A Comparison of Sufficiency Condtions for the Goldbach and the Twin Primes Conjectures ()
1. Introduction
Let and for. Let rapidly. When and
let denote the closed interval, a so-called major arc.
It is easily shown, for any choice of, that all the are disjoint and contained in the closed interval.
For each let be those points in which are not in any closed neighborhood (major arc)
of radius about any rational number, where and.
For each let be those points in which are not in any closed neighborhood (major arc)
of radius about any rational number, where and.
Let denote the number of ways (even) can be represented as a sum of two primes.
Let denote the number of twin primes less than or equal to.
In [1] the following two theorems are established:
Theorem 1.1 Under the generalized Riemann hypothesis with and if
, then for all even.
Theorem 1.2 Let and if
, then for all even.
In [2] the following two theorems are established:
Theorem 1.3 Under the assumption that Siegel zeros do not exist with and if
then for all even.
Theorem 1.4 Let and. If
, then for all even.
The proof of Theorem 1.3 and, in particular, the proof of Theorem 1.4 is very complicated.
In Section 6 of [2] it is shown, by a very complicated argument, that a particularly natural approach for eliminating the condition in Theorem 1.2 does not work.
As we will now see, the situation with regard to the twin prime conjecture is significantly, very less complicated. The reason is primarily because we only need to consider the Ramanujan sums rather than the sums, which appear in all of the theorems above, related to the Goldbach conjecture.
In Section 2 we will establish Theorem 1.5 Let and. if as goes to infinity in some suitable sequence.
2. A Proof of Theorem 1.5
We decompose the above integral
It is immediate by the prime numbers theorem that
By definition
where
Lemma 2.1 Let and. Then
This is Theorem 58 in [3] .
Lemma 2.2 Under the hypothesis of Lemma 2.1 we have
Proof. This follows immediately from Lemma 2.1 and the trivial inequalities and and the fact that if and, then.
Hence it is immediate that
By the change of variable we have
However, by (2.1) we have
Let so that if then
Let with the condition of summation and.
It is easy to see that
For we have
Hence for and
so that for some fixed we have:
But and by definition
so that it follows immediately that
Now summing over all we have
since by Theorem 327 page 267 in [3]
Hence
where. Now let
Lemma 2.3
Proof. [4] page 211.
It is immediate by Lemma 2.3 that
What remains to be done is to show is bounded away from 0.
Let
Since and are all multiplicative functions of, is a multiplicative function of. Also, by means of the trivial estimate on, namely 2, and a direct application of Theorem 327 page 267 in [5] we have
so that by Theorem 2 [3] page 3 we have
Hence
3. A Primitive Formulation of the Circle Method
1) Part I
We assume (even).
For each let.
Let be the number of representations of as the sum of two primes, each of which is less than.
Clearly,
We decompose this integral
where
Clearly, by Theorem 55 in [3] and the last paragraph on page 63 in [3] we have
By direct application of the easily established Equation (151) in [3]
(3.0)
and the Equation (204) in [3]
(3.1)
We have for
(3.2)
By the trivial inequalities and and the fact that if and, then with and, we have for
(3.3)
By the change of variable, we have
so that
(3.4)
Clearly
Also, the number of terms on the right hand side of (3.4) is and each term is greater than and less than 1 so that
Hence by definition of and Abel's lemma we have
so that
(3.5)
Hence,
(3.6)
Let. Then
so that
so that we have
Remark. Unfortunately, the integral cannot be; since for almost all, the integral is asymptotically where is the usual singular series.
We assume. For each let.
Let be the number of twin primes, each of which is less than.
Clearly,
We decompose this integral
where
where
Immediately, from (3.3) we have
By the change of variable, we have
so that
Let
Let
(3.7)
Clearly,
Also, the number of terms on the right hand side of (3.7) is and each term is greater than and less than 1 so that
Hence by definition of and Abel’s lemma we have
so that
(3.8)
Hence
(3.9)
Let Then;
so that
so that we have as goes to infinity in some suitable sequence.
2) Part II
For each (even) let be a prime in Let be those points in
which are not in and not in any closed interval of radius about any rational number where
.
Let be the number of representations of as the sum of two primes, which are limited to those primes in the arithmetic progressions mod.
Clearly,
We decompose this integral
where
Conjecture 1.
We now estimate.
But consider
so that uniformly
Hence for uniformly
So by (100) and (101) for
But since and since, we have by the inequalities immediately below (3.2) for
But
Hence for
(3.10)
By a change of variable we have
However,
Let
Hence, if,
By (3.5) we have
and
But
Hence, if,
so that
where
But by (3.6) we have
so that
so that
Let
By Theorem 272 in [5]
so that, since we assume.
Hence
so that if Conjecture 1 is true.
Let be the number of twin primes, each of which is in one of the arithmetic progressions mod defined above.
Clearly,
We decompose the integral
Conjecture 2.
infinity in some suitable sequence.
From (3.10) we have for
By change of variable we have
However,
Let
Hence, if
Clearly,
So that by (3.8)
and.
Hence, if,
so that
where
But by (3.6)
so that
Hence
Hence
so that if Conjecture 2 is true, as goes to infinity in a sequence that satisfies the conjecture.
4. Some Heuristics
Theorem 4.1 If
then every sufficiently large even integer is the sum of two primes, where is an exceptional set, whose measure goes to 0 with.
Proof. This is established in [1] .
Theorem 4.2 Let be an arbitrary fixed integer. Then
uniformly for almost all.
Proof. This deep result is immediate by (5-2) in [6] .
There is no compelling reason to assume Theorem 4.2 is not true for.
It is worthwile to investigate if Carleson’s proof can be modified to establish Theorem 4.2 with replaced with and replaced with.
In [7] Tao presents a heuristic argument to establish that the major arc contribution in the circle method is
. He states that his argument can be made rigorous.
However, it follows from the proof of Theorem 1.5 that the major arc contribution is not in any sequence of.
But it is well known that so that the contribution of the minor arc in the circle method approach to the twin primes conjecture (Theorem 1.5) is, which makes plausible that the required estimate of might be true.
It is plausible that in Theorem 1.3
where the latter integral is that of Theorem 1.5, which makes plausible that the required estimate of might be true.
Those, who seriously attempt Conjecture 2 have the advantage that there is some degree of freedom in the choice of and in the choice of for each; and the estimate is required only as goes to infinity in some suitable sequence.
Acknowledgements
I thank R. C. Vaughan for the Remark in Part I, Section 3.