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.