Unexpected Irregularities in the Distribution of
Prime Numbers
A N D R E W GRANVILLE*
D e p a r t m e n t of Mathematics
University of Georgia
Athens, Georgia 30602, USA
In 1849 t h e Swiss mathematician E N C K E wrote to G A U S S , asking whether he had
ever considered trying to estimate ir(x), t h e number of primes up to x, by some
sort of "smooth" function. On Christmas Eve 1849, GAUSS replied t h a t "he h a d
pondered this problem as a bof' and h a d come to t h e conclusion t h a t "at around
x, the primes occur with density 1/loga;." Thus, he concluded, ir(x) could b e
approximated by
Li(z)
r—
J2
log*
Ioga:
+ log2x + 0
log 3 a;
Comparing G A U S S ' s guess t o the best d a t a available today (due to D E L E G L I S E
and R I V A T ) , we have:
X
8
10
109
10 l ü
IO 11
IO 12
IO 13
IO 14
IO 15
IO 16
IO 17
IO1«
ir(x)
[U(x)-7r(x)}
5761455
50847534
455052511
4118054813
37607912018
346065536839
3204941750802
29844570422669
279238341033925
2623557157654233
24739954287740860
754
1701
3104
11588
38263
108971
314890
1052619
3214632
7956589
21949555
The number of primes, 7T(:E), u p to x.
This d a t a certainly seems to support G A U S S ' S prediction, because Li(x) — ir(x)
appears t o be no bigger t h a n a small power of TT(X). In 1859, R I E M A N N , in a
now famous memoir, illustrated how t h e question of estimating ir(x) could be
*)
The author is an Alfred P. Sloan Research Fellow; and is also supported, in part, by the
National Science Foundation.
Proceedings of the International Congress
of Mathematicians, Zürich, Switzerland 1994
© Birkhäuser Verlag, Basel, Switzerland 1995
Distribution of Primes
389
turned into a question in analysis: Define Ç(s) := Y2n>i n _ S ^ o r ^ e ( s ) > 1> a n d
then analytically continue Ç(s) to t h e rest of t h e complex plane. We have, for
sufficiently large x,
xb~E < max \w(y) - U(y)\
< xb+E
where b : =
sup
/3.
(1)
The (as yet unproven) Riemann Hypothesis ( R H ) asserts t h a t b = 1/2 (in fact
t h a t /? = 1/2 whenever C(ß+i"y) = 0 with 0 < ß < 1) leading t o t h e sharp estimate
ir(x)
= Li(a0 + O(a: 1 / 2 loga:) .
(2a)
It was not until 1896 t h a t H A D A M A R D and D E L A V A L L é E - P O U S S I N independently
proved t h a t ß < 1 whenever ((ß + ry) = 0, which implies the prime
theorem: t h a t is, G A U S S ' S prediction t h a t
TT(X) ~
Li(x)
number
Ioga:
In 1914, L I T T L E W O O D showed, unconditionally, t h a t
*(*) - u(x) = n± («v» logl^gx),
(2b)
the first proven "irregularities" in t h e distribution of primes. 1
As G A U S S ' S vague "density assertion" was so prescient, CRAA4ÉR [4] decided,
in 1936, to interpret GAUSS's statement more formally in terms of probability
theoiy, t o t r y t o make further predictions about t h e distribution of prime numbers:
let Z2,Zs,...
be a sequence of independent random variables with
Prob(Zn = l) = ^ —
log 11
a n d P r o b ( Z n = 0) = 1 -
Let S be the space of sequences T — z2,23,...
KT(X)
= YI
1
log 77,
and for each x > 2 define
Zn
'
2<n<x
T h e sequence P = ir2,7T3,... , where irn = 1 if and only if n is prime, belongs to S.
C R A M E R wrote: "In many cases it is possible to prove that, with probability 1, a
certain relation R holds for sequences in S ...
Of course we cannot in general
conclude that R holds for the particular sequence P, but results suggested in
this way may sometimes afterwards be rigorously proved by other methods." For
example C R A M E R was able t o show, with probability 1, t h a t
m^M^-LKy)!-^.1^^,
1
f(x-)
f{x) — tì±{9Ìx)) means that there exists a constant c > 0 such t h a t /(œ_j-) > cg{x+) and
< — cg(x-) for certain arbitrarily large values of x±.
