Journal of Number Theory 119 (2006) 284–296
www.elsevier.com/locate/jnt
On twin primes associated with
the Hawkins random sieve
H.M. Bui ∗ , J.P. Keating
School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK
Received 13 July 2005; revised 27 July 2005
Available online 4 January 2006
Communicated by J. Brian Conrey
Abstract
We establish an asymptotic formula for the number of k-difference twin primes associated with the
Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k = 1
was obtained by M.C. Wunderlich [A probabilistic setting for prime number theory, Acta Arith. 26 (1974)
59–81]. We here extend this to k 2 and generalize it to all l-tuples of Hawkins primes.
2005 Published by Elsevier Inc.
1. Introduction
The random sieve was introduced by Hawkins [4,5] as follows. Let S1 = {2, 3, 4, 5, . . .}. Put
P1 = min S1 . Every element of the set S1 \ {P1 } is then sieved out, independently of the others,
with probability 1/P1 , and S2 is the set of the surviving elements. In general, at the nth step,
define Pn = min Sn . We then use 1/Pn as the probability with which to delete the numbers in
Sn \ {Pn }. The set remaining is denoted by Sn+1 . The Hawkins sieve is essentially a probabilistic
analogue of the sieve of Eratosthenes. The sequences {P1 , P2 , . . . , Pn , . . .} of Hawkins primes
mimic the primes in the sense that their statistical distribution is expected to be like that of the
primes. The primes themselves correspond to one realization of the process.
A great deal is known about the Hawkins primes. For instance, the analogues of the prime
number theorem [5,6,9], Mertens’ theorem [6,9] and the Riemann hypothesis [7,8] are true with
* Corresponding author.
E-mail address: hm.bui@bristol.ac.uk (H.M. Bui).
0022-314X/$ – see front matter 2005 Published by Elsevier Inc.
doi:10.1016/j.jnt.2005.11.015
H.M. Bui, J.P. Keating / Journal of Number Theory 119 (2006) 284–296
285
probability 1. We here concern ourselves with the density of k-difference Hawkins twin primes
and its generalization to other l-tuples.
Instead of a sequence of probability spaces, as considered by Hawkins, Wunderlich [9] simplified the process in a single probability space. Let X be the space of all sequences of integers
greater than 1, i.e., X consists of all finite and infinite sequences. The class of all sets of those
sequences is Ω. For α ∈ X, we denote by αn the set of elements of α which are less than n, i.e.,
αn = α ∩ {2, 3, 4, . . . , n − 1} and α n = α \ αn .
Definition 1. An element E ∈ Ω is called an elementary set if there exists a sequence
{a1 , a2 , . . . , ak } ∈ X and an integer n > ak such that E consists of all the sequences α such
that αn = {a1 , a2 , . . . , ak }. E is denoted by {a1 , a2 , . . . , ak ; n}, and if k = 0, E = {· ; n} is the set
of all sequences whose elements are not less than n.
The probability function is now defined recursively on the class of elementary sets.
Definition 2. Define a non-negative real-valued function μ on the class of elementary sets as
follows:
(i) μ({· ; 2}) = 1,
(ii) μ({a1 , . . . , ak , n; n + 1}) = ki=1 (1 − a1i )μ({a1 , . . . , ak ; n}),
(iii) μ({a1 , . . . , ak ; n + 1}) = (1 − ki=1 (1 − a1i ))μ({a1 , . . . , ak ; n}).
For any α ∈ X, the analogue of the k-difference twin prime counting function is defined as
ΠX,X+k (x; α) = #{j x: j ∈ α and j + k ∈ α}.
Wunderlich [9] showed that ΠX,X+1 (x) ∼ x/(log x)2 almost surely, which is an analogue of
Hardy and Littlewood’s famous conjecture concerning the distribution of the twin primes [3]. The
absence of the twin prime constant factor here is due to the drawback of the probabilistic setting
of the random sieve that it contains little arithmetical information about the primes. Though the
result is not unexpected, it is not easy to establish, as it is, for example, in Cramer’s model [1],
where every number n is independently deleted with probability 1/ log n. In Section 2, we follow
the lines of Wunderlich [9] and extend the result to k = 2,
Theorem 1. Almost surely
ΠX,X+2 (x) ∼
x
.
