Download On a Density for Sets of Integers 1 Introduction 2 A

On a Density for Sets of Integers
R. J. Cintra
L. C. Rêgo
Depto. de Estatı́stica, UFPE
52050-740, Recife, PE
E-mail: [email protected] [email protected]
H. M. de Oliveira
R. M. Campello de Souza
Depto. de Eletrônica e Sistemas, UFPE
52050-740, Recife, PE
E-mail: [email protected] [email protected]
Abstract A relationship between the Riemann
zeta function and a density on integer sets is
explored. Several properties of the introduced
density are derived.
Key Words Number theory, probability theory,
arithmetization.
1
In [1], Bell and Burris bring a good exposition
on the Dirichlet density. The Dirichlet density is
defined as the limit of the ratio between two Dirichlet series. Let A ⊂ B be two sets. The generating
series of A is given by
A(s) =
Introduction
∞
X
N (A, n)
,
ns
n=1
(6)
Several measures for the density of sets have been where N (A, n) is a counting function that returns
discussed in the literature [1–6]. Presumably the the number of elements in A of norm n [1]. The
most employed tool for evaluating the density of sets Dirichlet density is then expressed by
is the asymptotic density, also referred to as natural
density. The asymptotic density is expressed by
A(s)
D(A) = lim
,
(7)
s↓α B(s)
kA ∩ {1, 2, 3, . . . , n}k
d(A) = lim
,
(1)
n→∞
n
where B(s) is the generating series of B and α is an
provided that such a limit does exist. The symbol abscissa of convergence [1]. In [6], we also find the
k · k denotes the cardinality, and A is a set of inte- Dirichlet density defined as
gers. Analogously, the lower and upper asymptotic
X 1
densities are defined by
D(A) = lim(s − 1)
,
(8)
s↓1
ns
kA ∩ {1, 2, 3, . . . , n}k
n∈A
d(A) = lim inf
,
(2)
n→∞
n
whenever the limit exists. This density admits lower
kA ∩ {1, 2, 3, . . . , n}k
d̄(A) = lim sup
,
(3) and upper versions, simply by replacing the above
n
n→∞
limit by lim inf and lim sup, respectively.
The aim of this work is to investigate the properThe asymptotic density is said to exist if and only
if both the lower and upper asymptotic densities do ties of the Dirichlet density as defined in Equation 7
in the particular case where the set B is the set of
exist and are equal.
Although the asymptotic density does not always natural numbers. This induces a density based on
exist, the Schnirelmann density [2, 3] is always well- the Riemann zeta function.
defined. The Schnirelmann density is defined as
kA ∩ {1, 2, 3, . . . , n}k
.
(4)
n
Interestingly, this density is highly sensitive to the
initial elements of sequence. For instance, if 1 6∈ A,
then δ(A) = 0 [7].
Another interesting tool is the logarithmic density [4–6]. Let A = {a1 , a2 , a3 , . . .} be a set of integers. The logarithmic density of A is given by
P
1
δ(A) = inf
n≥1
`d(A) = lim
n→∞
ai ≤n ai
log n
.
2
A Density for Sets of Integers
In this section, we investigate the particular case
of the Dirichlet function, applied for subsets of the
natural numbers. In this case, taking into consideration the usual norm, where the norm of any natural
number is equal to its absolute value, the counting
function N (·, ·) becomes
n
if n ∈ A,
N (A, n) = 1,
(9)
(5)
0, otherwise.
Definition 1 The density dens of a subset A ⊂ N Proof: dens(Ac ) = dens(N−A) = dens(N)−dens(N∩
is given by
A) = 1 − dens(A).
P∞
N (A, n) n1s
proposed
(10) Proposition 2 (Monotonicity) The
dens(A) , lim Pn=1
∞
1
density is a monotone function, i.e., if A ⊂ B, then
s↓1
n=1 N (N, n) ns
P
dens(A) ≤ dens(B).