390
Andrew Granville
which corresponds well with the estimates in (2); and if true for T — P implies
RH, by (1).
GAUSS'S assertion was really about primes in short intervals, and so is best
applied to tr(x + y) —ir(x), where y is "small" compared to x. The binomial random
variables Zn are, more-or-less, the same for all integers n in such an interval. If we
take y = À log x so that the "expected" number of primes À in the interval is fixed
then we would expect that the number of primes in such intervals should follow
a Poisson distribution. Indeed, we can prove that for any fixed À > 0 and integer
k > 0, we have
+ A log x) — irT(x) = k} ~ e~x—X
(3)
k\
as X —> oo, with probability 1 for T e S. In 1976, GALLAGHER [11] showed that
this holds for the sequence of primes (that is, for P) under the assumption of
a reasonable "uniform" version of HARDY AND LITTLEWOOD'S Prime k-tuplets
conjecture [14]. This conjecture is the case where we take each fj(x) to be a linear
polynomial in SCHINZEL [22]
HYPOTHESIS H. Let F = {fi(x),f2(x),...
, fk(x)} be a set of irreducible polynomials with integer coefficients. Then the number of integers n < x for which
each | fj (n) \ is prime is
#{integers x < X :
*,(,)
= {Cr +
• * -
TTT(X
°(1),iog|/.W|log IAWI-.log IAWI
* , -
n
p prime
0 - ^ ) / K ) ' .
N
*-
/
/
\
r /
and UJF(P) counts the number of integers n, in the range 1 < n < p, for which
fi(n)f2(n)...fk(n)=0
(modp). 2 - 3
Estimates analogous to (2a) should hold for the number of primes in intervals
of various lengths, if we believe that what almost always OCCLUS in S should also
hold for P. Specifically, if 10 log2 x < y < x then
7TT(x + y) - TTT(X) = Li(z + y) - Li(z) + 0(y^2)
(4)
with probability 1 for T G S. In 1943 SELBERG [23] showed that primes do, on the
whole, behave like this by proving, under the assumption of RH, that
ir(x + y) -ir(x)
-
(5)
Ioga;
for "almost all" integers x, provided y /log x —> oo as x —> oo.
MONTGOMERY [17] has shown that one can deduce estimates about primes
in short intervals by understanding local distribution properties of the zeros of
CM:
2
Elementary results on prime ideals guarantee that the product defining Cp converges if
the primes are taken in ascending order.
3
T h e asymptotic formula proposed here for ITF(X) has a "local part" Cp, which has a factor
corresponding to each rational prime p, and an "analytic part" x/Tli l°ë\fi(x)\This reminds
one of formulae that arise when counting points on varieties.
Distribution of P r i m e s
P A I R C O R R E L A T I O N C O N J E C T U R E ( P C ) . Assume
average number of zeros 1/2 + ry' of ((s)
with 7 < 7 ; < 7 + 27rci'/log(|7| + 1), is
391
R H . For any fixed a > 0, the
"close" to a given zero 1/2 + ry, that is
(6a)
Inspired by t h e work of M O N T G O M E R Y and others, G O L D S T O N [12] showed t h a t
(6a) holds for eveiy fixed a > 0 if and only if for every fixed ß > 0,
.. .
TJ
s \ 2 dx
• ^ ) - r )
^
a
log2T
~ ^ ^ ^ '
(6b)
where ^ ( z ) : = ^ „ m ^ l o g p . Because (6b) is predicted by C R A M E R ' S model, thus
so is P C . D Y S O N predicted an analogous density function for t h e correlation of
7i-tuples of zeros of C( s )> 4 which presumably may b c shown t o be equivalent t o
estimates for primes in short intervals, and thus b e predicted b y C R A M E R ' S model.
