Idea Transcript
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 2. April 1989
PRIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI. III E. BOMBIERI, J. B. FRIEDLANDER, AND H. IWANIEC
1.
INTRODUCTION
For q a positive integer and a an integer prime to q, we let n(x;q,a)=
L
p5,x p=a(modq)
count those primes up to x in the arithmetic progression a modulo q. One expects these arithmetic progressions to contain their due proportion of primes, namely n(x; q ,a) '" n(x)! 0, and A > O. Suppose P satisfies (A,) and (A 2 ). Then there exist constants c and Ao such that provided P satisfies (A~) and
Q2 X -'LAo < N log njIog z . By Lemma 1, E(x ,~) ~
1 E -:Ej(x ,~), 6
j=1 ]
where E j is a sum similar to E but with A(n)/Iogn replaced by tj(n). It clearly suffices to prove that, for 1 ~ j ~ 6 , and for ~ contained in (Q/2, Q] , (3.2)
1 2
Ej(X ,~) ~ {Kj«(J - 2) xL
-I
2 '" + O(xL -3 (Ioglogx»} ~
qEI!'
I 4>( ) q
+ O(xL -A ),
and the main theorem then follows with 6 I K=E-:K j
.
j=1 ]
Throughout the proof we shall be dealing with various sums ~ n) •.•• In, I . These sums will always be restricted to n I ' ... ,n j that satisfy the constraints
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PRIMES IN ARITHMETIC PROGRESSIONS
n 1 •.• n j '" x, n 1 ~
Z , ...
219
,n j ~ z. Any additional constraints will be specifi-
cally mentioned but these will not. We also adopt the convention that all implied constants are absolute, unless specific mention is made to the contrary, with the unique exception that in error terms of the form O(xL -A) the implied constants (0 or «) may depend on a and A. 4.
THE CASES
Let (4.1 )
,,(z)
~
11)x,z;q,a)=
1~ j
~
3
1 1-¢(q)
nl···nj=a(q)
,,(z)
~
1.
(nl···n J ,q)=1
In [4], we dealt with the cases j ~ 3 by using ([4, Lemma 2]) results for the divisor function r j in arithmetic progressions. These results cannot be immediately applied to D.j because of the condition (n i ' P (z)) = 1 and in [4] this condition was removed via a sieve-theoretic fundamental lemma. In the present case, our choice of z is much larger and we first reduce to (n i , P(w)) = 1 with
w = Q2X-IL Ao
(4.2)
(where Ao is as in Theorem 1) by means of the Buchstab identity: (4.3)
D.j(x,z;q,a)=D.j(x,w;q,a)-
L
D.jp(x,p;q,a),
w5,p X I / 7 , we get, on combining Lemmata 2
-
L...J (Ad - Ad )Td(q) «xL
~D
-I 3-s",
s
~.~q
(d .q)=1
and by (4.2) this is easily seen to be «xL -1(0 - !)2 E where «5(A) may depend on Ao and hence on A. Combining this with (4.6) and (4.7) we have (4.8)
'"
.
L...JIA/x,w,q,a)l«xL
qE.
-I
1 2",
I
L...J "'( )
I
+ X 7/8
qE .(1!ifJ(q))
(o-!) L...JifJ() +x qE. q
I-J(A)
+ Xl-J(A)
,
.
Combining (4.8) with (4.3) and (4.5) we get (3.2) for j = 1,2,3. We remark that (4.8) is obviously easy to improve but that this bound will in any case occur later (in §7) in a way that is essential from the point of view of our method.
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222
E. BOMBIERI, J. B. FRIEDLANDER, AND H. IWANIEC
5.
4
THE CASES
~
j
~
6:
A SUBDIVISION
Definition. The sequence (nl ' ... ,n j ) is "exceptional" of type B, if it can be partitioned into r subsets whose products d l , ••• ,d" d l ~ d2 ~ ... ~ d" satisfy one of the following:
( B4 )
d l ~ I1d2 '
r = 4,
= 5,
r
(B5 )
(B6 )
d3 ~ I1d4 '
r
= 6,
with 11 = w 2 • A sequence (n l ' ••• ,n) is "regular" if it is not exceptional of any type. We may by symmetry restrict to sequences with n l ~ n 2 ~ ... ~ nj • A regular sequence is called "smoothable" if j = 4, n l ~ n 2 ~ n3 ~ n 4 , and one of the following holds:
(5.1 ) or X
(5.2)
1/6-6
<
n4