This was my final paper for 18.099, seminar in discrete analysis, jointly with Sammy Luo and Ryan Alweiss.
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.
In this paper, we prove the following result:
Theorem 1 (Vinogradov)
Every sufficiently large odd integer is the sum of three prime numbers.
In fact, the following result is also true, called the “weak Goldbach conjecture”.
Theorem 2 (Weak Goldbach conjecture)
Every odd integer 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 9). 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 and then exhausting the remaining cases via a computer search.
As for unconditional proofs, Vinogradov was able to eliminate the dependency on the generalized Riemann hypothesis in 1937, which is why the Theorem 1 bears his name. However, Vinogradov’s bound used the ineffective Siegel-Walfisz theorem; his student K. Borozdin showed that is large enough. Over the years the bound was improved, until recently in 2013 when Harald Helfgott claimed the first unconditional proof of Theorem 2, see here.
In this exposition we follow Hardy and Littlewood’s approach, i.e. we prove Theorem 1 assuming the generalized Riemann hypothesis, following the exposition of Rhee. An exposition of the unconditional proof by Vinogradov is given by Rouse.
We are going to prove that
and is the von Mangoldt function defined as usual. Then so long as , the quantity will be bounded away from zero; thus (1) will imply that in fact there are many ways to write as the sum of three distinct prime numbers.
The sum (1) is estimated using Fourier analysis. Let us define the following.
Note that .
For and we define
Then we can rewrite (1) using as a “Fourier coefficient”:
Proof: We have
In order to estimate the integral in Proposition 5, we divide into the so-called “major” and “minor” arcs. Roughly,
- The “major arcs” are subintervals of centered at a rational number with small denominator.
- The “minor arcs” are the remaining intervals.
These will be made more precise later. This general method is called the Hardy-Littlewood circle method, because of the integral over the circle group .
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 . 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 , where is a complex function and a nonnegative real one, means , a statement about the magnitude of . Likewise, simply means that for some , for all sufficiently large .
3.1. Dirichlet characters
In what follows, denotes a positive integer.
The Dirichlet characters form a multiplicative group of order under multiplication, with inverse given by complex conjugation. Note that is a primitive th root of unity for any , thus 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 . In particular, they satisfy an orthogonality relationship
and thus form an orthonormal basis for functions .
3.2. Prime number theorem for arithmetic progressions
The generalized Chebyshev function is defined by
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
where (thus is the constant function ). Similarly, Dirichlet’s theorem actually asserts that any ,
However, the error term in these estimates is quite poor (more than for every ). However, by assuming the Riemann Hypothesis for a certain “-function” attached to , we can improve the error terms substantially.
Theorem 9 is the strong estimate that we will require when putting good estimates on , and is the only place in which the generalized Riemann Hypothesis is actually required.
3.3. Gauss sums
For a Dirichlet character modulo , the Gauss sum is defined by
We will need the following fact about Gauss sums.
3.4. Dirichlet approximation
We finally require Dirichlet approximation theorem in the following form.
4. Bounds on
In this section, we use our number-theoretic results to bound .
First, we provide a bound for if is a rational number with “small” denominator .
Let . Assuming Theorem 9, we have
where denotes the Möbius function.
Proof: Write the sum as
First we claim that the terms (and ) contribute a negligibly small . To see this, note that
- The number has distinct prime factors, and
- If , then .
So consider only terms with . To bound the sum, notice that
by the orthogonality relations. Now we swap the order of summation to obtain a Gauss sum:
Thus, we swap the order of summation to obtain that
Now applying both parts of Lemma 11 in conjunction with Theorem 9 gives
We then provide a bound when is “close to” such an .
Let and . Assuming Theorem 9, we have
Proof: For convenience let us assume . Let . Let us denote , so by Lemma 13 we have . We have
Thus if is close to a fraction with small denominator, the value of is bounded above. We can now combine this with the Dirichlet approximation theorem to obtain the following general result.
Suppose and suppose for some with . Assuming Theorem 9, we have
for any .
Proof: Apply Lemma 14 directly.
5. Estimation of the arcs
for brevity in this section.
Recall that we wish to bound the right-hand side of (2) in Proposition 5. We split 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
5.1. Setting up the arcs
For and , , we define
These will be the major arcs. The union of all major arcs is denoted by . The complement is denoted by .
Equivalently, for any , consider as in Theorem 12. Then if and otherwise.
is composed of finitely many disjoint intervals with . The complement is nonempty.
Proof: Note that if and then .
In particular both and are measurable. Thus we may split the integral in (2) over and . This integral will have large magnitude on the major arcs, and small magnitude on the minor arcs, so overall the whole interval it will have large magnitude.
5.2. Estimate of the minor arcs
First, we note the well known fact . Note also that if as in the last section and is on a minor arc, we have , and thus .
As such Corollary 3.3 yields that .
using the well known bound . This bound of 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
By Proposition 17 we can split the integral over each interval and write
Then we apply Lemma 14, which gives
Now, we can do casework on the side of that lies on.
- If , we have , and , because certainly we have .
- If on the other hand , we have obviously, and .
As such, we obtain
in either case. Thus, we can write
just using . Now, we use
This enables us to bound the expression
But the integral over the entire interval is
Considering the difference of the two integrals gives
for some absolute constant .
For brevity, let
The inner sum is bounded by . So,
which converges since for some . So
Now, since , , and are multiplicative functions of , and unless is squarefree,
When is odd,
so that we have
6. Completing the proof
Because the integral over the minor arc is , it follows that
Consider the set of integers with . We must have , and for each such there are at most possible values of . As such, .
and similarly for and . Notice that summing over is equivalent to summing over composite , so
where the sum is over primes . This finishes the proof.