
X
n≤x
1≤m≤x−1
+ e(xβ) Err(x, α) − e(0) Err(0, α)

 

X
X
µ(q) 
e(nβ) + 
≤
kβk Err(m, α) + Err(0, α) + Err(x, α)
φ(q)
n≤x
1≤m≤x−1


µ(q)  X
√
e(nβ) + (1 + x kβk) O qx(log qx)2
φ(q)
n≤x
as desired.
Thus if α is close to a fraction with small denominator, the value of S(x, α) is
bounded above. We can now combine this with the Dirichlet approximation theorem
to obtain the following general result.
Corollary 4.3. Suppose M = N 2/3 and suppose |α − a/q| ≤
1 with q ≤ M . Assuming Theorem 3.4, we have
S(x, α) 5
x
+ x 6 +ε
ϕ(q)
for any ε > 0.
Proof. Apply Lemma 4.2 directly.
5
Estimation of the arcs
We’ll write
def
f (α) = S(N, α) =
X
n≤N
for brevity in this section.
8
Λ(n)e(nα)
1
qM
for some gcd(a, q) =
Recall that we wish to bound the right-hand side of (2) in Proposition 2.3. We
split [0, 1] into two sets, which we call the “major arcs” and the “minor arcs.” To
do so, we use Dirichlet approximation, as hinted at earlier.
In what follows, fix
M = N 2/3
(4)
10
K = (log N ) .
5.1
(5)
Setting up the arcs
Definition 5.1. For q ≤ K and gcd(a, q) = 1, 1 ≤ a ≤ q, we define
1
a .
M(a, q) = α ∈ T | α − ≤
q
M
These will be the major arcs. The union of all major arcs is denoted by M. The
complement is denoted by m.
Equivalently, for any α, consider q = q(α) ≤ M as in Theorem 3.7. Then α ∈ M
if q ≤ K and α ∈ m otherwise.
Proposition 5.2. M is composed of finitely many disjoint intervals M(a, q) with
q ≤ K. The complement m is nonempty.
3
Proof. Note that if q1 , q2 ≤ K and a/q1 6= b/q2 then qa1 − qb2 ≥ q11q2 qM
.
In particular both M and m are measurable. Thus we may split the integral in
(2) over M and m. This integral will have large magnitude on the major arcs, and
small magnitude on the minor arcs, so overall the whole interval [0, 1] it will have
large magnitude.
5.2
Estimate of the minor arcs
First, we note the well known fact φ(q) q/ log q. Note also that if q = q(α)
as in the last section and α is on a minor arc, we have q > (log N )10 , and thus
φ(q) (log N )9 .
N
As such Corollary 3.3 yields that f (α) φ(q)
+ N .834 (logNN )9 .
9
Now,
Z
Z
f (α)3 e(−N α) dα ≤
|f (α)|3 dα
m
m
Z 1
N
|f (α)|2 dα
(log N )9 0
Z 1
N
=
f (α)f (−α) dα
(log N )9 0
X
N
=
Λ(n)2
(log N )9
n≤N
N2
(log N )8
,
P
2
using the well known bound n≤N Λ(n)2 logNN . This bound of (logNN )8 will be
negligible compared to lower bounds for the major arcs in the next section.
5.3
Estimate on the major arcs
We show that
Z
f (α)3 e(−N α)dα M
N2
G(N ).
2
By Proposition 5.2 we can split the integral over each interval and write
Z
M
X
f (α)3 e(−N α) dα =
q≤(log N )10
Z
X
1/qM
f (a/q + β)3 e(−N (a/q + β)) dβ.
−1/qM
1≤a≤q
gcd(a,q)=1
Then we apply Lemma 4.2, which gives
3
X
µ(q)
f (a/q + β)3 = 
e(βn)
φ(q)
n≤N

2
X
p
µ(q)
+
e(βn) O (1 + kβkN ) qN log2 qN
φ(q)
n≤N


2
X
p
µ(q)
+
e(βn) O (1 + kβkN ) qN log2 qN
φ(q)
n≤N
3
p
+ O (1 + kβkN ) qN log2 qN .

Now, we can do casework on the side of N −.9 that kβk lies on.
10
P
2
1
.9
• If kβk N −.9 , we have
n≤N e(βn) |e(β)−1| kβk N , and (1 +
√
kβkN ) qN log2 qN N 5/6+ε , because certainly we have kβk < 1/M =
N −2/3 .
P
• If on the other hand kβk N −.9 , we have n≤N e(βn) N obviously, and
√
O(1 + kβkN ) qN log2 qN ) N 3/5+ε .
As such, we obtain
3
X
µ(q)
f (a/q + β)3 
e(βn) + O N 79/30+ε
φ(q)

n≤N
in either case. Thus, we can write
Z
f (α)3 e(−N α) dα
M
X
=
Z
X
q≤(log N )10
1≤a≤q
gcd(a,q)=1
X
X
q≤(log N )10
1≤a≤q
gcd(a,q)=1
=
1/qM
f (a/q + β)3 e(−N (a/q + β)) dβ
−1/qM
3

X
µ(q)

e(βn) + O N 79/30+ε  e(−N (a/q + β)) dβ
φ(q)
−1/qM
Z
1/qM

n≤N

X
=
q≤(log N )10

µ(q)