1
ns
= lim Pn∈A
(11)
∞
1
s↓1
Proof: We have that
n=1 ns
P
1
P
s
1
= lim n∈A n , s > 1,
(12)
s
s↓1
ζ(s)
dens(B) = lim n∈B n
(19)
s↓1
ζ(s)
P
P
1
1
if the limit exists. The quantity ζ(·) denotes the
s
s
= lim n∈A n + lim n∈B−A n (20)
Riemann zeta function [8].
s↓1
s↓1
ζ(s)
ζ(s)
P
1
Proposition 1 The following assertions hold true:
s
≥ lim n∈A n
(21)
s↓1
ζ(s)
1. dens(N) = 1
= dens(A).
(22)
2. dens(A) ≥ 0 (nonnegativity)
¤
3. if dens(A) and dens(B) exist1 and A ∩ B = ∅,
then dens(A ∪ B) exists and is equal to
Proposition 3 (Finite Sets) Every finite subset
of N has density zero.
dens(A) + dens(B) (additivity).
Proof: Let A be a finite set. For s > 1, it yields
Proof: Statements 1 and 2 cab be trivially checked.
X 1
X 1
The additivity property can be derived as follows:
≤
= constant < ∞.
(23)
s
P
n
n
1
n∈A
n∈A
s
dens(A ∪ B) = lim n∈A∪B n
(13)
s↓1
ζ(s)
Thus,
P
P
1
1
n∈A ns
n∈B ns
P
= lim
+ lim
1
s↓1
s↓1
constant
ζ(s)
ζ(s)
n∈A ns
dens(A)
=
lim
≤ lim
= 0. (24)
(14)
s↓1
s↓1
ζ(s)
ζ(s)
= dens(A) + dens(B).
(15)
¤
¤ As a consequence, sets A ⊂ N of nonzero density
must be infinite.
Corollary 1 The density of the null set is zero.
Corollary 4 The density of a singleton is zero.
Proof: In fact, dens(∅) = 0, since 1 = dens(N) =
dens(∅ ∪ N) = dens(∅) + dens(N) = dens(∅) + 1. Proposition 4 (Union) Let A and B be two sets
¤ of integers. Then the density of A ∪ B is given by
Corollary 2 dens(B−A) = dens(B)−dens(A∩B).
Proof: The proof is straightforward:
dens(A ∪ B) = dens(A) + dens(B) − dens(A ∩ B).
(25)
dens(B) = dens((A ∩ B) ∪ (Ac ∩ B))
= dens(A ∩ B) + dens(Ac ∩ B)
= dens(A ∩ B) + dens(B − A),
(16) Proof: Observe that A ∪ B = A ∪ (B − A), and
(17) A ∩ (B − A) = ∅. Then, it follows directly from the
properties of the proposed density measure that
(18)
(26)
since A ∩ B and Ac ∩ B are disjoint.
¤ dens(A ∪ B) = dens(A ∪ (B − A))
= dens(A) + dens(B − A)
(27)
Corollary 3 For every A, dens(Ac ) = 1−dens(A),
=
dens(A)
+
dens(B)
−
dens(A
∩
B).
where Ac is the complement of A.
(28)
1 For ease of exposition, in the following results, we assume
that the densities of the relevant sets always do exist.
¤
Proposition 5 (Heavy Tail) Let
A
= have that ∀N dens(A) = dens(AN
2 ). For the same
{a1 , a2 , . . . , aN −1 , aN , aN +1 , . . .}.
If A1
= reason, ∀N , dens(A ⊕ 1) = dens(AN
2 ⊕ 1).
{a1 , a2 , . . . , aN −1 } and A2 = {aN , aN +1 , . . .},
Note that, for s > 1, if kn ≥ n + 1, then the
following inequalities hold:
then
P
P
P
1
1
1
dens(A) = dens(A2 ).
(29)
s
s
s
n∈AN
n∈AN
n∈AN
2 (kn)
2 (n+1)
2 n
≤
≤
.
ζ(s)
ζ(s)
ζ(s)
Proof: Observe that A = A1 ∪A2 and A1 and A2 are
(38)
disjoint. Therefore, dens(A) = dens(A1 )+dens(A2 ).
Since A1 is a finite set, dens(A1 ) = 0.
