A Comparison of Sufficiency Condtions for the Goldbach and the Twin Primes Conjectures ()
1. Introduction
Let
and
for
. Let
rapidly. When
and ![](https://www.scirp.org/html/htmlimages\1-5300658x\36a5e5a3-37d8-458c-9b82-b9b8e5580f15.png)
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
.
![](https://www.scirp.org/html/htmlimages\1-5300658x\7ca97cbf-c6fb-4c23-8559-129ae343c687.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\bcb9d063-7ef1-4907-b7fa-7a6246b2c77b.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\ae55f48d-3086-4779-86da-50396b9cb48d.png)
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
![](https://www.scirp.org/html/htmlimages\1-5300658x\56bb4fd1-a2bd-4914-926e-945d1a01db43.png)
We decompose the above integral
![](https://www.scirp.org/html/htmlimages\1-5300658x\d7a1d67a-7735-4c5a-b8a6-49c82886fc6a.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\d3d75b1a-3443-4cf3-ba09-beeacfa5a5d4.png)
It is immediate by the prime numbers theorem that
![](https://www.scirp.org/html/htmlimages\1-5300658x\eafb5c60-b804-4b0a-b763-f62bc66776ef.png)
By definition
![](https://www.scirp.org/html/htmlimages\1-5300658x\7c56d8a6-f841-411f-8e74-f16ba2274f10.png)
where
![](https://www.scirp.org/html/htmlimages\1-5300658x\153e94c5-b663-4e5f-9003-9cc7a0e45fb7.png)
Lemma 2.1 Let
and
. Then
![](https://www.scirp.org/html/htmlimages\1-5300658x\d06c4515-892a-4986-86c7-478fa2ff194e.png)
This is Theorem 58 in [3] .
Lemma 2.2 Under the hypothesis of Lemma 2.1 we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\7a6bcaa1-6400-4b40-a83c-40c3716a176f.png)
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
![](https://www.scirp.org/html/htmlimages\1-5300658x\6cee1de1-9d0c-41f4-90d6-cbeae43d0ae4.png)
By the change of variable
we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\92a9d8e2-510c-4e0e-9d04-a1f70b1e971d.png)
However, by (2.1) we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\faf731af-e97b-4f88-9df0-273f2548a139.png)
Let
so that if
then
![](https://www.scirp.org/html/htmlimages\1-5300658x\c5e4c4da-067a-4cbb-8e28-220daef3cca6.png)
Let
with the condition of summation
and
.
It is easy to see that
![](https://www.scirp.org/html/htmlimages\1-5300658x\815ec161-e559-46c8-acb5-a9ee4ce4fc52.png)
For
we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\22442e7c-2ec6-4c76-935a-9686fa7b5dda.png)
Hence for
and ![](https://www.scirp.org/html/htmlimages\1-5300658x\f80b692f-5438-4fdc-8773-9dffcdd62ac4.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\0d5cb3d2-a65e-4864-a520-8d969c65be61.png)
so that for some fixed
we have:
![](https://www.scirp.org/html/htmlimages\1-5300658x\08347940-1158-40e3-8d2f-80ff01dba3c6.png)
But
and by definition
![](https://www.scirp.org/html/htmlimages\1-5300658x\f9bc811f-9398-4fd0-b764-a613e2d3e733.png)
so that it follows immediately that
![](https://www.scirp.org/html/htmlimages\1-5300658x\3f3b09ff-e90c-42a0-a94f-3a5bfe932b73.png)
Now summing over all
we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\7e607080-ee0a-47b7-ba13-f3549a45bc3c.png)
since by Theorem 327 page 267 in [3]
![](https://www.scirp.org/html/htmlimages\1-5300658x\2cece82e-04fe-4605-9082-416edd7b80d9.png)
Hence
![](https://www.scirp.org/html/htmlimages\1-5300658x\71fb3189-1802-42b1-867b-3ce51816a686.png)
where
. Now let
![](https://www.scirp.org/html/htmlimages\1-5300658x\7603483f-d811-490a-9bed-d339a8548a86.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\660adeb6-5713-451f-8b7d-3cfebf3e515b.png)
Lemma 2.3
![](https://www.scirp.org/html/htmlimages\1-5300658x\ba91a61f-bdda-4073-8f67-aa31f5162d43.png)
Proof. [4] page 211.
It is immediate by Lemma 2.3 that
![](https://www.scirp.org/html/htmlimages\1-5300658x\31221795-c00b-42a8-adf7-3c4c04bf9664.png)
What remains to be done is to show
is bounded away from 0.
