LIMIT POINTS OF NORMALIZED CONSECUTIVE PRIME GAPS
D. A. GOLDSTON AND A. H. LEDOAN
Abstract. Let pn denote the nth prime number. We find some new intervals in which there must
exist infinitely many limit points of the sequence {(pn+1 − pn )/ log n} of normalized differences between
consecutive prime gaps.
1. Introduction
Let dn = pn+1 − pn denote the nth consecutive prime gap. We examine the limit points of the sequence
{(pn+1 − pn )/ log n} of normalized differences between consecutive prime gaps. Since log pn ∼ log n, some
authors equivalently look at the limit points of the sequence of {(pn+1 − pn )/ log pn }.
What is known?
1) ∞ is a limit point, Westzynthius 1931.
2) There is a limit point less than 1 − c. Erdős 1940.
3) The measure of the set of limit points is positive. Ricci, Erdős (1954,1955?)
4) The measure of the set of limit points in [T, T ] is T as T → ∞. Hildebrand and Maier 1988
5) 0 is a limit point. GPY 2005
6)The interval [0, c] are all limit points, where c is a very small ineffective constant. Pintz 2013 using
Zhang’s theorem.
2. Erdős’ Method
Erdős’ method not only obtains a positive measure for the set of limit points, but obtains explicit
intervals within which there must be an infinite number of limit points. For example, the interval [ 81 , 2]
must contain infinitely many limit points.
We will examine dn when pn is in the interval (x, 2x] as x → ∞. By the prime number theorem
X
X
x
1∼
, and
dn ∼ x,
as x → ∞.
log x
x<pn ≤2x
x<pn ≤2x
Lemma 1. For any fixed real number M > 1, we have for any > 0 and x sufficiently large that
X
1
x
(2.1)
1≥ 1−
−
.
M
log x
x<pn ≤2x
dn <M log x
Proof. Since
X
x<pn ≤2x
dn ≥M log x
1≤
X
x<pn ≤2x
dn
1
=
M log x
M log x
x<pn ≤2x
we see
X
x<pn ≤2x
dn <M log x
1=
X
x<pn ≤2x
1−
X
x<pn ≤2x
dn ≥M log x
X
1≥
dn ≤
1+ x
,
M log x
1
x
1−
−
.
M
log x
Date: October 26, 2013.
2000 Mathematics Subject Classification. Primary 11N05; Secondary 11P32, 11N36.
Key words and phrases. Prime numbers; Consecutive Prime Differences.
Research of the first author was supported in part by NSF grant DMS-1104434.
2
D. A. GOLDSTON AND A. H. LEDOAN
For our next lemma we need a sieve upper bound for prime pairs, which may be stated in the form
X
x
(2.2)
1 ≤ (C + ) S(k) 2 ,
log x
x<p≤2x
p+k prime
where
(2.3)
with
(2.4)

Y p − 1


, if k is even, k 6= 0;
 2C
p−2
S(k) =
p|k

p>2


0,
if k is odd;
Y
C=
1−
p>2
1
(p − 1)2
= 0.66016 . . .,
and C is a constant. (C = 4 is due to Bombieri and Davenport [1], and there are somewhat smaller values
now known to hold.)
Lemma 2. If C is a number for which (2.2) holds, we have for any > 0 and x sufficiently large that
X
x
(2.5)
.
1 ≤ (C + ) (b − a)
log x
x<pn ≤2x
dn
a≤ log
n ≤b
Proof. Since
X
k≤N
S(k) ∼ N,
see [1], the sum on the left-hand side of (2.5) is bounded by
X
X
1=
a log x≤k≤b log 2x
x<p≤2x
a log x≤p0 −p≤b log 2x




