Proceedings of the International Congress of Mathematicians
Vancouver, 1974
The Distribution of Sequences in Arithmetic Progressions
C. Hooley
1. Introduction. The subject of prime numbers in arithmetic progressions has
certainly been of interest since Legendre enunciated his celebrated theorem on
ternary quadratic forms in 1785, his demonstration having assumed that there
exist primes in any arithmetic progression whose terms are coprirne to the common
difference. Although Gauss subsequently established Legendre's theorem unconditionally by other means, Legendre's method was vindicated by Dirichlet when the
latter proved in 1837 the famous theorem to the effect that admissible arithmetical
progressions—that is to say, those whose terms are coprirne to the common difference—contained infinitely many primes. Subsequently it has been realized that
Legendre's theorem is but one of many interesting arithmetical theorems that are
related to the theory of primes in arithmetic progressions, there being several important unproved conjectures for whose solution we await further developments in
the latter theory. In like manner there are important applications to the theory of
numbers of results about the distribution in arithmetic progressions of sequences
other than that of the primes.
After briefly summarizing the more important earlier work in the subject in order
to put the main substance of our survey in historical perspective, we propose to
discuss some recent developments which not only have already had some application but which also seem to be of interest when viewed purely as part of the theory
of sequences in arithmetic progressions. To this end it is appropriate to introduce
at once the customary notation
% (x; a9 k) =
2
1,
0 (x; a9k) =
p&x\p=a mod k
E
log p9
p^x\p=a mod k
where the summations are over positive primes/?, it being familiar that it is normal©1975, Canadian Mathematical Congress
357
358
C. HOOLEY
ly immaterial whether results are expressed in terms of it(x\ a9 k) or 0(x; a9 k).1
We also always assume that (a, k) = 1, and define E(x; a, k) by
E(x; a9 k) = 0(x\ a9 k) — xj<j>(k).
Dirichlet's result, which may be expressed in our notation as 6(x; a9 k) -» oo as
x -> oo, was followed by the asymptotic formula
(1)
0(x',a9k)~x/<j>(k)
obtained by de Vallée Poussin in 1896. Later work centred round the two questions
of the degree of accuracy with which (1) represented 0(x; a, k) and of the uniformity
of (1) with respect to k. The Siegel-Walfisz theorem, for example, gave
(2)
E(x \a9k) = 0(x exp( - c Vïôgx))
4
for k < log' *, while a result due to Titchmarsh gave
E(x;a9k) = 0(xl/z log2*)
on the supposition of the extended Riemann hypothesis (to which, for brevity, we
hereafter refer as E.R.H.). Here we should remark in passing that it never seems to
have been subsequently noticed that Titchmarsh's result can be easily improved to
E(x; a9 k) = 0(xU2 logxlog {2x1/2/0(*O})
(^ < *1/2)>
in which form it leads to the validity of (1) for large values of x whenever k is
less than about x1/2/log x. These results were augmented and complemented in the
middle of the past decade by the important Bombieri-Vinogradov theorem, which
asserted that, for any assigned positive constant A9 there exists a positive constant
B such that2
S
mQ
bd
\E(y; a,k)\ <x log-^4*
0<a£k;y£x '
'
for Q < x1/2log~Bx. It would be superfluous for us to dwell here on the importance
of this theorem in view of Professor Chandrasekharan's comments on Professor
Bombieri's work in his article in these PROCEEDINGS. Suffice it then to mention that
the theorem asserts, roughly speaking, that (1) holds almost always for values of
k nearly up to xin9 the consequence being that it has often proved to be an effective
unconditional substitute for Titchmarsh's result in applications.
The results so far described lose all significance when k > x1/2. If, however, we
waive the requirement of asymptotic equality and are content with meaningful
upper bounds, then we have the useful Brun-Titchmarsh theorem, which can be
proved in the form
(3)
% (x\ a9 k) < (2 + e)x/{<f>(k) log (2x/k)}
by an easy application of Selberg's sieve method. Thus, for values of k as large as
xl~v9 the sum %(x\ a, k) is of the expected order of magnitude x/{<f>(k)log x}9
although the constant implicit in the statement becomes large as t] becomes small.