C R A M E R ' S model does seem to accurately predict what we already believe t o
be true about primes for more substantial reasons. 5 To b e sure, one can find small
discrepancies 6 b u t t h e probabilistic model usually gives one a strong indication of
the t r u t h . C R A M E R made one conjecture, based on his model, t h a t does not seem
to be attackable by other methods: if p\ — 2 < p2 = 3 < p 3 = 5 < . . . is t h e
sequence of prime numbers then
max ( p n + 1 -pn)
~ log 2 x.
pn<X
This statement (or t h e weaker 0(log 2 x)) is known as "Cramer's
is some computational evidence to support it:
conjecture";
Pn
Pn+1 ~Pn
(Pn+l - Pn)/log* Pn
31397
370261
2010733
20831323
25056082087
2614941710599
19581334192423
72
112
148
210
456
652
778
.6715
.6812
.7025
.7394
.7953
.7975
.8177
there
Record-breaking gaps between primes, u p t o 10 14
4
Based somewhat surprisingly on the fact that (6a) also describes the distribution of eigenvalues of random Hermitian matrices, which arise in physics as models for various naturally
occurring quanta. The "Fourier transforms" of these pair correlation and 7i-tuple correlation
conjectures for the zeros of Ç(s) are now known to hold in a natural restricted range, under the
assumption of RH (see [17] and [21]).
5
Though HARDY AND LITTLEWOOD [14] remarked thus on probabilistic models: "Probability is not a notion of pure mathematics,
but of philosophy or physics".
6
As has been independently pointed out to me by SELBERG, MONTGOMERY, and PINTZ:
for example, PINTZ noted that the mean square of \ip(y) — y\ for y < ce, is ^» x2b~E (with b as in
(1)), in fact x x assuming RH, whereas the probabilistic model predicts X x logx.
Andrew Granville
392
In 1985 M A I E R [16] surprisingly proved t h a t , despite S E L B E R G showing (5) holds
"almost all" t h e time when y = l o g 5 x (assuming R H ) for fixed B > 2, it cannot
hold all of t h e time for such y. This not only radically contradicts what is predicted
by t h e probabilistic model, but also what most researchers in t h e field had believed
to b e t r u e , whether or not they had faith in t h e probabilistic model. Specifically,
M A I E R showed t h e existence of a constant 6B > 0 such t h a t for occasional, but
arbitrarily large, values of x+ and z _ ,
-B-l
7 r ( z + + l o g * z + ) - * " ( * + ) > (l + A^Jlog^ - s x+
and
?v(x-+logBX-)-TT(X-)
(7)
< (1 - < 5 ß ) l o g ß - 1 :
Outline of the Proof. There are ~ e~1x/ log z integers < x, all of whose prime
factors are > z, provided z is not too large. Among these we have all but ir(z)
of t h e primes < x, and so the probability t h a t a randomly chosen such integer is
prime is ~ e 7 log zj log x. Thus, in a specific interval (x,x-\-y\ we should "expect"
~ e7<Ë> log z/ log x primes, where <£> is t h e number of integers in t h e interval t h a t are
free of prime factors < z. Now if we can select our interval so t h a t $ ^ ^~1y/ log z
then our new prediction is not t h e same as t h a t in (5).
If x is divisible by P = YiP<z P ^ n e n
<Ê> = $(y, z) := # { 1 <n<y
: p\n => p > z} ~ Lü(U)
log z
for y = zu with u fixed (see [3]), where CJ(U) = 0 if 0 < u < 1 and satisfies
the differential-delay equation uu(u) = 1 + J"" u(t) dt if u > 1. Obviously
lim u _ > 0 0 cj(ii) = e - 7 . IWANIEC showed t h a t LJ(U) — e ~ 7 oscillates, crossing zero
either once or twice in every interval of length 1. Thus, if we fix u > B, chosen
so t h a t u(u) > e Y or < e _ 7 (as befits t h e case of (7)), select y = log x and
z — y1/™, and "adjust" x so t h a t it is divisible by P, then we expect (5) to be
false.
To convert this heuristic into a proof, M A I E R considered a progression of
intervals of t h e form (rP, rP -\- y], with R < r < 2R for a suitable value of R.
Visualizing this as a "matrix", with each such interval represented by a different
row, we see t h a t t h e primes in t h e matrix are all contained in those columns j for
which (j, P) = l.