¤
In other words, for the above inequality to be valid,
we must have
Now consider the following operation m ⊗ A ,
n+1
{ma | a ∈ A, m ∈ N}. This can be interpreted as a
k≥
,
∀n ∈ AN
(39)
2 .
dilation operation on the elements of A.
n
In [4, 5, 9], Erdös et al. examined the density of
the set of multiples m ⊗ A, showing the existence of For instance, let k = (aN + 1)/aN . Consequently,
a logarithmic density equal to its lower asymptotic we establish that
density. Herein we investigate further this matter, P
1
P
P
1
1
a +1
evaluating the proposed density of sets of multiples.
n∈AN
s
2 ( N
s
s
n∈AN
n∈AN
(n+1)
aN n)
2
2 n
≤
≤
.
ζ(s)
ζ(s)
ζ(s)
Proposition 6 (Dilation) Let A be a set, such as
(40)
dens(A) > 0. Then
Taking the limit as s ↓ 1, the above upper bound
(30) becomes
P
1
s
n∈AN
Proof: This result follows directly from the defini2 n
lim
= dens(AN
(41)
2 ) = dens(A).
tion of the proposed density:
s↓1
ζ(s)
P
1
Now examining the lower bound and using the din∈A (mn)s
dens(m ⊗ A) = lim
(31) lation property, we have that
s↓1
ζ(s)
P
P
1
1
1
dens(m ⊗ A) =
1
dens(A).
m
= lim
s↓1
=
ms
n∈A ns
ζ(s)
1
dens(A).
m
(32)
lim
s↓1
a +1
n∈AN
2 ( N
n)s
a
N
ζ(s)
aN
dens(AN
2 )
aN + 1
aN
=
dens(A).
aN + 1
(33)
=
¤
Let A⊕m , {a+m | a ∈ A, m ∈ N}. This process
is called a translation of A by m units [10, p.49].
Our aim is to show that the proposed density is Since ∀² > 0, ∃N such that
translation invariant, i.e., dens(A ⊕ m) = dens(A),
aN
dens(A) > dens(A) − ²,
m > 0. Before that we need the following lemma.
aN + 1
(42)
(43)
(44)
(45)
Lemma 1 (Unitary Translation) Let A be a and as
set, such as dens(A) > 0. Then
P
dens(A ⊕ 1) = dens(A).
(34) lim
s↓1
Proof: Let A = {a1 , a2 , . . . , aN −1 , aN , aN +1 , . . .}.
We can split A into two disjoint sets as shown below:
A = {a1 , a2 , . . . , aN −1 , aN , aN +1 , . . .}
= {a1 , a2 , . . . , aN −1 } ∪ {aN , aN +1 , . . .}
N
= AN
1 ∪ A2 ,
(35)
(36)
(37)
N
where AN
1 is a finite set and A2 is a ‘tail’ set starting
at the element aN . By the Heavy Tail property, we
1
s
n∈AN
2 (n+1)
ζ(s)
= dens(AN
2 ⊕ 1) = dens(A ⊕ 1),
(46)
it follows that ∀² > 0 we have that
dens(A) − ² ≤ dens(A ⊕ 1) ≤ dens(A).
(47)
Therefore, letting ² → 0, it follows that
dens(A ⊕ 1) = dens(A).
(48)
¤
Proposition 7 (Translation Invariance) Let A As matter of fact, any
P∞criterion that ensure the conbe a set, such as dens(A) > 0. Then
vergence of lims↓1 n=1 a1s can be taken into conn
sideration. In the next result, we utilize the Ratio
dens(A ⊕ m) = dens(A),
(49) Test for convergence of series.
where m is a positive integer.
Corollary 5 Let A = {a1 , a2 , . . . , an , an+1 , . . .}. If
n
limn→∞ aan+1
< 1, then dens(A) = 0.
Proof: The proof follows by finite induction. We
have already proven that dens(A ⊕ 1) = dens(A). Proof: Observe that for s > 1
Therefore, we have that
∞
X 1
X 1
X
1
<
=
.
(52)
s
n
n
a
dens(A ⊕ (m + 1)) = dens((A ⊕ m) ⊕ 1) (50)
n=1 n
n∈A
n∈A
= dens(A ⊕ m) = dens(A).
