On the Distribution of the Number of
Goldbach Partitions of a Randomly Chosen
Positive Even Integer
Ljuben Mutafchiev 1
Department of Mathematics and Science
American University in Bulgaria
Bllagoevgrad, Bulgaria
Institute of Mathematics and Informatics
Soa, Bulgaria
Abstract
Let P = fp1 ; p2 ; :::g be the set of all odd primes arranged in increasing order. A
Goldbach partition of the even integer 2k > 4 is a way of writing it as a sum of two
primes from P without regard to order. Let Q(2k ) be the number of all Goldbach
partitions of the number 2k . Assume that 2k is selected uniformly at random from
the interval (4; 2n]; n > 2, and let Yn = Q(2k ) with probability 1=(n 2). We prove
Yn
that the random variable
2 converges weakly, as n ! 1, to a uniformly
n=( 12 log n)
distributed random variable in the interval (0; 1). The method of proof uses sizebiasing and the Laplace transform continuity theorem.
Keywords: Goldbach partition, limiting distribution.
1
Email: [email protected]
1
Introduction
Let P = fp1; p2; :::g be the sequence of all odd primes arranged in increasing
order. A Goldbach partition of the even integer 2k > 4 is a way of writing it
as a sum of two primes pi; pj 2 P without regard to order. The even integer
2k = pi + pj ; i j; is called a Goldbach number. Let Q(2k) denote the number
of the Goldbach partitions of the number 2k. In 1742 C. Goldbach conjectured
that Q(2k) 1 for all k > 2. This problem still remains unsolved (for more
details, see, e.g., [4, Section 2.8, p. 594], [8, Section 4.6] and [9, Chapter VI]).
Let 2n be the set of all Goldbach partitions of the even integers from the
interval (4; 2n]; n > 2. The cardinality of this set is obviously
j 2n j=
X
2<kn
(2k):
Q
(1)
In this paper, we consider two random experiments. In the rst one, we
select a partition uniformly at random from the set 2n, i.e., we assign the
probability 1= j 2n j to each Goldbach partition of an even integer from the
interval (4; 2n]. An important statistic (random variable) of this experiment
is the Goldbach number 2Gn 2 (4; 2n] partitioned by this random selection.
Its probability mass function (pmf) is given by
( ) = jQ(2k)j if x = 2k 2 (4; 2n];
fGn x
2n
(2)
and zero elsewhere.
Remark 1. It was established in [7] that
2
2n ;
j 2n j log
2n
n
! 1:
(3)
The proof is based on a classical Tauberian theorem due to Hardy-LittlewoodKaramata [3, Chapter 7]. Recently, in a private communication, Kaisa Matomaki
[5] showed me a shorter and direct proof of (3) that uses only the Prime Number Theorem [4, Section 17.7] and partial summation.
In the second random experiment, we select an even number 2k 2 (4; 2n]
with probability 1=(n 2). Let Yn be the statistic, equal to the number Q(2k)
of its Goldbach partitions. Obviously, the pmf of Yn is
1 if x = Q(2k); 2k 2 (4; 2n];
fYn (x) =
(4)
n 2
and zero elsewhere.
The main goal of this paper to study the limiting distribution of the random
variable Yn. In Section 2 we show that Yn, appropriately normalized, converges
weakly, as n ! 1, to a random variable that is uniformly distributed in the
interval (0; 1).
The method of proof uses size biasing and the Laplace transform continuity
limit theorem [2, Chapter XIII, Section 1].
2
The Limiting Distribution of Yn
The rst step is to determine the asymptotic of the expected value of Yn.
From (1), (3) and (4) it follows that
1 X Q(2k) 2n = 1 b ; n ! 1;
(5)
E(Yn ) =
2n 2 n
n 2
log
2<kn
where
bn
=
n
1 log n2 ;
2
2
n> :
(6)
Next, we will introduce the concept of size-biasing. The denition we
present below is given in [1, Section 4.2]. Suppose that X is a non-negative
random variable with nite mean and distribution function F . The notation
X is used to denote a random variable with distribution function given by
xF (dx)
F (dx) =
; x > 0:
(7)
The random variable X and the distribution function F are called size-biased
versions of X and F , respectively. We note that by standard arguments (see
[1, formula (4.3)]) (7) is equivalent to
E(Xg(X )) = (E(X ))E(g(X ))
(8)
for all bounded measurable functions g : R+ ! R.