RP + l
{R+1)P+1
RP + 2
{R+l)P + 2
AP + 3
(Ä + 1)P + 3
{R + 2)P + 2
{i,j)th entry:
(R + i)P + j
•••
RP + y
(R+l)P + y
•••
(2R-l)P
{R + 2)P+1
(R + 3)P+1
(2R - 1)P + 1
The "Maier m a t r i x " for ir(x + y) — TV(X)
{2R - 1)P + 2
+y
Distribution of Primes
393
Now, for any integer q > 1, the primes are roughly equi-distributed amongst those
arithmetic progressions a (mod q) with (a, q) — 1: in fact up to x we expect that
the number of such primes
*(*;*«) ~ -±±.
(8)
If so, then the number of primes in the j t h column, when (j, P) = 1, is
*(2X;P,j)-K{X;P,j)
~ ^ p j ^
where X = RP.
To get the total number of primes in the matrix we sum over all such j , and then
we can deduce that, on average, a row contains
${Viz)
1
RP
E (ß(P) log P P ~
y
P
1
l oi gr :^i^7( Pm) ul o. gDPD
P
t Jv y
Mi
7 / N Ï/
v
~ e7<j(u)'logPP
primes. MAIER'S result follows provided we can prove a suitable estimate in (8).
D
In general, it is desirable to have an estimate like (8) when x is not too large
compared to q. It has been proved that (8) holds uniformly for
(i) All q < log 0 x and all (a, q) = 1, for any fixed B > 0 (SIEGEL-WALFISZ).
(ii) All q < y ^ / l o g 2 + E x and all (a,q) = 1, assuming GRH. 7 In fact (8) then
holds with error term O (y/xlog2(qx)).
(iii) Almost all q < v / ï / l o g 2 + E x and all (a,q) = 1 (BOMBIERI-VINOGRADOV). 8
(iv) Almost all q < a;1/2+o(i) w j t n ^^^ _ ^ for ß x e c j a ^ Q (BOMBIERIFRIEDLANDER-IWANIEC, FOUVRY).
(v) Almost all q < x/ log 2+E x and almost all (a, g) = 1 (BARBAN-DAVENPORTHALBERSTAM, MONTGOMERY, HOOLEY).
Thus, when GRH is true, we get a good enough estimate in (8) with R = P 2
to complete MAIER'S proof. However MAIER, in the spirit of the BOMBIERIVINOGRADOV theorem, showed how to pick a "good" value for P (see [8, Prop.
2]), so that (8) is off by, at worst, an insignificant factor when R is a large, but
fixed, power of P (thus proving his result unconditionally).
In [15], HILDEBRAND AND MAIER extended the range for y in the proof above,
establishing that there are arbitrarily large values of x for which (4) fails to hold for
some y > exp ((loga:) 1 / 3_£ ); and, assuming GRH, for some y > exp ((loga;) 1 / 2- ^).
Moreover, they show that such intervals (x,x -f y] occur within every interval
[X,2X].9
It is plausible that (5) holds uniformly if logy/ log logo; —> oo as x —> oo; and
that (4) holds uniformly for T = P if y > exp ((loga;) 1 / 2 "^) (at least, we can't
7
The Generalized Riemann Hypothesis (GRH) states that if ß-\-ry is a zero of any Dirichlet
L-function then ß < 1/2.
8
This result is often referred to as "GRH on average".
9
A far better localization than those obtained in any proof of (2b).
394
Andrew Granville
disprove these statements as yet). We conjecture, presumably safely, that (4) and
(5) hold uniformly when y > xE.
One can show that there are more than x/exp ((logx) CB ) integers x± < x
satisfying the unexpected inequalities in (7). Although this may not be enough to
upset (6b), it surely guarantees that the error term there will not be as small as
might have been hoped. Thus, we should not expect the pair correlation conjecture
to hold with as much uniformity as might have been believed. Evidence that this is
so may be seen in the computations represented by [19, Figures 2.3.1,2,3]: the pair
correlation function for the nearest 106 zeros to the 10 12 th zero fits well with (6a)
for a < .8 and then has larger amplitude for .8 < a < 3; also, the pair correlation
function for the nearest 8 x 106 zeros to the 10 20 th zero fits well with (6a) for
a < 3 and then has larger amplitude for 3 < a < 5.