The result does not, however, reflect the anticipated size of %(x\ a, k) for larger
values of k such as x log~rx.
*0r the associated </>(x; a, k) whose use here we avoid.
The condition (a, k) = 1 remains implicit in all summations involving k.
z
DISTRIBUTION OF SEQUENCES IN PROGRESSIONS
359
2. The Brun-Titchmarsh theorem for the larger values of k. We conclude from the
above summary that the extension of Bombieri's theorem to cover values of Q as
large as x1"5 is a desideratum in the theory. In the absence of such a generalization,
it is therefore of interest to investigate whether (3) can be substantially improved for
nearly all values of A: up to a limit almost as large as x. Recently, in this direction,
the author [2] proved the following
THEOREM 1. Let a be afixednonzero integer. Then almost all numbers k have the
property that
% (x; a, k) < (4 + e)x/{<f>(k) log k]
for all values ofx exceeding k log34/:.
If the exponent 34 in log k is replaced by a larger number B9 then the method used
also leads to sharp estimates for the measure of the exceptional set of k in which
the inequality fails.
To interpret the result, we should note in particular that when x is large we obtain
TC(X; a, k) < (8 + e)x /{<j)(k) log x} for nearly all k between xl/2 and x log-34*, an
inequality in which 8 can be replaced by 5 by the method given in [2] and an earlier
paper [1]. Thus, in a suitable sense, we have achieved an inequality for %(x\a9 k)
of the required order of magnitude for values of A: as large as x log-34*.
A few comments on the method are in place. In Selberg's upper bound method
we use a nonnegative function of n which is equal to 1 when 77 is a prime number
exceeding £ and which is of the form ]£d\npd9 where pd = 0 if d > £2. Thus, by the
usual reasoning,
(4)
*(*;«,*) S - Ì +
K>
s
£p« = -f
n^x; n=amodk din
in e
K
+
ir£
•*£ + <*&,
K> (d,k)=l
a
in which we are constrained to limit £ to (x/k) ~ in order to obtain a meaningful
result, there being a consequent diminution in the efficacy of the ensuing inequality
when k is large. Yet we would suspect that a better result is true because the
estimate 0(£2) for the remainder term in (4) is probably too large. We therefore
consider, along the lines of Linnik's dispersion method, an expression of the type
Q<k^2Q
Jt^2Q \n^x; n=amod k d\n
K (d,k)=l
«
/
in order to show that the remainder term is usually smaller and that therefore
larger values of £ can usually be chosen. We should notice here that pd must be
independent of k so that the minimal property inherent in Selberg's method cannot
be retained. This, however, is not important, and we have here an example of the
enveloping sieve—a term due to Linnik to express a similar application of the idea
by the author to the Hardy-Littlewood problem about numbers as the sum of
two squares and a prime.
3. Theorems of the Barban-Davenport-Halberstam type. Results about larger
moduli k are also supplied by theorems of the Barban-Davenport-Halberstam type,
360
C. HOOLEY
which are concerned with the adjusted variance
G(x9k) = S
E2(x\a,k)
0<<zêk
and the sum
H(x, Q) = S G(x9 k).
The fundamental result in this theory is the following theorem, due essentially
to the independent work of Barban and of Davenport and Halberstam. This is
stated here in the improved form given by Gallagher.
THEOREM
2. For Q ^ x and for any positive constant A, we have
H(x, Q) = 0(Qx log x) + 0(x2 log^x).
The main importance of the theorem of course lies in its assertion that almost
all moduli k between x l o g ~ ^ and x are such that (1) holds for almost all residue
classes a, modulo k.
Previously, apart from earlier large sieve results to prime moduli which were the
harbingers of this theorem, the only other known theorem of this type was the conditional estimate
(5)
E
E2(x; a9 k) = 0(x log4*)
that was obtained by Turân on E.R.H. Although the latter is weaker in some respects than Theorem 1, the author [3] has noted that it can be utilized in combination with Gallagher's method in order to obtain
THEOREM
3. For Q ^ x9 we have
H(x9 Q) = 0(Qx log x) + 0(x*<2 log3 x)
on E.R.H.
Thus (5) can certainly be improved on average for k > x1/2. Later, however, we
shall see that such improvements can be effected in a more precise sense over certain
ranges of k.