Let
![](https://www.scirp.org/html/htmlimages\1-5300658x\ff0b4ccf-deea-4f24-94fb-91f20b9410bb.png)
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
![](https://www.scirp.org/html/htmlimages\1-5300658x\7473c1c3-54b4-4513-bb36-e61813fd346b.png)
so that by Theorem 2 [3] page 3 we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\028aa64f-c542-4bd4-b96a-97fc560b6fb4.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\ec0ec377-ee4c-4139-907f-af6468655e69.png)
Hence
![](https://www.scirp.org/html/htmlimages\1-5300658x\26f3d15c-5f9d-4c07-a104-401339fa5b36.png)
3. A Primitive Formulation of the Circle Method
1) Part I
We assume
(even)
. ![](https://www.scirp.org/html/htmlimages\1-5300658x\382e2518-b156-416a-9b88-f6ffa9a75e06.png)
For each
let
.
Let
be the number of representations of
as the sum of two primes, each of which is less than
.
Clearly,
![](https://www.scirp.org/html/htmlimages\1-5300658x\343c0699-58f9-4bf8-b99e-3608298b3aca.png)
We decompose this integral
![](https://www.scirp.org/html/htmlimages\1-5300658x\a865720a-2ef2-489c-919f-b6220925075c.png)
where
![](https://www.scirp.org/html/htmlimages\1-5300658x\2fc92768-330d-49db-a67e-00302164f961.png)
Clearly, by Theorem 55 in [3] and the last paragraph on page 63 in [3] we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\8c631062-5a3a-40f0-a568-272338da0cf1.png)
By direct application of the easily established Equation (151) in [3]
(3.0)
and the Equation (204) in [3]
(3.1)
We have for ![](https://www.scirp.org/html/htmlimages\1-5300658x\ab0558de-bb51-4ea5-9551-23ad463d3cbd.png)
(3.2)
By the trivial inequalities
and
and the fact that if
and
, then
with
and
, we have for ![](https://www.scirp.org/html/htmlimages\1-5300658x\63c19454-2b3e-439a-b8f8-e3e38fce6128.png)
(3.3)
By the change of variable
, we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\31e7ae33-ebdf-4d5d-b590-f31416b8df3b.png)
so that
![](https://www.scirp.org/html/htmlimages\1-5300658x\fbd23f87-26ab-44f1-a1fd-d216f88e05df.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\72bea3bd-733f-4f06-8261-a1c82316f75b.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\3115e970-a84b-4e47-b998-6c8ba64dca1d.png)
(3.4)
Clearly
![](https://www.scirp.org/html/htmlimages\1-5300658x\77516eb8-1c82-42eb-b343-f8d059286705.png)
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
![](https://www.scirp.org/html/htmlimages\1-5300658x\71cfcf62-1e34-4b9b-b966-c4f5f23a24db.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\b54da3a8-22e2-4b5c-9e8e-57b5e824a789.png)
Hence by definition of
and Abel's lemma we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\6ffc0633-244e-43b0-8ff5-3548a89d3c46.png)
so that
(3.5)
Hence,
(3.6)
Let
. Then ![](https://www.scirp.org/html/htmlimages\1-5300658x\b7bb8c59-bbea-42a1-87b7-d003bcfb6972.png)
so that
![](https://www.scirp.org/html/htmlimages\1-5300658x\d1962f0c-1b65-4341-87b4-18efad5a8f3b.png)
so that we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\bc6482cc-7019-4a0c-be15-4b590d658565.png)
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,
![](https://www.scirp.org/html/htmlimages\1-5300658x\2c5d0b45-c6b1-4f9e-898c-0d4e3783fcb8.png)
We decompose this integral
![](https://www.scirp.org/html/htmlimages\1-5300658x\06fdb1fb-91bf-41a7-bd3c-1c5ac24a7ab1.png)
where
![](https://www.scirp.org/html/htmlimages\1-5300658x\00bbea4e-0e5d-4733-b9e1-fde111b66564.png)
where
![](https://www.scirp.org/html/htmlimages\1-5300658x\016b4155-664f-403d-9b1d-adf71d021e31.png)
Immediately, from (3.3) we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\37cbf60f-41a2-4a22-b591-4907d1914e41.png)
By the change of variable
, we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\544ceca2-3441-4d3d-9430-0472546ca334.png)
so that
![](https://www.scirp.org/html/htmlimages\1-5300658x\3f04a402-9dfa-4c76-a7b9-33565663ad8f.png)
Let
![](https://www.scirp.org/html/htmlimages\1-5300658x\ae3791a7-9ca3-4cbb-aedc-984b7ad8273e.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\dd0ab86d-3b7d-47ff-bf25-ea7eecdfa77a.png)
Let
(3.7)
Clearly,
![](https://www.scirp.org/html/htmlimages\1-5300658x\dcbd400c-5856-429a-b43e-aee90e24cb47.png)
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
![](https://www.scirp.org/html/htmlimages\1-5300658x\9e432a7e-b699-4ba6-ae4c-f6e48fdac157.png)
Hence by definition of
and Abel’s lemma we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\68e9c039-1f37-4f13-b904-4a1e200a8938.png)
so that
(3.8)
Hence
(3.9)
Let
Then
;
so that
![](https://www.scirp.org/html/htmlimages\1-5300658x\cbbc37dd-cabf-4b4f-aa51-6de7c6071d44.png)
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
.