M A I E R ' S work suggests that CRAMER'S model should be adjusted to take into
account divisibility of n by "small" primes. 10 It is plausible to define "small" to
mean those primes up to a fixed power of log n. Then we are led to conjecture that
there are infinitely many primes pn with p n +i — pn > 2e _ 7 log 2 p n , contradicting
11
CRAMER'S conjecture!
If we analyze the distribution of primes in arithmetic progressions using a
suitable analogue of CRAMER'S model, then we would expect (8), and even
ir(x;q,a) = 5 4 + o f f - V
m
log(gz)] ,
(8')
to hold uniformly when (a, q) = 1 in the range
q<Q = x/logBx,
(9)
for any fixed B > 2. However the method of M A I E R is easily adapted to show that
neither (8) nor (8') can hold in at least part of the range (9): for any fixed B > 0
there exists a constant 6B > 0 such that for any modulus q, with "not too many
small prime factors", there exist arithmetic progressions a± (mod q) and values
x
± £ [(ß(q) log Qi 20(c) log q] such that
n(x+;q,a+)
> (l + 5 f l ) ^ ± 2
and 7r(^_;g,a_) < ( l _ Ä f l ) ? f e l . (10)
The proof is much as before, though now using a modified "Maier matrix" :
10
O n e has to be careful about the meaning of "small" here, because if we were to take into
account t h e divisibility of n by all primes up to y/n, then we would conclude t h a t there are ~
e *x/\ogx primes up to x.
11
It is unclear what the "correct conjecture" here should be because, to get at it with this
approach, we would need more precise information on "sifting limits" than is currently available.
395
Distribution of P r i m e s
RP
(R+1)P
RP + q
(R+l)P + q
{R + 2)P
(R+2)P
(R + 3)P
':
(2Ä - 1)P
(2R -l)P
RP + 2q
{R+l)P + 2q
•••
RP + yq
(R+l)P + yq
•••
(2R-l)P
+q
{i,j)th entry:
(R + i)P + jq
+q
+ yq
The Maier matrix for n(yq; q, a)
The BOMBIERI-VINOGRADOV theorem is usually stated in a stronger form than
above: for any given A > 0, there exists a value B = B(A) > 0 such that
V^ max max n(y\q,a)
-
TT(Z/)
*{q)
<
log X
(11)
where Q = y ^ / l o g x. It is possible [6] to take the same values of R and P in
the Maier matrix above for many different values of q, and thus deduce that there
exist arbitrarily large values of a and x for which
£>*«-$}
»z;
(12)
Q<q<2Q
(<7,a)=l
thus refuting the conjecture that for any given A > 0, (11) should hold in the
range (9) for some B = B(A) > 0. In [10] we showed that (11) even fails with
Q = a;/exp ((A - e)(loglogz) 2 /(loglogloga;)) .
We also showed that (8;) cannot hold for every integer a, prime to q, for
(i) Any q > xjexp ((logz) 1 / 5 - 6 ).
(ii) Any q > a;/exp ((logx)1/3"8) that has < 1.5 log log log ç distinct prime
factors < logg. 12
(iii) Almost any q E (y,2y], for any y > xj exp ((loga:) 1 / 2 _ £ ).
Moreover, under the assumption of GRH we can improve the values 1/5 and 1/3
in (i) and (ii), respectively, to 1/3 and 1/2.
It seems plausible that (8) holds uniformly if log(a;/ç)/log logg —> oo as
q —> oo; and that (11) holds uniformly for Q < x/exp ((loga:) 1 / 2+E ). At least we
can't disprove these statements as yet, though we might play it safe and conjecture
only that they hold uniformly for q, Q < xl~E.
Notice that in the proof described above, the values of a increase with x,
leaving open the possibility that (8) might hold uniformly for all (a, q) = 1 in the
12
Which, by the TURAN-KUBILIUS inequality, includes "almost all" integers.