(log x)2
Theorem 1 requires rather more work than [9, Theorem 4], but the idea is similar and straightforward. Nevertheless, it is clear from the proof for k = 2 that as k increases, the calculations will
become extremely complicated, and the proof for the general case using Wunderlich’s method
is likely to be extremely messy. In Section 3, we therefore develop a different approach and
establish the following theorem.
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Theorem 2. Almost surely, for any fixed integer k, as x → ∞,
ΠX,X+k (x) ∼
x
.
(log x)2
As we note in Section 3, our approach extends straightforwardly to l-tuples of Hawkins primes
to yield:
Theorem 3. Let 0 < k1 < k2 < · · · < kl−1 and denote by ΠX,X+k1 ,...,X+kl−1 (x; α) the number of
m x such that the set {m, m + k1 , . . . , m + kl−1 } ⊂ α. Then as x → ∞, almost surely
x
.
(log x)l
ΠX,X+k1 ,...,X+kl−1 (x) ∼
An immediate corollary of this theorem is:
Corollary 1. For any positive integers d, l, and l 2, almost surely, as x → ∞,
ΠX,X+d,...,X+(l−1)d (x) ∼
x
,
(log x)l
which is reminiscent of a recent theorem of Green and Tao [2] on the existence of arbitrarily long
arithmetic progressions in the primes, proved using powerful techniques from analytic number
theory, combinatorics and ergodic theory.
2. Proof of Theorem 1
We begin the proof by stating a lemma from [9].
Lemma 1. For r, s, t non-negative integers, r t, define
Mk = 1 +
k−2
1
j =1
j
.
Then
s+1
n
c(t − r − 1, r)ns+1
ks
n
ns+1
c(1, r)ns+1
+ ··· +
+O
=
+
r+1
t−1
Mkr
(s + 1)Mnr
Mnt
Mn
Mn
k=2
=
s+1
t−r−1
ns+1
ns+1
n
,
c(j,
r)
+
+
O
r
r+j
(s + 1)Mn
Mnt
(s + 1)Mn
j =1
where c(j, r) = r(r + 1) · · · (r + j − 1)/(s + 1)j +1 .
As in [9], we define
ym (α) =
j <m, j ∈α
1
1−
.
j
H.M. Bui, J.P. Keating / Journal of Number Theory 119 (2006) 284–296
287
Then P(m ∈ α) = ym (α), and if we let Cn be the set of all sequences containing n,
μ(Cn ) = E(yn ) =
yn dμ.
Wunderlich then obtained the asymptotic formula for the kth moment of yn , which is an analogue
of Mertens’ theorem,
k
1
1
E ym = k + O
.
k+2
Mm
Mm
Some simple calculations give
1
1 3
1
2
P(m ∈ α, m + 2 ∈ α) = 1 −
ym
1−
ym (α) .
(α) −
m
m+1
m
Define the auxiliary function
ΠX,X+2 (x; α) =
mx, m∈α
m+2∈α
ym (α) −
−1
1 2
1
1−
ym (α)
.
m+1
m
In what follows, we write Π(x; α) for ΠX,X+2 (x; α), and if f : R → R, we define the usual
difference operator applied to f by f (m) := f (m + 1) − f (m). We have
Π(m; α) =
ym+1 (α) −
0
1
m+2
1−
1
m+1
2
ym+1
(α)
−1
if m + 1 ∈ α, m + 3 ∈ α,
otherwise.
(1)
Hence
E Π(m + 1) − E Π(m) = 1 −
1
E(ym+1 ).
m+1
Thus
n
1
1
1
1
1−
E(ym+1 ) =
+O
3
m+1
m
Mm
Mm
m=3
m=2
n
n
n
=
+ 2 +O
.
Mn Mn
Mn3
n−1
1−
E Π(n) =
(2)
We now wish to estimate the variance of Π(n). It is easy to see from (1) that
E Π 2 (m) = 2 1 −
1
E ym+1 Π(m) + O(1).
m+1
(3)
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H.M. Bui, J.P. Keating / Journal of Number Theory 119 (2006) 284–296
i
It is necessary to find another recursion for ym+1
(α)Π (m; α). We have
i
ym+2
(α)Π (m + 1; α)
⎧
1
1
1 i i
2
⎪
1 − m+1
ym+1 Π(m; α) + ym+1 (α) − m+2
1 − m+1
ym+1
(α)
⎪
⎪
⎪
⎨
if m + 1 ∈ α, m + 3 ∈ α,
=
i i
1
⎪
1 − m+1 ym+1 Π(m; α) if m + 1 ∈ α, m + 3 ∈
/ α,
⎪
⎪
⎪
⎩ i
ym+1 Π(m; α)
if m + 1 ∈
/ α.