![](https://www.scirp.org/html/htmlimages\1-5300658x\34f08f4c-88a3-4eb9-a301-9f58fb4ef01a.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\aed301b7-2209-4665-88bc-f7081be885f0.png)
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,
![](https://www.scirp.org/html/htmlimages\1-5300658x\0bccb057-c7a3-42b1-92c1-98917839a073.png)
We decompose this integral
![](https://www.scirp.org/html/htmlimages\1-5300658x\2916b8c0-4246-4afe-9911-7c2b2633ae88.png)
where
![](https://www.scirp.org/html/htmlimages\1-5300658x\2c807a70-371c-40ab-a239-af0fd7dd821b.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\33fa5cee-24a5-478a-898c-bd317f7c422e.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\2cafefb7-47dd-43fa-b61c-44dc3a1eec3d.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\fea10f9b-68a7-46fb-8e39-3818f755cb02.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\5552055d-aa5b-465b-a647-42dfa3d8e39a.png)
Conjecture 1.
![](https://www.scirp.org/html/htmlimages\1-5300658x\4c683c36-14ef-4879-95c9-9941aa715216.png)
We now estimate
.
![](https://www.scirp.org/html/htmlimages\1-5300658x\88c68860-3710-4b1d-b113-88df5beb1499.png)
But consider
![](https://www.scirp.org/html/htmlimages\1-5300658x\5aa54e49-97a0-4d04-a7d2-9c3ae10f3433.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\7fd2d1a6-ab01-4c7e-81d3-6d147e7ff526.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\c01921f8-92ae-47c1-a789-1c5c409e4052.png)
so that uniformly ![](https://www.scirp.org/html/htmlimages\1-5300658x\8a0a08ce-25a2-45a4-8320-447d3b09842e.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\dae183bb-05bd-4ce0-8b54-79ef2436137e.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\4901523c-644e-4a1b-9221-5cdad8a3d55a.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\02ca1dc7-55e6-4525-901a-ca137e4d895b.png)
Hence for
uniformly ![](https://www.scirp.org/html/htmlimages\1-5300658x\a7d3aa72-7a0f-4e24-84df-25eca95fd441.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\fb6e6a5e-48bb-4169-a1b1-bc5f23ada8d9.png)
So by (100) and (101) for ![](https://www.scirp.org/html/htmlimages\1-5300658x\d7a67419-5647-4e1b-a8e7-18b9e8c9c8ee.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\5e02700f-552d-4217-ad2c-a5cc0a99a4e7.png)
But since
and since
, we have by the inequalities immediately below (3.2) for ![](https://www.scirp.org/html/htmlimages\1-5300658x\82c89577-8b45-47f3-bb58-63243172e477.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\7b0b9ad2-0c81-4357-8a60-6ccc8fce4c0d.png)
But
![](https://www.scirp.org/html/htmlimages\1-5300658x\97bb5936-e1a8-4fd3-a358-afc2b292adb5.png)
Hence for ![](https://www.scirp.org/html/htmlimages\1-5300658x\412a54d5-f47a-43b9-9c62-0b8a34797840.png)
(3.10)
By a change of variable
we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\c39ba011-70f1-49ee-9d4c-37b4741e0feb.png)
However,
![](https://www.scirp.org/html/htmlimages\1-5300658x\82281439-0bf9-41e6-b3cb-17cdfc5ff637.png)
Let
![](https://www.scirp.org/html/htmlimages\1-5300658x\fc497fa8-dfa5-480d-b2bb-f9b185f35889.png)
Hence, if
,
![](https://www.scirp.org/html/htmlimages\1-5300658x\f6ebec02-6345-4fc7-9967-ed4b55abbf74.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\fa7212b4-f04d-42ac-a6fb-204b3ee859c1.png)
By (3.5) we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\9fd02e97-cd2d-4770-931f-dfb9b332f207.png)
and
![](https://www.scirp.org/html/htmlimages\1-5300658x\12f10396-ef81-44a6-a485-fe9e131ebb8a.png)
But
![](https://www.scirp.org/html/htmlimages\1-5300658x\9a70f9b3-0a87-4ee8-9f54-5e826cdd98a3.png)
Hence, if
,
![](https://www.scirp.org/html/htmlimages\1-5300658x\9a4255fe-122f-44fb-8497-06019e5df191.png)
so that
![](https://www.scirp.org/html/htmlimages\1-5300658x\2a9340c6-931f-40bd-8ae7-f0f0563c6009.png)
where
![](https://www.scirp.org/html/htmlimages\1-5300658x\ff3f0504-301b-4661-ade7-523dd965d702.png)
But by (3.6) we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\d6b6f05e-bc08-4fd3-b4cf-98b5748e53ec.png)
so that
![](https://www.scirp.org/html/htmlimages\1-5300658x\1e7f092d-b5b1-481a-82e7-154f504be5a8.png)
so that
![](https://www.scirp.org/html/htmlimages\1-5300658x\08ec3da3-c99b-4217-aaf0-267b249941f1.png)
Let
![](https://www.scirp.org/html/htmlimages\1-5300658x\f69dbde1-7c21-410c-ae02-4fa20def85a4.png)
By Theorem 272 in [5]
![](https://www.scirp.org/html/htmlimages\1-5300658x\587090ac-cc48-43b8-9fbb-d74c6f771515.png)
so that
, since we assume
.