X
x<p≤2x
p+k prime


1

X
x
S(k)
2
log x a log x≤k≤b log 2x
x
.
≤ (C + ) (b − a)
log x
≤ (C + )
We now are ready to prove Erdős’ result on limit points. Let
dn
dn
S(M ) =
:
<M
and S ∗ (M ) = {c : c is a limit point of S(M )} .
log n log n
For a measurable set A we denote by m(A) the Lebesgue measure of A.
Theorem 1 (Erdős). For any > 0, we have
m(S ∗ (M )) ≥
1
C
1
1−
− .
M
1
Since the interval [0, C1 1 − M
− 2 has shorter length then the lower bound of the measure of S ∗ (M )
in the theorem, we conclude that for any M > 1, the interval
1
1
1−
− 2 , M
C
M
contains infinitely many limit points. The interval [ 81 , 2] mentioned above is obained by taking M = 2
and a C < 4.
LIMIT POINTS OF PRIME GAPS
3
Proof. First observe that S ∗ (M ) is a closed and bounded set since it is a set of limit points contained in
the interval [0, M ]. We construct a sequence of open covers of S ∗ (M ) by
1
1
∗
Ok = Ic (k) = (c − , c + ) : c ∈ S (M ) ,
k
k
and letting
[
Qk =
Ic (k)
c∈S ∗ (M )
we have that
S ∗ (M ) =
∞
\
Qk .
k=1
∞
Notice {Qk }k=1 is an infinite decreasing sequence of measurable sets with Qk+1 ⊂ Qk and m(E1 ) ≤ M +2.
Hence by a well-known theorem [7]
!
∞
\
m(S ∗ (M )) = m
Qk = lim m(Qk ).
k→∞
k=1
Given any > 0, we can find a sufficiently large number k0 such that
m(S ∗ (M )) ≥ m(Qk0 ) − .
By the Heine-Borel Theorem, S ∗ (M ) has a finite subcover taken from Ok0 , i.e. there are r intervals
{Ic1 (k0 ), Ic2 (k0 ), . . . , Icr (k0 )}
such that
∗
S (M ) ⊂
r
[
i=1
Ici (k0 ) ⊂ Qk0 .
These open intervals will not normally all be disjoint from each other, but since the union of two overlapping open intervals is an open interval containing both of the original intervals, we obtain by combining
overlapping open intervals a smaller number of disjoint open intervals with the same union as the original
intervals. Thus we have obtained disjoint intervals I1 , I2, . . . Is , s ≤ r, such that
S ∗ (M ) ⊂
s
[
i=1
I i ⊂ Qk 0 ,
and on letting `(I) denote the length of the interval I, we obtain
m(S ∗ (M )) ≥ m(
s
[
i=1
Ii ) − =
s
X
i=1
`(Ii ) − .
By Lemma 2
`(Ii ) ≥
and since for all n ≥ n0 each
dn
log n
1
(C + ) logx x
X
1,
x<pn ≤2x
dn
log n ∈Ii
is in an interval Ii , we have for x sufficiently large
m(S ∗ (M )) ≥
1
(C + ) logx x
X
1
x<pn ≤2x
dn <M log x
and we conclude by Lemma 1 that
1
m(S (M )) ≥
C
∗
1
1−
− .
M
− ,
4
D. A. GOLDSTON AND A. H. LEDOAN
3. Bombieri and Davenport’s Method
The Erdős method obtains an interval containing 1 with infinitely many limit points. The HildebrandMaier method produces infinitely many limit points in intervals consisting of large numbers. Pintz’s
recent result obtain limit points in a very small interval [0, c] with c ineffective. We now prove that there
are explicit intervals inside the interval (0, 1) which contain infinitely many limit points.
In 1965 Bombieri and Davenport proved that
lim inf
n→∞
1
pn+1 − pn
≤
log pn
2
by using the Hardy-Littlewood Circle method together with the recently proved Bombieri-Vinogradov
theorem. By incorporationg Erdős’ method into their method they improved this to
√
pn+1 − pn
2+ 3
lim inf
≤
= 0.46650 . . . .
n→∞
log pn
8
We make use of Bombieri and Davenport’s main result in the form
!
H
X
X
1
1
N
j
(1 − )
1 ≥ ( − )(H − log N ) 2 .
H
2
2
log N
j=1
N<p≤2N
p0 −p=j
Hence
X
S(H) :=
N<p≤2N
0<p0 −p≤H
1
1
N
1 ≥ ( − )(H − log N ) 2
2
2
log N
We write
S(H) =
X
X
1
N<pn ≤2N
0<pn+j −pn ≤H
r
X
X
1+
j≥1
=
j=1
N<pn ≤2N
0<pn+j −pn ≤H
X
j>r
X
1
N<pn ≤2N
0<pn+j −pn ≤H
:= S1 (H) + S2 (H).
Since trivially
S1 (H) ≤ r
X
1,
N<pn ≤2N
0<pn+1 −pn ≤H
we conclude
(3.1)
X
N<pn ≤2N
0<pn+1 −pn ≤H
1≥
1
r
1
1
N
( − )(H − log N ) 2 − S2 (H) .
2
2
log N
We now need to bound S2 (H). Note
S2 (H) =
X
1.
N<pn ≤2N
(pn ,pn +H] contains ≥ r + 1 primes
µ
Gallagher [3] considered a very similar expression and proved, with H = λ log N , µ ≥ 4λ, and k = [ 4λ
],
where [x] is the integer part of x, that
X
1 ≤ (1 + )k2−k N
N<n≤2N
[n,n+H] contains ≥ µ primes
LIMIT POINTS OF PRIME GAPS
5
Gallagher’s argument applies equally well to bound S2 (H); we include this argument for completeness.
We have
X
X
X
(π(pn + H) − π(pn ))k =
1
N<pn ≤2N
N<pn ≤2N pn <p1 ,...,pk ≤pn +H
=
k
X
X
σ(k, j)
1≤d1 <d2 <...<dj ≤H
j=1
X
1,
N<pn ≤2N