−1
Since
⎧
2
1
1
1
3
⎪
⎨ P(m + 1 ∈ α, m + 3 ∈ α) = 1 − m+1 ym+1 (α) − m+2 1 − m+1 ym+1 (α) ,
1
1
1
3
2
P(m + 1 ∈ α, m + 3 ∈
/ α) = ym+1 (α) − 1 − m+1
ym+1
(α) − m+2
1 − m+1
ym+1
(α) ,
⎪
⎩
P(m + 1 ∈
/ α) = 1 − ym+1 (α),
we easily obtain
i
E ym+1
Π(m) = 1 −
1
m+1
i+1
i+1
− 1− 1−
E ym+1
1
m+1
i
i+1
E ym+1
Π(m) .
4 Π (k)) = O(kE(y 4 )), we have
Taking i = 3, summing from 1 to n − 1, and using E(yk+1
k+1
n
n
1
3
n
4
.
E yn+1 Π(n) = O
E ym = O
=O
4
Mm
Mn4
m=2
m=2
Letting i = 2 in (4),
2
2
Π(m + 1) − E ym+1
Π(m)
E ym+2
3
2
3
3
1
1
E ym+1
Π (m) .
= 1−
E ym+1
− 1− 1−
m+1
m+1
Summing from 1 to n − 1, we obtain
n
n
1
2
1 3 3
1−
E yn+1 Π(n) =
E ym + O
4
m
Mm
m=2
m=2
n
n
1
1
n
n
.
=
=
+
O
+
O
3
4
Mm
Mm
Mn3
Mn4
m=2
m=2
We are now ready to find E(ym+1 Π(m)). Letting i = 1 in (4),
E ym+2 Π (m + 1) − E ym+1 Π(m) = 1 −
1
m+1
2
2
E ym+1
−
2
1
E ym+1
Π(m) .
m+1
(4)
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289
So
n
n
1 2 2
1 2
1−
E yn+1 Π(n) =
E ym −
E ym Π (m − 1)
m
m
m=2
m=2
n
n
n
1
n
1
1
n
n
=
−
+O
= 2 + 3 +O
.
2
3
4
Mm
Mm
Mm
Mn
Mn
Mn4
m=2
m=2
m=2
Substituting this into (3), we have
n
2
m
m
m
1
+ O(n)
+ 3 +O
2 1−
E Π (n) =
2
4
m
Mm
Mm
Mm
m=2
2
2
2
2
n
n
n2
n
n
=2
+2
+
+O
+O
2Mn2 2Mn3
Mn4
2Mn3
Mn4
2
n2
2n2
n
= 2 + 3 +O
.
Mn
Mn
Mn4
From (2) and (5) we deduce that
⎧
⎨ E(Π(n)) =
n
Mn
+ Mn2 + O
n
⎩ Var(Π (n)) = O n24 .
M
n
Mn3
,
n
Theorem 2 in [9] then implies that
Π(n) ∼
n
.
log n
Now we define
rm (α) =
Then
ΠX,X+2 (n; α) =
1 if m ∈ α, m + 2 ∈ α,
0 otherwise.
rm (α)
mn
rm (α)
=
1
y (α) − m+1 1 −
mn m
=
am (α)bm (α),
1
m
2 (α)
ym
ym (α) −
1 2
1
1−
y (α)
m+1
m m
mn
where
am (α) =
ym (α) −
rm (α)
1
1
m+1 1 − m
2 (α)
ym
(5)
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H.M. Bui, J.P. Keating / Journal of Number Theory 119 (2006) 284–296
and
1 2
1
bm (α) = ym (α) −
1−
y (α).
m+1
m m
Let A0 (α) = 0 and Am (α) =
ΠX,X+2 (n; α) =
m
j =1 aj (α).
Using Abel summation,
am (α)bm (α) = An (α)bn (α) −
m<n
mn
Am (α) bm+1 (α) − bm (α) .
Since An (α) = Π(n; α), An bn ∼ (n/ log n)(1/ log n) ∼ n/(log n)2 . The result follows if we can
show that
n
.