S
q
φ(q)3 

X
1≤a≤q
gcd(a,q)=1

Z



e(−N (a/q))
1/qM
−1/qM
3

X

e(βn) e(−N β) dβ 

n≤N
+ O N 59/30+ε .
just using M ≤ N 2/3 . Now, we use
X
e(βn) =
n≤N
1
1 − e(βN )
.
1 − e(β)
kβk
This enables us to bound the expression
Z
1−1/qM
3

X

1/qM
n≤N
Z
1−1/qM
e(βn) e(−N β)dβ −3
kβk
1/qM
11
Z
1/2
dβ = 2
1/qM
β −3 dβ q 2 M 2 .
But the integral over the entire interval is

3
Z 1 X
Z


e(βn) e(−N β)dβ =
0
1
X
e((a + b + c − N )β)
0 a,b,c≤N
n≤N
X
1(a + b + c = N )
a,b,c≤N
N −1
=
.
2
Considering the difference of the two integrals gives

3
Z 1/qM X
N2

q 2 M 2 + N (log N )c N 4/3 ,
e(βn) e(−N β) dβ −
2
−1/qM
n≤N
for some absolute constant c.
For brevity, let
Sq =
X
e(−N (a/q)).
1≤a≤q
gcd(a,q)=1
Then

Z
f (α)3 e(−N α) dα =
M
=
=
µ(q)
S 
3 q
φ(q)
10
q≤(log N )
+ O N 59/30+ε
X
N2
2
N2
2
X
q≤(log N )10
X
q≤(log N )10
Z
1/qM
−1/qM
3

X

e(βn) e(−N β) dβ 
n≤N
µ(q)
Sq + O((log N )10+c N 4/3 ) + O(N 59/30+ε )
φ(q)3
µ(q)
+ O(N 59/30+ε ).
φ(q)3
.
The inner sum is bounded by φ(q). So,
X
X
µ(q) 1
≤
S
,
q
3
2
q>(log N )10 φ(q)
q>(log N )10 φ(q)
which converges since φ(q)2 q c for some c > 1. So
Z
M
f (α)3 e(−N α) dα =

∞
N 2 X µ(q)
Sq + O(N 59/30+ε ).
2
φ(q)3
q=1
12
Now, since µ(q), φ(q), and
P
1≤a≤q e(−N (a/q))
gcd(a,q)=1
are multiplicative functions of q,
and µ(q) = 0 unless q is squarefree,
∞
Y
X
µ(p)
µ(q)
1
+
S
=
S
q
p
φ(q)3
φ(p)3
p
q=1
=
!
p−1
X
1
1−
e(−N (a/p))
(p − 1)3
Y
p
a=1
!
p−1
X
1
1−
=
(p · 1(p|N ) − 1)
3
(p
−
1)
p
a=1
Y
Y
1
1
=
1+
1−
(p − 1)2
(p − 1)3
Y
p-N
p|N
= G(N ).
So,
Z
f (α)3 e(−N α) dα =
M
When N is odd,
Y
G(N ) =
1−
p|N
1
(p − 1)2
Y
p-N
N2
G(N ) + O(N 59/30+ε ).
2
1
1+
(p − 1)3
so that we have
Z
f (α)3 e(−N α) dα M
Y m − 2 m 1
≥
= ,
m−1m−1
2
m≥3
N2
G(N ),
2
as desired.
6
Completing the proof
Because the integral over the minor arc is o(N 2 ), it follows that
X
a+b+c=N
Z
Λ(a)Λ(b)Λ(c) =
1
f (α)3 e(−N α)dα 0
N2
G(N ) N 2 .
2
1
Consider the set SN of integers pk ≤ N with k > 1. We must have p ≤ N 2 ,
and for each such p there are at most O(log N ) possible values of k. As such,
|SN | π(N 1/2 ) log N N 1/2 .
13
Thus
X
Λ(a)Λ(b)Λ(c) (log N )3 |S|N log(N )3 N 3/2 ,
a+b+c=N
a∈SN
and similarly for b ∈ SN and c ∈ SN . Notice that summing over a ∈ SN is equivalent
to summing over composite a, so
X
X
Λ(p1 )Λ(p2 )Λ(p3 ) =
Λ(a)Λ(b)Λ(c) + O(log(N )3 N 3/2 ) N 2 ,
p1 +p2 +p3 =N
a+b+c=N
where the sum is over primes pi . This finishes the proof.
References
[1] David Rhee, Proof of Vinogradov’s Three Primes Theorem, 2008. http://
etotheipi.weebly.com/uploads/8/1/2/0/8120131/_threeprimes.pdf
[2] Nicholas Rouse, Vinogradov’s Three Prime Theorem, 2013. http://math.
uchicago.edu/~may/REU2013/REUPapers/Rouse.pdf
[3] Harold A. Helfgott, The ternany Goldbach conjecture is true, 2013. http://
arxiv.org/abs/1312.7748
[4] Deshouillers, Jean-Marc; Effinger, Gove W.; Te Riele, Herman J. J.; Zinoviev,
Dmitrii (1997). A complete Vinogradov 3-primes theorem under the Riemann
hypothesis. Electronic Research Announcements of the American Mathematical
Society 3 (15): 99-104.
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