pn +hi prime, 1≤i≤j
where σ(k, j) are Stirling numbers of the 2nd kind. The inner sum can be written as
X
1
N<n≤2N
n prime
n+hi prime, 1≤i≤j
which corresponds to a prime j + 1 tuple with shifts H0 = {0, h1, h2 , . . . , hj }. Applying the sieve upper
bound
X
N
1 ≤ (2k k! + )S(H) k ,
log N
N<n≤2N
n+hi prime,1≤i≤k
we obtain the bound
X
N<n≤2N
n prime
n+hi prime, 1≤i≤j
1 ≤ (2j+1 (j + 1)! + )S(H0 )
N
log
j+1
N
.
Since Lemma 6 and the argument at the end of the proof of Lemma 2 in the Jumping Champion paper,
X
1≤d1 <d2 <...<dj ≤H
S(H0 ) ∼ S({0})
Hj
Hj
=
,
j!
j!
we conclude, since σ(k, j) ≤ k k , for N sufficiently large and H = λ log N , and provided 2λk ≥ 1,
X
N<pn ≤2N
(π(pn + H) − π(pn ))k ≤
k
X
σ(k, j)(2j+1 (j + 1)! + )
j=1
≤ (1 + o(1))2k(k + 1)(2λk)k
Hj
N
j! logj+1 N
N
log N
Since
X
N<pn ≤2N
(pn ,pn +H] contains ≥ µ primes
1≤
X
N<pn ≤2N
S2 (H) ≤ (1 + o(1))2k(k + 1)
π(pn + H) − π(pn )
µ
2λk
r+1
k
k
,
N
.
log N
We now take (r + 1)/λ ≥ 4, and choose k = [(r + 1)/4λ], so that k ≥ 18 (r + 1)/λ, and also
Hence (3.1) becomes
X
1
1
1
N
1≥
( − )(λ − ) − 2k(k + 1)2−k
.
r
2
2
log N
N<pn ≤2N
0<pn+1 −pn ≤λ log N
By Lemma 2 we can bound the terms in the sum where
X
1
1
1≥
( − )(λ −
r
2
N<pn ≤2N
b log N≤pn+1 −pn ≤λ log N
pn+1 − pn ≤ b log N , which gives
1
N
) − 2k(k + 1)2−k − rbC
.
2
log N
2λk
r+1
≤
1
.
2
6
D. A. GOLDSTON AND A. H. LEDOAN
From this we conclude that given a value of λ > 1/2 we can choose r and thus k sufficiently large and
then b sufficiently small so that the interval [b, λ] has infinitely many limit points. As an example, we
choose λ = 3/4, r = 41, k = 14, C = 4, and then the right-hand side is
1 1
( − 4202−14 − 164b − /4),
41 8
1
1
and this is positive if b = 2000
. Hence the interval [ 2000
, 3/4] contains infinitely many limit points.
4. More stuff for paper
In 1976 Gallagher [3] showed, assuming the Hardy-Littlewood prime tuple conjecture, that the prime
numbers are distributed according to a Poisson distribution around their average spacing. Specifically,
he showed on this conjecture that letting h ∼ λ log N as N → ∞,
n
o e−λ λk
N, as N → ∞.
(4.1)
1 ≤ n ≤ N : (n, n + h] contains exactly k primes ∼
k!
In particular, it is a simple deduction from (4.1) with k = 0 and the prime number theorem that, letting
pn denote the nth prime,
X
N
(4.2)
1 ∼ e−λ
, as N → ∞.
log N
pn+1 ≤N
pn+1 −pn ≥λ log n
In this paper we will prove an extension of this last result. Our result is that, assuming the same
Hardy-Littlewood conjecture used for (4.2), then the Poisson distribution extends down to the individual
differences between consecutive primes.
Theorem. Assume the Hardy-Littlewood Conjecture (2.5) holds uniformly in the range (??). Then, for
integers d with d ∼ λ log x as x → ∞,
X
x
,
(4.3)
N (x, d) :=
1 ∼ e−λ S(d)
(log x)2
pn+1 ≤x
pn+1 −pn =d
Here S(d) is the familar singular series for the number of prime pairs differing by d, and p will always
in this paper denote a prime. Our theorem shows that the Poisson density for consecutive primes gets
superimposed on the Hardy-Littlewood conjecture for prime pairs.
References
[1] E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. Ser. A 293 (1966), 1–18.
[2] P. Erdős and E. G. Straus, Remarks on the differences between consecutive primes, Elem. Math. 35 (1980), no. 5,
115–118.
[3] P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), 4–9; Corrigendum, Mathematika 28 (1981), no. 1, 86.
[4] D. A. Goldston and A. H Ledoan, The Jumping Champion Conjecture, submitted.
[5] H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, no. 4, Academic Press,
London, New York, San Francisco, 1974.
[6] G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a
sum of primes, Acta Math. 44 (1923), no. 1, 1–70. Reprinted as pp. 561–630 in Collected Papers of G. H. Hardy, Vol.
I (edited by a committee appointed the London Mathematical Society), Clarendon Press, Oxford, 1966.
[7] Real Analysis, 3rd Edition.
Department of Mathematics, San José State University, MacQuarrie Hall, One Washington Square, San
José, CA 95192-0103, USA
E-mail address : goldston@math.sjsu.edu
Department of Mathematics, University of Rochester, Hylan Building, Rochester, NY 14627-0138, USA
E-mail address : ledoan@math.rochester.edu