Am |bm+1 − bm | = o
(log n)2
m<n
Firstly,
Am (α) =
j m, j ∈α
j +2∈α
=
3
2
yj (α) −
j m, j ∈α
j +2∈α
1
j +1
1
,
yj (α)
1
1−
1
j
yj2 (α)
j m, j ∈α
j +2∈α
1
yj (α) − 31 yj (α)
which is O(m/ log m) from [9, Theorem 4].
Secondly,
bm+1 (α) − bm (α)
1
1 2
1
1
2
= ym+1 (α) −
1−
ym+1
1−
ym (α) .
(α) − ym (α) −
m+2
m+1
m+1
m
Since
ym+1 (α) =
1 − m1 ym (α) =
ym (α)
m−1
m ym (α)
if m ∈ α,
otherwise,
we obtain
So
bm+1 (α) − bm (α) =
1
− ym (α) + 3(m−1) y 2 (α)
m
m(m+1)(m+2) m
m−2
y 2 (α)
m(m+1)(m+2) m
bm+1 (α) − bm (α)
1
m ym (α)
1
2
m(m+1) ym (α)
if m ∈ α,
otherwise.
if m ∈ α,
otherwise.
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H.M. Bui, J.P. Keating / Journal of Number Theory 119 (2006) 284–296
Hence
m<n
Am (α)bm+1 (α) − bm (α)
Am (α)bm+1 (α) − bm (α) =
m<n
m∈α
+
m<n
m∈α
/
Am (α)bm+1 (α) − bm (α)
1
1
2
Am (α)ym (α) +
Am (α)ym
(α).
m
m(m
+
1)
m<n
m<n
m∈α
m∈α
/
Thus
m<n
Am (α)bm+1 (α) − bm (α) = O
=O
1
(log m)2
+O
1
(log m)2
+ O(1),
m<n
m∈α
m<n
m∈α
m<n
m∈α
/
1
m(log m)3
which is easily seen to be O(n/(log n)3 ) from the analogue of prime number theorem for
Hawkins random sieve. The result follows.
3. Proof of Theorems 2 and 3
In this section, we take m[i1 , i2 , . . . , il ] ∈ α, where i1 < i2 < · · · < il , to mean m +
{i1 , i2 , . . . , il } ⊂ α, and m + h ∈
/ α for all h ∈ [i1 , il ] \ {i1 , i2 , . . . , il }.
Lemma 2. Given a non-negative integer l and 0 = i0 < i1 < i2 < · · · < il < il+1 = k, define
T[0,i1 ,i2 ,...,il ,k] (n) :=
mn
1.
m[0,i1 ,i2 ,...,il ,k]∈α
Then T[0,i1 ,i2 ,...,il ,k] (n) ∼ n/(log n)l+2 almost surely.
Proof. We simply write T (n) for T[0,i1 ,i2 ,...,il ,k] (n). Let An be the event n ∈ α and Bn be the
complement of An , i.e., Bn = Acn . We then have
Pm = P (m + 1)[0, i1 , i2 , . . . , il , k] ∈ α
= P(Am+k+1 Bm+k . . . Bm+2+il Am+1+il . . . Bm+2 Am+1 ).
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H.M. Bui, J.P. Keating / Journal of Number Theory 119 (2006) 284–296
By the chain rule
Pm = P(Am+1 )
× P(Bm+2 |Am+1 ) . . . P(Bm+i1 |Bm+i1 −1 . . . Bm+2 Am+1 )
× P(Am+1+i1 |Bm+i1 . . . Bm+2 Am+1 )
× ···
× P(Am+k+1 |Bm+k . . . Bm+2+il Am+1+il . . . Bm+2 Am+1 )
= ym+1
i1 −1
1
ym+1
m+1
1
ym+1
×
1−
m+1
× ···
1
1
1
1−
··· 1 −
ym+1 ,
×
1−
m + 1 + il
m + 1 + il−1
m+1
× 1− 1−
or, in short,
l
1−
Pm =
j =0
1
m + 1 + ij
l+1−j
l+2
ym+1
l
j =0
l−j
1−
1−
h=0
il+1−j −il−j −1
1
ym+1
.
m + 1 + ih
Since
l
j =0
l−j
1−
1−
h=0
il+1−j −il−j −1
1
ym+1
m + 1 + ih
2
ym+1
+ O ym+1
(il+1−j − il−j − 1)ym+1 + O
=1−
m
j =0
2
ym+1
= 1 − (k − l − 1)ym+1 + O
+ O ym+1
,
m
l
we have
l+2
Pm = ym+1
− (k − l
l+3
− 1)ym+1
+O
l+3
ym+1
m
l+4
+ O ym+1
.