In 1970 Montgomery [10] obtained a striking improvement in Theorem 2 in which
the upper bound was replaced by an asymptotic equality. This work had, however,
been partially anticipated by Barban, who had enunciated the result for the special
case Q = x. Montgomery's results, as improved and augmented by his later work,
are given by
THEOREM
(i)
(ii)
on E. R. H.
4. For Q ^ x9 we have
H(x9 Q) = Qx log Q + 0(Qx) + 0(x2 log"**),
H(x9 Q) = Qx log ß + 0(Qx) + 0(xi<*+*)
The proof depends intrinsically on the equation
G(x9 k) =
S
X'•2
62(x; a9 k) - - f ^ + 0(x2 log-^x),
DISTRIBUTION OF SEQUENCES IN PROGRESSIONS
361
in which the first sum on the right-hand side is equal to
(6)
2 log2/?+ 2 2
p^x
(1)
2
OKa^k p'<p^x\p=p'^a
log/? log/?'
mod k
= x log x + 0(x) + 2
2
log/? log//,
p-p'-ik^x
the summation in the final sum being over /?, p'9 and positive integers /. Montgomery then writes this final sum as
(8)
2 {J(x,lk) + K(x,lk)}9
mx/k
where J (x, m) is the usual heuristic estimate for the sum
2
log/? log/?'.
The contribution of J(x9 Ik) to the problem through (7) and subsequent summation over k is then in principle easy to assess, whereas the effect of K(x9 Ik) is
handled by Lavrik's theorem on the mean-square value of K(x9 m).
Montgomery's treatment lies deep because Lavrik's theorem is of the same order
of sophistication as Vinogradov's theorem. The author [3], however, has developed
an alternative proof that depends only on the comparatively simple large sieve
inequality and theSiegel-Walfisz theorem (2). In sketching the ideas behind this
proof, we note first that Theorem 2 implies that it suffices to estimate the contribution to H(x9 Q) due to values of k exceeding a number Qt that is not much smaller
than Q. Summation of (7) over these values of A: then gives rise to a sum of the form
2
log/? log/?'
p-p'=lkêx
in which the variables of summation are /?, p'9 l9 k and in which / in particular is
subject to the condition / < x/Q\. Since this sum possesses a certain symmetry in
terms of /, k9 it is then possible to utilize in reverse the transformation that took (6)
into (7) save that / and k exchange rôles. The modulus / in the counterpart of the
final sum in (6) being small, the estimations can then be completed by appealing
to (2).
Theorem 4 by no means exhausts the potentialities of either method, there being
several applications to which we shall presently refer. Since, however, the two
methods differ in character, it frequently happens in any given situation that the
balance of advantage lies decisively with one of the methods.
The author's method, for example, leads to the following improved version of the
second part of Theorem 4 [4].
THEOREM 5. On E. R. H. we have
H(x9 Q) = Qxlog Q + 0(Qx) + 0(x3/2+*)
for Q^x.
It also leads to
THEOREM 6. We have
2 £ 2 E\x\ a9 Jfc)' = o(Q*'2x*/2 log3/2x) + 0(x* log-^jc)
362
C. HOOLEY
provided that Q/x -> 0 as x -» oo.
These results when considered together are tantalizing in that they suggest that
(9)
G (x9 k) ~ x log k
and that
E(x; a, k)
{(* log kW (*)}I'«
may have a distribution function, subject to any obvious qualifications that may
have to be made. Yet the evidence supplied is weakened because in the theorems
quoted so far the value of x remains constant with respect to the variables of
summation. It is, therefore, of interest that further supporting evidence for such
conjectures is supplied by the following theorem, which can be derived by a development of Montgomery's method [6]. This is quoted in conditional form for
effect, although the same method leads to a much weaker unconditional version.
THEOREM 7. On E.R.H. we have
(i)
2 bd
G(y9k)~Qx\ogQ
provided that x 4/5+e < Q <L x;
(ii) almost all numbers k have the property that
G(x9 k) = xlogk + 0(xlogi/2k)
for all x such that k ^ x g k*n+£.