(51)
According to the Ratio Test [11, p.68], the series on
¤ the right-hand side of the above inequality converges
One possible application for a density on a set of whenever
natural numbers A is to interpret it as the chance
1
an
of choosing a natural number in A when all natural
an+1
= lim
< 1.
(53)
lim
1
numbers are equally likely to be chosen. Interestn→∞ an+1
n→∞
an
ingly, as we show next the above measure of uncerP
tainty does not obey all the axioms of Kolmogorov
1
since it is not σ-additive. Additionally, we show Thus, by series dominance, it follows that n∈A ns
that it is impossible to define a finite σ-additive also converges. And finally, this implies that the
¤
translation invariant measure on (N, 2N ). This re- density of A must be zero.
sult emphasizes an important point that there are Lemma 2 (Powers of the Set Elements) Let
reasonable measures of uncertainty that do not sat, . . .} and m > 1 be an integer. The
isfy the formal standard definition of a probability A = {a1 , a2m
m
measure.
density of A , {am
1 , a2 , . . .} is zero.
Theorem 1 There is no σ-additive measure de- Proof: Notice that, for s > 1 and m > 1,
fined on the measurable space (N, 2N ) such that:
∞
∞
∞
X
X
X
1
1
1
<
<
<∞
1. 0 < µ(N) < ∞; and
(am )s
am
nm
n=1
2. µ is translation invariant.
n
n=1
n
(54)
n=1
Thus, it follows from the preceding discussion that
¤
Proof: Suppose that µ is translation invariant. Then dens(A) = 0.
every singleton set must have the same measure. Let Corollary 6 The density of the set of perfect
ω1 < ω2 be any natural numbers. Since {ω1 }⊕(ω2 − squares is zero.
ω1 ) = {ω2 }, we have
Corollary 7 (Geometric Progressions) Let
µ({ω2 }) = µ({ω1 } ⊕ (ω2 − ω1 )) = µ({ω1 }),
G = {ar, ar2 , ar3 , . . .}, where a and r > 1 are
positive integers. Then
where the last equality follows from translation indens(G) = 0.
(55)
variance.
Let c = µ({1}). If c = 0, then µ(N) 6= 0 =
P∞
µ({i}).
Thus
µ
is
not
σ-additive.
If
c
>
0,
i=1
P∞
Proof: Notice that, for r > 1,
then µ(N) < ∞ = i=1 µ({i}), which also implies
∞
∞
∞ µ ¶n
X
that µ is not σ-additive.
¤
1 X 1
1 X 1
1
=
<
<∞
(arn )s
as n=1 rns
as n=1 r
n=1
Proposition 8 (Criterion for Zero Density)
(56)
Let A
{a1 , a2 , . . . , an , an+1 , . . .}.
If
P∞ =
lims↓1 n=1 a1s converges, then dens(A) = 0.
Thus, it also follows from the preceding discussion
n
that dens(G) = 0.
¤
Proof: It follows directly from the definition of
Let Mp = {p, 2p, 3p, . . .} be an arithmetic prodens(A).
¤ gression, where p ∈ N. For example, M2 =
{2, 4, 6, . . .}, M5 = {5, 10, 15, . . .}, etc. Regarding
Definition 2 (Sparse Set) A zero density set is the cardinality of these sets, we have that kMp k =
said to be a sparse set.
ℵ0 .
Proposition 9 (Arithmetic Progressions) For
a fixed integer p, the density of the set Mp is given
by
1
(57)
dens(Mp ) = .
p
0.8
1
1
dens(N) = . (58)
p
p
¤
Consequently, we have, for instance, dens(M1 ) =
dens(N) = 1, dens(M2 ) = 1/2, dens(M5 ) = 1/5 etc.
One can derive a physical interpretation for the
density of Mp . Consider discrete-time signals Mp [n]
characterized by a sequence of discrete-time impulses associated to the sets Mp . The signals Mp [n]
are built according to a binary function that returns
one if n is a multiple of p, and zero otherwise. For
example, we can have M3 [n] and M5 [n], associated
to M3 and M5 , as depicted in Figure 1. In terms of
signal analysis, dens(Mp ) has some correspondence
to the average value of Mp [n].