Hence
![](https://www.scirp.org/html/htmlimages\1-5300658x\7990b74d-6f98-4109-b552-cd27443d3496.png)
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,
![](https://www.scirp.org/html/htmlimages\1-5300658x\cb0b2bcf-54c9-42b8-bd97-06e5ef1c7d62.png)
We decompose the integral
![](https://www.scirp.org/html/htmlimages\1-5300658x\79fad702-d521-445b-954e-f4689c705f6e.png)
Conjecture 2.
![](https://www.scirp.org/html/htmlimages\1-5300658x\2cddfa48-242d-45fa-be87-bd565e2a861d.png)
infinity in some suitable sequence.
From (3.10) we have for ![](https://www.scirp.org/html/htmlimages\1-5300658x\89f0aae4-cbe8-4d7b-ba6e-8257596ba209.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\8019bf41-3ec9-4356-8a3e-aa7e40496111.png)
By change of variable
we have
![](https://www.scirp.org/html/htmlimages\1-5300658x\126be794-f8db-4705-87d1-775d3d91f1b1.png)
However,
![](https://www.scirp.org/html/htmlimages\1-5300658x\426db10a-e537-4411-85dd-dba25835c05f.png)
Let
![](https://www.scirp.org/html/htmlimages\1-5300658x\110dee7c-ef19-43e1-9d6f-cdf35750a178.png)
Hence, if ![](https://www.scirp.org/html/htmlimages\1-5300658x\c9469e10-3ddf-4cd7-81f4-871f5c88300a.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\a247200b-77c8-4855-aaa7-d021d6d5616f.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\e6a43c08-7e44-46af-932a-0b9e5c78f630.png)
Clearly,
![](https://www.scirp.org/html/htmlimages\1-5300658x\dff7242f-d514-4c97-87f9-5cc948ebe018.png)
So that by (3.8)
![](https://www.scirp.org/html/htmlimages\1-5300658x\91fa3324-eb13-49a4-89d2-2f7945e0d7de.png)
and
.
![](https://www.scirp.org/html/htmlimages\1-5300658x\44090b72-8101-4338-b889-d4ebf9dc02e5.png)
Hence, if
,
![](https://www.scirp.org/html/htmlimages\1-5300658x\c7b5227d-ea5e-4c0d-84df-50e978db5683.png)
so that
![](https://www.scirp.org/html/htmlimages\1-5300658x\1a25bb09-9b5d-4432-84ad-797f059ceb05.png)
where
![](https://www.scirp.org/html/htmlimages\1-5300658x\15967151-fda5-4862-860d-a011d58ec8fb.png)
But by (3.6)
![](https://www.scirp.org/html/htmlimages\1-5300658x\0e2c93e9-1e05-4395-8ad9-d9bad1a3dff3.png)
so that
![](https://www.scirp.org/html/htmlimages\1-5300658x\c5cf5987-f907-46b0-9b72-4bc713cb2ee7.png)
Hence
![](https://www.scirp.org/html/htmlimages\1-5300658x\948e9ecf-ee8a-4a45-bb3a-5f77caee471c.png)
![](https://www.scirp.org/html/htmlimages\1-5300658x\f4c934f0-8204-4149-8fed-22c1e3e63d08.png)
Hence
![](https://www.scirp.org/html/htmlimages\1-5300658x\0d43eb9c-d3f1-4f52-8f26-4e43c720cb63.png)
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
![](https://www.scirp.org/html/htmlimages\1-5300658x\fee6e284-aec5-417f-bf8f-75da499f3fc4.png)
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
![](https://www.scirp.org/html/htmlimages\1-5300658x\b7a5e814-9901-4400-bd4a-b08ba07fa305.png)
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
![](https://www.scirp.org/html/htmlimages\1-5300658x\5686e82c-b173-4c84-a073-0ec0bc14f0b1.png)
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.