This theorem should be compared with Uchiyama's interesting bound [11]
2 2
bd E2(y;a9k) = 0(ß*log 3 *) + 0(x2 log-**),
k^Q 0<a^k
l^yûx
an improved form of which is given by the following
THEOREM 8. We have
2 2 bd E2(y; a, k) = 0(Qx log x)9
ifx log-i* g Q ^ x9
k Q 0<a k a y
- - -*
= 0(Qx log x (log log x))\ ifx l o g " ^ < Q < x log-**.
Here it would be desirable to discover whether we could dispense with the factor
(log log x)2 in the second estimate.
Before quitting the subject of the Barban-Davenport-Halberstam theorem, we
remark on the apparent anomaly whereby theorems of this type have so far only
been obtained for values of k larger than x1/2 while the asymptotic formulae described in §1 are only significant for values of k less than x1/2. Modest progress for
the smaller values of k can, however, be made with theorems of the type considered
in this section. We can prove [7], for example, the following theorem, which is consistent with the conjecture (9).
THEOREM 9.
On E.R.H. we have
j fl';*> dt = 0Gog*logifc).
DISTRIBUTION OF SEQUENCES IN PROGRESSIONS
363
We can also obtain an extension of Turân's estimate (5) that is related to Theorem 8 [8].
THEOREM
10. On E.R.H. we have
2
bd E2 (y; a9 k) = 0(x log4*)
for k g x.
4. Other sequences. We end with a brief discussion on the application of these
ideas to other sequences.
In considering possible generalizations of the Barban-Davenport-Halberstam
theorem to other sequences, we should observe that the original form of the
theorem implies a weakish form of the prime number theorem for arithmetic progressions. Likewise a sequence cannot possess a Barban-Davenport-Halberstam
property unless it is well distributed among arithmetic progressions with small
moduli. However, it can be shown that an analogue of the theorem always holds
provided that this obvious necessary condition obtains [5].
The Bombieri-Vinogradov theorem, on the other hand, is not susceptible to an
analogous generalization unless additional stringent hypotheses about the sequence
are made. Here we confine our remarks to the square-free numbers, which perhaps
constitute the case next in interest after the primes.
Let S(x; a9 k) be the number of square-free numbers not exceeding x that
are congruent to a mod k. Then Prachar has obtained an asymptotic formula for
S(x; a, k) that is significant forfc ^ x2/3~e9 while Orr has subsequently derived a
Bombieri-Vinogradov type theorem for S(x; a, k) in which, however, the range of
significance is still limited to values of k not exceeding about x2n. Although
Prachar's formula is almost certainly true for k ^ *1_e, the problem of extending
the range of validity beyond k = x2n seems to present considerable difficulty.
Partial progress, however, has been made by the author [9] by means of
Let Q ^ x*n~B. Then, for a positive proportion of moduli k satisfying
Q < k < 2g, we have
THEOREM 11.
for all residue classes a mod k coprirne to k,
References
1. C, Hooley, On the Brun-Titchmarsh theorem. I, J. Reine Angew. Math. 255 (1972), 60-79.
MR 46 #3463.
2.
; On the Brun-Titchmarsh theorem. II, Proc. London Math. Soc. (3) 30 (1975), 114-128.
3.
, On the Barban-Davenport-Halberstam theorem. I, J. Reine Angew. Math, (to appear).
4.
, On the Barban-Davenport-Halberstam theorem. II, J.London Math. Soc. (2) 9 (1975),
625-636.
5.
, On the Barban-Davenport-Halberstam theorem. Ill, J. London Math. Soc. (to appear).
364
C. HOOLEY
6.
, On the Barban-Davenport-Halberstam theorem. IV, J. London Math. Soc. (to appear).
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, On the Barban-Davenport-Halberstam theorem. V, Proc. London Math. Soc. (to
appear).
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, On the Barban-Davenport-Halberstam theorem. VI, J. London Math. Soc. (to
appear).
9.
, A note on square-free numbers in arithmetic progressions, Bull. London Math. Soc.
8 (1975).
10. H. L.Montgomery, Primes in arithmetic progressions, Michigan Math. J. 17(1970), 33-39.
MR 41 #1660.
11. S. Uchiyama, Prime numbers in arithmetic progressions, Math. J. Okayama Univ. 15 (1971/
72), 187-196. MR 47 #8464.
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