0.6
0.4
0.2
0
0
6
9
12
15
18
21
(a)
1.2
1
0.8
0.6
0.4
0.2
0
0
3
3
n
M5[n]
dens(Mp ) = dens(p ⊗ N) =
1
M3[n]
Proof: Note that Mp = p ⊗ N. Thus, according to
Proposition 6, we have that
1.2
5
10
15
20
25
n
Density of Particular Sets
(b)
In this section, some special sets have their density Figure 1: Discrete-time signals M [n] and M [n] as3
5
examined. We focus our attention in three notable sociated to the sets M and M , respectively.
3
5
sets: (i) set of prime numbers, (ii) Fibonacci sequence, and (iii) set of square-free integers.
3.1
Now consider the following inequalities:
Set of Prime Numbers
Let P = {2, 3, 5, 7, 11, 13, 17, 19, 23, . . .} be the set
of prime numbers.
P∞
ln ζ(1 + ²) − k=2
ζ 0 (1 + ²)
0≤
=
ζ(1 + ²)
ζ(1 + ²)
(62)
Proposition 10 The prime numbers set is sparse.
Proof: Utilizing the
P definition of−sthe prime zeta
function, ζ 0 (s) ,
i : pi is prime pi , we have that
=
0
(s)
. Additionally observe that
dens(P ) = lims↓1 ζζ(s)
ζ 0 (s) < ∞ for s > 1. Taking in account that
ln ζ(s) =
∞
X
ζ 0 (ks)
k=1
k
≤
ln ζ(1 + ²)
−
ζ(1 + ²)
(59)
lim
∞
X
ζ 0 (k(1 + ²))
(60)
k
= ζ 0 (1 + ²) +
∞
X
k=2
k=2
ζ(1 + ²)
. (63)
(64)
As ² → 0, we have that ζ(1 + ²) → ∞, therefore
,
for ² > 0 we have that
k=1
∞ ζ 0 (k(1+²))
X
k
ln ζ(1 + ²)
.
ζ(1 + ²)
²→0
ln ζ(1 + ²) =
ζ 0 (k(1+²))
k
(61)
(65)
because limx→∞ ln(x)/x = 0. Therefore, we can apply the Squeeze Theorem once more, and find that
ζ 0 (1 + ²)
= 0.
²→0 ζ(1 + ²)
lim
0
ζ (k(1 + ²))
.
k
ln ζ(1 + ²)
= 0,
ζ(1 + ²)
(66)
¤
Thus the first elements of S are given by
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, . . .
1.2
1
Proposition 12 The density of the set of square1
free integers is ζ(2)
.
p[n]
0.8
P∞
Proof: It is known [14] that
n=1
thus it follows easily that:
0.2
P
1
s
dens(S) = lim n∈S n
0
0
5
10
15
20
25
30
s↓1
ζ(s)
n
P∞ |µ(n)|
s
= lim n=1 n
Figure 2: Discrete-time signal P [n] corresponding
s↓1
ζ(s)
to the set of prime numbers.
0.6
0.4
= lim
Since the density is zero, it indicates that the associated discrete-time signal P [n], as shown in Figure 2, has null average value.
3.2
s↓1
=
ζ(s)
ζ(2s)
ζ(s)
= lim
s↓1
1
6
= 2.
ζ(2)
π
|µ(n)|
ns
=
ζ(s)
ζ(2s) ,
(70)
(71)
1
ζ(2s)
(72)
(73)
¤
Fibonacci Sequence
The Fibonacci sequence constitutes another interComputational Results
esting subject of investigation. Fibonacci numbers 4
are constructed according to the following recursive
equation u[n] = u[n − 1] + u[n − 2], n > 1, for u[0] = Definition 3 Let A be a set of positive integers.
T
u[1] = 1, where u[n] denotes the nth Fibonacci num- The truncated set A is defined by
ber [12, p.160]. This procedure results in the FiAT = A ∩ {1, 2, 3, . . . , T − 1, T },
(74)
bonacci set F = {1, 2, 3, 5, 8, 13, 21, 34, 55, . . .}.