396
Andrew Granville
range (9) if we fix a.13 However, in [8] we observed that when a is fixed one can
suitably modify the Maier matrix, by forcing the elements of the second column
to all be divisible by P :
1
1
1 + (RP - 1)
1+ ((Ä+1)P-1)
1
1+
1 + 2(RP - 1)
l + 2((fl+l)P-l)
•••
l +
l+
y(RP-l)
y((R+l)P-l)
{(R+2)P-1)
(i,j)th entry:
l+j((R +
i)P-l)
1
:
1
1 + ((2Ä - 1)P - 1)
•••
l+y{{2R-l)P-l)
The Maier matrix for ir(yq; q,l)
Notice that the jth column here is now part of an arithmetic progression with
a varying modulus, namely 1 — j (mod jP). With this type of Maier matrix we
can deduce that, for almost all 0 < \a\ < x/log x (including all fixed a / 0 ) ,
there exist q G (xj log x, 2x/log x], coprirne to a, for which (8) does not hold.
However (8) cannot be false too often (like in (12)), because this would contradict
the BARBAN-DAVENPORT-HALBERSTAM theorem. So for which a is (8) frequently
false? It turns out that the answer depends on the number of prime factors of a:
In [9], extending the results of [2], we show that for any given A > 1 there exists
a value B = B(A) > 0 such that, for any Q < x / l o g ß x and any integer a that
satisfies 0 < \a\ < x and has <C log log x distinct prime factors, 14 we have
(x;q,a)
Q<q<2Q
(q,a)=l
-
n(x)\
mi
<
log X
(13)
On the other hand, for every given A,B > 0, there exists Q < z / l o g ^ z and
an integer a that satisfies 0 < \a\ < x and has < (loglog^) 6 / 5 + e distinct prime
factors, for which (13) does not hold (and assuming GRH we may replace 6/5 + e
here by 1 + e).
Finding primes in (x, x + y] is equivalent to finding integers n < y for which
f(n) is prime, where f(t) is the polynomial t-\-x. Similarly, finding primes < x that
belong to the arithmetic progression a (mod q) is equivalent to finding integers
n <y := x/q for which f(n) is prime, where f(t) is the polynomial qt + a. Define
the height h(f) of a given polynomial f(t) — Y2i c%k% to be h(f) := > / ^ c 2 . In the
cases above, in which the degree is always 1, we proved that we do not always get
the asymptotically expected number of prime values f(n) with n <y = log 0 h(f),
13
14
Which would be consistent with the BARBAN-DAVENPORT-HALBERSTAM theorem.
Which includes almost all integers a once the inexplicit constant here is > 1 (by a famous
result of H A R D Y AND RAMANUJAN) .
Distribution of Primes
397
for any fixed B > 0. In [7] we showed that this is true for polynomials of arbitrary
degree d, which is somewhat ironic because it is not known that any polynomial of
degree > 2 takes on infinitely many prime values, nor that the prime values are ever
"well-distributed". NAIR AND PERELLI [18] showed that some of the polynomials
d
FR(IT) — fi + RP attain more than, and others attain less than, the number of
prime values expected in such a range, by considering the following Maier matrix:
*Wi(i)
FR(2)
FR+I(2)
FR(3)
*WI(3)
•••
•••
F„(y)
FR+1(y)
•••
F2R-i{y)
FR+2(1)
FR+3(1)
FR+2{2)
F2R-I(1)
(i,j)th entry :
FR+i(j)
The Maier matrix for 71^(7/)
F2R-I(2)
Notice that the j t h column here is part of the arithmetic progression j d (mod P).
Using Maier matrices it is possible to prove "bad equi-distribution" results
for primes in other interesting sequences, such as the values of binary quadratic
forms, and of prime pairs. For example, if we fix B > 0 then, once x is sufficiently
large, there exists a positive integer k < Ioga: such that there are at least 1 + 6B
times as many prime pairs p,p + 2k, with x < p < x + log 0 a:, as we would
expect from assuming that the estimate in Hypothesis H holds uniformly for n <^C
logBh((t + x)(t + (x + 2k))) .