Let fcngn>2 be an arbitrary numerical sequence and let Wn = cnYn; n >
2. If Wn is the size-biased version of Wn and FWn denotes its distribution
function, then
FWn (dx) = FGn (dx);
(9)
where FGn is the distribution function of the variable Gn. To show this,
we recall that both FWn and FGn are jump functions. Their pmf's (Radon-
Nikodym derivatives with respect to the Lebesgue measure) can be obtained
from (7), (4), (2) and (1). After obvious cancellations, we have
1 c Q(2k )
n 2 n
P
fWn (x) = 1
c
2<kn Q(2k )
n 2 n
= jQ(2k)j = fGn (x); if x = 2k 2 (4; 2n];
(10)
2n
where fWn denotes the pmf of Wn. Noting that fWn (x) = fGn (x) = 0 for
x 6= 2k 2 (4; 2n], we see that (9) follows immediately from (10).
Let
1 Y ; n > 2;
Zn =
(11)
n
b
n
where the
scaling factor 1=bn is dened by (6). Now, we set in (10) cn =
1 log n2 and divide the numerator and denominator of its right-hand side by
2
n. To nd an identity for the expectations similar to (8), we multiply both
sides of (9) by g(x) = e x=n; > 0; and integrate with respect to x from 0 to
1. Using the above modication of (10), for xed n, we obtain
E(Zn e
Zn
) = (E(Zn))E(e
Gn =n
);
2
n> :
(12)
The limit of the rst factor in the right-hand side of (12) follows immediately from (5) and (11). We have
1
lim
E
(
Zn ) = :
n!1
2
(13)
The second factor in the right-hand side of (12) is the Laplace transform
of the random variable Gn=n. Its limiting distribution, as n ! 1, was found
in [7]. (The proof is based on the asymptotic equivalence (3).) We state this
result in the following separate lemma.
The sequence of random variables fGn =ngn1 converges weakly,
! 1, to the random variable U = max fU1; U2g, where U1 and U2 are two
Lemma 2.1
as n
independent copies of a uniformly distributed random variable in the interval
(0; 1).
Clearly, the probability density function of the random variable
U equals 2x, if 0 < x < 1, and zero elsewhere. The r th moment of U is
2 ; r = 1; 2; :::. The Laplace transform of the random variable U1 (uniformly
r +2
Remark 2.
distributed in the interval (0; 1)) is
1
'() =
e
0
;
(14)
> ;
while the Laplace transform of U is
2 (1 e e ) = 2'0();
2
0
(15)
> :
To nd the limit of the second factor in the right-hand side of (12), we
combine the result of Lemma 2.1 with the continuity theorem for Laplace
transforms (see, e.g., [2, Chapter XIII, Section 1]). Using (15) and (14), we
deduce that
lim E(e Gn=n) = 2'0(); > 0:
(16)
n!1
Let FZn be the distribution function of the random variable Zn (see (11)
and (6)) and let
( )=
'n Z
1
0
e
x
( ) = E(e
FZn dx
Zn
);
0
> ;
(17)
be its Laplace transform. For xed n 1, we can dierentiate the integral
in (17) with respect to since the integrand obtained after a formal dierentiation is bounded continuous function (see also [2, Chapter XIII, formula
(2.5)]). We thus have
'0n () = E(Zn e Zn ); > 0:
(18)
Hence, combining (12), (13), (16) and (18), we see that
lim '0n() = '0(); > 0:
(19)
n!1
Further, we note that, for any s > 0 and xed n,
Z
0
s
'0n d
( ) = 'n(s)
'n
(0):
Moreover, (12), (13) and (18) imply that the sequence f'0n()gn>2 is uniformly
bounded. So, by Lebesgue dominated convergence theorem, (19) and (14),
lim ('n(s)
n!1
=
Z
s
0
'n
(0)) = nlim
!1
Z
s
0
1
'0 ()d = '(s) 1 =
'0 ()d =
n
e
s
s
1:
Z
s
0
lim
'0 ()d
!1 n
n
Applying again the continuity theorem for Laplace transforms, we obtain the
following limit theorem.
Theorem 2.2 The sequence fZn = Yn =bn gn>2 , with bn given by (6), converges
weakly, as n
! 1, to the random variable U1, which is uniformly distributed
(0; 1).
Theorem 1 shows that the number of Goldbach partitions of even integers
2k 2 (4; 2n] is typically of order abn = log4an2 n ; where 0 < a < 1 (see (6)). This
informal evidence in favor of the Goldbach's conjecture does not give us any
rigorous argument to prove it. Finally, we note that in number theory special
interest is paid on asymptotic estimates for the probability P(Yn = 0) (see,
e.g., [9, Chapter VI]). For instance, Montgomery and Vaughan [6] have shown
that, for suciently large n, there exist a positive (eectively computable)
constant < 1, such that P(Yn = 0) (2n) .
in the interval
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