Proposition 11 The Fibonacci set is sparse.
where T is a positive integer.
Proof: It is known that the sum of the reciprocals
of the Fibonacci numbers converges to a constant
(reciprocal Fibonacci constant), whose value is approximately 3.35988566 . . . [13]. Consequently, for
s > 1, we have that
P∞
1
In other words, AT contains the elements of A which
are no greater than T .
The concept of truncated sets can furnish computational approximations for the numerical value of
some densities. Consider, for example, the quantity
dens(F ) = lim
s↓1
P∞
1
< lim
s↓1
3.3
n=1 u[n]
ζ(s)
n=2 u[n]s
ζ(s)
= lim
s↓1
(67)
d0 (AT ) ,
kAT k
.
kNT k
(75)
0
3.35988566 . . .
= 0. (68) This expression d (·) can be interpreted in a freζ(s)
quentist way as the ratio between favorable cases
and possible cases. Clearly, the asymptotic density
¤ can be expressed in terms of d0 (·):
Set of Square-free Integers
d(A) = lim d0 (AT ).
(76)
T →∞
The set of square-free numbers can be defined as
In a similar fashion, we can consider a version of
S , {n ∈ N | |µ(n)| = 1}, where µ(·) is the Möbius
the discussed density for truncated sets as defined
function, which is given by
below. First, observe that discussed density dens(·)

1,
if
n
=
1,
can be expressed as the following double limit

2
0,
if
p
|n,
for
some
prime
number
p,
P
µ(n) =
1
(−1)r , if n is the product of distinct
T ns
dens(A) = lim lim Pn∈A
.
(77)
prime numbers.
T
1
s↓1 T →∞
(69)
n=1 ns
Restricted to the class of set for which the above
limits can have their order interchanged, we find
that
P
1
T ns
dens(A) = lim lim Pn∈A
(78)
T
1
dens’(MT2 ), d’(MT2 )
0.5
T →∞ s↓1
n=1 ns
!
1
T
n∈A n
.
PT 1
n=1 n
ÃP
= lim
T →∞
(79)
0.4
0.3
0.2
0.1
Definition 4 (Approximate Density) The ap0 0
1
2
3
10
10
10
10
proximate density for a truncated set AT is given
T
by
P
Figure 3: Computational result for the approximate
1
T n
density for the truncated prime set M2T , according
.
(80)
dens0 (AT ) = Pn∈A
T
1
to (i) the function dens0 (P T ) (bold curve), and (ii)
n=1 n
the function d0 (P T ) (thin line). Observe the conAgain, whenever the limit order of Equation 77 can vergence to 1/2.
be interchanged, we have that
dens(A) = lim dens0 (AT ).
(81) the number of primes that do not exceed x, is to utilize the approximation for π(T ) given by the Prime
0
The quantity dens (·) furnishes a computationally Number Theorem [12, 15, p.336]:
feasible way to investigate the behavior of dens(·).
π(T ) ≈ li(T ),
(84)
In the following, we obtain computational approximations for the asymptotic density and the dis- where li(·) denotes the logarithmic integral [16]. The
cussed density. The approximate densities are nu- curves displayed in Figure 4 correspond to the calmerically evaluated as T increases in the range from culation of dens0 (P T ), d0 (P T ) and li(T )/T . As ex1 to 1000. Now we investigate (i) arithmetic pro- pected, all curves decay to zero.
gressions, (ii) the set of prime numbers, and (iii) the
Fibonacci sequence.
T →∞
4.3
4.1
Arithmetic Progressions
Considering an arithmetic progressions Mq , we have
that:
1
d(Mq ) = lim d0 (MqT ) = ,
(82)
T →∞
q
and
Now consider the truncated Fibonacci set F T =
{1, 2, 3, 5, 8, 13, 21, . . . , T }. The result of calculating
d0 (F T ) =
and
T →∞
Prime numbers sets
Consider the truncated set of prime numbers P T =
{p : p is a prime, p < T ). One may easily compute
both d0 (P T ) and dens0 (P T ) for any finite T .