From the definition of T (m),
T (m + 1) − T (m) =
1 if (m + 1)[0, i1 , i2 , . . . , il , k] ∈ α,
0 otherwise.
H.M. Bui, J.P. Keating / Journal of Number Theory 119 (2006) 284–296
293
Hence
l+3
E(ym+1
)
l+3
l+2
l+4
E T (m + 1) − E T (m) = E ym+1
− (k − l − 1)E ym+1
+O
+ O E ym+1
m
1
1
k−l−1
1
+O
= l+2 −
+O
l+3
l+4
l+3
Mm+1
Mm+1
Mm+1
mMm+1
k−l−1
1
1
.
= l+2 −
+
O
l+3
l+4
Mm+1
Mm+1
Mm+1
Summing from 1 to n − 1 yields
E T (n) =
=
n
Mnl+2
n
Mnl+2
(l + 2)n
(k − l − 1)n
n
+O
Mnl+3
Mnl+4
(k − 2l − 3)n
n
.
−
+O
l+3
Mn
Mnl+4
+
Mnl+3
−
(6)
The next step is to estimate the variance of T (n). As in the case of the previous theorem, we
i
need to establish a recursion for ym+1
T (m). For this we have
⎧ i
y
T (m)
⎪
⎪
⎨ m+1
1 i i
i
1 − m+1
ym+1 (T (m) + 1)
ym+2
T (m + 1) =
⎪
⎪
⎩
i
1
i
1 − m+1
ym+1
T (m)
if m + 1 ∈
/ α,
if (m + 1)[0, i1 , i2 , . . . , il , k] ∈ α,
otherwise.
Since
P(m + 1 ∈
/ α) = 1 − ym+1 ,
P((m + 1)[0, i1 , i2 , . . . , il , k] ∈ α) = Pm ,
we deduce that
i
i
T (m + 1) = E ym+1
T (m)(1 − ym+1 ) + 1 −
E ym+2
+ 1−
1
m+1
i
1
m+1
i
i
E ym+1
T (m) + 1 Pm
i
E ym+1
T (m)(ym+1 − Pm ) ,
or, equivalently,
i
E ym+1 T (m) = 1 −
1
m+1
i
i
E ym+1 Pm − 1 − 1 −
1
m+1
i
Letting i = l + 4, and recalling that
l+4
l+2
l+3
Pm = ym+1
− (k − l − 1)ym+1
+ O ym+1
,
i+1
E ym+1
T (m) .
(7)
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H.M. Bui, J.P. Keating / Journal of Number Theory 119 (2006) 284–296
we obtain
l+4
2l+6
E ym+1
T (m) = O E ym+1
=O
1
2l+6
Mm+1
.
So
n
l+4
E yn+1 T (n) = O
m=2
1
2l+6
Mm
=O
n
Mn2l+6
.
Letting i = l + 3 in (7), and summing from 1 to n − 1, we have
n
l+3
T (n) =
E yn+1
m=2
1
+O
2l+5
Mm
n
m=2
1
2l+6
Mm
=
n
Mn2l+5
+O
.
2l+6
n
Mn
Finally, substituting i = l + 2 in (7),
l+2
E ym+1
T (m) = 1 −
=
1
m+1
1
2l+4
Mm+1
l+2
− (k + 1)
l+2
E ym+1
Pm − 1 − 1 −
1
2l+5
Mm+1
+O
1
2l+6
Mm+1
1
m+1
l+2
l+3
E ym+1
T (m)
.
Thus
n
l+2
E yn+1
T (n) =
1
2l+4
Mm
m=2
=
=
n
Mn2l+4
n
Mn2l+4
− (k + 1)
+ (2l + 4)
n
1
2l+5
Mm
m=2
n
Mn2l+5
− (k − 2l − 3)
+O
− (k + 1)
n
Mn2l+5
n
n
m=2
1
2l+6
Mm
n
+O
Mn2l+5
Mn2l+6
n
.
+O
Mn2l+6
Now, back to the variance of T (n),
T 2 (m + 1) =
T 2 (m) + 2T (m) + 1 if (m + 1)[0, i1 , i2 , . . . , il , k] ∈ α,
T 2 (m)
otherwise.