We have now seen that the asymptotic formula in Hypothesis H fails when
x is an arbitrary fixed power of log h(F)(:= ^ ^ log h(fi)), for many different nontrivial examples F. Presumably the asymptotic formula does hold uniformly as
log xj log log h(F) —> 00. However, to be safe, we only make the following prediction:
CONJECTURE. Fix e > 0 and positive integer k. The asymptotic formula in
Hypothesis H holds uniformly for x > h(F)E as h(F) —> 00.
Our work here shows that the "random-like" behavior exhibited by primes in many
situations does not carry over to all situations. It remains to discover a model that
will always accurately predict how primes are distributed, as it seems that minor
modifications of CRAMER'S model will not do. We thus agree that:
"It is evident that the primes are randomly distributed hut, unfortunately,
we don't know what random7 means." — R.C. VÀUGHAN (February 1990).
398
Andrew Granville
Final r e m a r k s
Armed with M A I E R ' S ideas it seems possible to construct incorrect conclusions
from, more or less, any variant of C R A M E R ' S model. This flawed model may still
be used t o make conjectures about t h e distribution of primes, b u t one should b e
very cautious of such predictions!
T h e r e are no more t h a n 0(x2/log3B
x) arithmetic progressions a (mod q),
with 1 < a < q < z / l o g ^ r r a n d (a, q) = 1, for which (8) fails, by t h e BARBAND A V E N P O R T - H A L B E R S T A M theorem. However, our methods here may be used t o
show t h a t (8) does fail for more t h a n a; 2 /exp ((logz) e ) such arithmetic progressions.
Maier's matrix has been used in other problems too: K O N Y A G I N recently used
it t o find unusually large gaps between consecutive primes. M A I E R used it t o find
long sequences of consecutive primes, in which there are longer than average gaps
between each pair. SHIU has used it t o show t h a t every arithmetic progression a
(mod q) with (a, q) = 1 contains arbitrarily long strings of consecutive primes.
BALOG [1] has recently shown t h a t t h e prime fc-tuplets conjecture holds "on
average" (in t h e sense of t h e B O M B I E R I - V I N O G R A D O V Theorem). 1 5
As we saw in t h e table above, Li(x) > TT(X) for all x < 10 1 8 . However, (2b)
implies t h a t this inequality does not persist for ever; indeed, it is reversed for
some x < IO 3 7 0 ( T E R I E L E ) . Recently, however, R U B I N S T E I N AND S A R N A K [20] 16
showed t h a t it does hold more often t h a n not, in t h e sense t h a t there exists a
constant 6 « 1/(4 • 10 6 ) such t h a t t h e (logarithmically scaled) proportion of x
for which 7v(x) >Li(a;) exists a n d equals 6. Such biases m a y also be observed in
arithmetic progressions, in t h a t there are "more" primes belonging t o arithmetic
progressions t h a t are quadratic nonresidues t h a n those t h a t are quadratic residues.
In particular they prove t h a t 7r(:c;4,3) > ir(x;A, 1) for a (logarithmically scaled)
proportion 0.9959 . . . of t h e time.
Delicate questions concerning t h e distribution of prime numbers still seem
to be very mysterious. It may b e t h a t by taking into account divisibility by small
primes we can obtain a very accurate picture; or it m a y b e t h a t there are other
phenomena, disturbing t h e equi-distribution of primes, t h a t await discovery .. .
"Mathematicians
have tried in vain to discover some order in the sequence
of prime numbers but we have every reason to beheve that there are some
mysteries which the human mind will never penetrate. "
— L. E U L E R (1770).
Acknowledgments: I'd like t o t h a n k R e d Alford, Nigel Boston, John Friedlander,
Dan Goldston, K e n Ono, Carl Pomerance, and Trevor Wooley for their helpful
comments on earlier drafts of this paper.
15
Which was improved to the exact analogue of (11) by MiKAWA for k = 2, and by KAWADA
for all k > 1.
16
A11 of their results are proved assuming appropriate conjectures such as RH, GRH, and
that the zeros of t h e relevant L-functions are linearly independent over (Q).
Distribution of Primes
399
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[20]
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