An alternative path for the estimation of
)
d0 (P T ) = π(T
T , where the function π(x) represents
1
n
1
n=1 n
dens (F ) = Pn∈F
T
(83)
Both quantities limits converge to the same quantity 1/q. Intuitively, we have that the chance of
“selecting” a multiple of q among all integers is 1/q.
This is exactly the density of Mq . Figure 3 shows
the result of computational calculations of d0 (·) and
dens0 (·) for M2T .
kF T k
kNT k
P
0
dens(Mq ) = lim dens0 (MqT ).
4.2
Fibonacci Sequence
T
T
(85)
,
(86)
are shown in Figure 5. Both curves tend to zero as
T grows. This fact is expected since we have already
verified that dens(F ) = 0. The small convergence
rate of dens0 (·) is typical of problems that deals with
the harmonic series.
5
Conclusion
A density for infinite sets of integers was proposed.
It was shown that the suggested density shares several properties of usual probability. A computational simulation of the proposed approximate density for finite sets was addressed. An open topic is
the characterization of sets for which the limiting
value of the approximate density equals their density.
[3] R. L. Duncan. Some continuity properties of
the Schnirelmann density II. Pacific Journal
of Mathematics, 32(1):65–67, 1970.
dens’(PT), d’(PT), li(T)/T
0.8
0.6
[4] Paul Erdös. On the density of some sequences
of integers. Bulletin of the American Mathematical Society, 54(8):685–692, August 1948.
0.4
[5] H. Davenport and P. Erdös. On sequences of
positive integers. Journal of the Indian Mathematical Society, 15:19–24, 1951.
0.2
0 0
10
1
2
10
10
3
10
T
Figure 4: Computational result for the approximate
density for the truncated prime set P T , according
to (i) the function dens0 (P T ) (bold curve), (ii) the
function d0 (P T ) (thin line), and (iii) the approximation based on the Prime Number Theorem (dotted
line).
1
dens’(FT), d’(FT)
[7] Ivan Niven and Herbert S. Zickerman. An Introduction to the Theory of Numbers. John Wiley & Sons, Inc., 1960.
[8] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. Academic Press,
New York, 4a. edition, 1965.
[9] P. Erdös, R. R. Hall, and G. Tenebaum. On the
densities of sets of multiples. Journal für die
reine und angewandte Mathematik, 454:119–
141, 1994.
0.8
0.6
[10] Walter Rudin. Real and Complex Analysis.
McGraw-Hill, 1970.
0.4
0.2
0 0
10
[6] Rudolf Ahlswede and Levon H. Khachatrian.
Number theoretic correlation inequalities for
Dirichlet densities. Journal of Number Theory,
63:34–46, 1997.
[11] Stephen Abbott.
Springer, 2001.
1
2
10
10
Understanding Analysis.
3
10
T
[12] David M. Burton. Elementary Number Theory.
McGraw-Hill, 1998.
Figure 5: Density of the truncated Fibonacci set:
[13] R. André-Jeannin. Irrationalité de la somme
dens0 (F T ) (bold line) and d0 (F T ) (thin line).
des inverses de certaines suites récurrentes. In
C. R. Acad. Sci. Paris Sér. I Math., volume
308, pages 539–541, 1989.
Acknowledgments
The first author would like to thank Dr. P. B. Oliva,
Department of Computer Science, Queen Mary,
University of London, for many insightful commentaries and suggestions.
References
[14] M. R. Schroeder. Number Theory in Science
and Communication, volume 7 of Springer Series in Information Sciences. Springer, 3rd edition, 1997.
[15] H. Riesel. Prime Numbers and Computer Methods for Factorization. Birkhäuser, 1985.
[16] M. Abramowitz and I. Segun. Handbook of
Mathematical Functions. Dover, New York,
[1] Jason P. Bell and Stanley N. Burris. Dirich1968.
let density extends global asymptotic density in multiplicative systems.
Preprint
published at htt://www.math.uwaterloo.ca/
~snburris/htdocs/MYWORKS/, 2006.
[2] R. L. Duncan. Some continuity properties of
the Schnirelmann density. Pacific Journal of
Mathematics, 26(1):57–58, 1968.