Therefore
E T 2 (m + 1) = E T 2 (m)(1 − Pm ) + E T 2 (m) + 2T (m) + 1 Pm
= E T 2 (m) + 2E T (m)Pm + E(Pm ).
(8)
295
H.M. Bui, J.P. Keating / Journal of Number Theory 119 (2006) 284–296
And hence
l+2
l+3
E T 2 (m) = 2E ym+1
T (m) − 2(k − l − 1)E ym+1
T (m)
l+4
l+2
+ O E ym+1 T (m) + O E ym+1 .
From (8), we obtain
m
m
−
2(k
−
l
−
1)
+
O
2l+4
2l+5
2l+5
2l+6
Mm
Mm
Mm
Mm
2m
2(2k − 3l − 4)m
m
.
= 2l+4 −
+O
2l+5
2l+6
Mm
Mm
Mm
E T 2 (m) = 2
m
m
− (k − 2l − 3)
So
n
E T 2 (n) =
m=2
=
2m
2l+4
Mm
n2
− (2k − 3l − 4)
+
(l + 2)n2
n
m=2
2m
2l+5
Mm
− (2k − 3l − 4)
+O
n
Mnl+2
⎩ Var(T (n)) = O
−
(k−2l−3)n
Mnl+3
n2
Mn2l+6
+O
.
n2
n
Mnl+4
m
2l+6
Mm
2
n
+O
Mn2l+6
m=2
Mn2l+4
Mn2l+5
Mn2l+5
(2k − 4l − 6)n2
n2
n2
.
= 2l+4 −
+O
Mn
Mn2l+5
Mn2l+6
Combining (6) with (9), we have
⎧
⎨ E(T (n)) =
n
(9)
,
Theorem 2 in [9] again yields T (n) ∼ n/(log n)l+2 as asserted.
The proof of Theorem 2 now follows immediately from Lemma 2 by noting that
ΠX,X+k (x) =
k−1
T[0,i1 ,i2 ,...,il ,k] (x)
l=0 0<i1 <i2 <···<il <k
= T[0,k] (x) +
k−1
T[0,i1 ,i2 ,...,il ,k] (x)
l=1 0<i1 <i2 <···<il <k
Similarly for Theorem 3,
= 1 + o(1)
x
x
.
+
O
(log x)2
(log x)3
x
(log x)l+1
x
x
.
+
O
= 1 + o(1)
(log x)l
(log x)l+1
ΠX,X+k1 ,...,X+kl−1 (x) = T[0,k1 ,k2 ,...,kl−1 ] (x) + O
✷
296
H.M. Bui, J.P. Keating / Journal of Number Theory 119 (2006) 284–296
Acknowledgment
J.P.K. is supported by an EPSRC Senior Research Fellowship.
References
[1] H. Cramer, On the order of magnitude of the differences between consecutive prime numbers, Acta Arith. 2 (1937)
23–28.
[2] B. Green, T. Tao, The primes contain arbitrarily long arithmetic progressions, math.NT/0404188.
[3] G.H. Hardy, J.E. Littlewood, Some problems in “Partitio Numerorum” III: On the expression of a number as a sum
of primes, Acta Math. 44 (1923) 1–70.
[4] D. Hawkins, The random sieve, Math. Mag. 31 (1957/1958) 1–3.
[5] D. Hawkins, Random sieves II, J. Number Theory 6 (1974) 192–200.
[6] C.C. Heyde, On asymptotic behaviour for the Hawkins random sieve, Proc. Amer. Math. Soc. 56 (1976) 277–280.
[7] C.C. Heyde, A log log improvement to the Riemann hypothesis for the Hawkins random sieve, Ann. Probab. 6 (5)
(1978) 870–875.
[8] W. Neudecker, D. Williams, The “Riemann hypothesis” for the Hawkins random sieve, Compos. Math. 29 (1974)
197–200.
[9] M.C. Wunderlich, A probabilistic setting for prime number theory, Acta Arith. 26 (1974) 59–81.
Further reading
[10] W. Neudecker, On twin “primes” and gaps between successive “primes” for the Hawkins random sieve, Math. Proc.
Cambridge Philos. Soc. 77 (1975) 365–367.
[11] P. Ribenboim, The Book of Prime Number Records, second ed., Springer, New York, 1989.
[12] M.C. Wunderlich, The prime number theorem for the random sequences, J. Number Theory 8 (1976) 369–371.
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