Internat. J. Math. & Math. Sci.
VOL. 13 NO 2 (1990) 383-386
383
TAU NUMBERS, NATURAL DENSITY, AND HARDY AND WRIGHT’S THEOREM 437
ROBERT E. KENNEDY and CURTIS N. COOPER
Department of Mathematics and Computer Science
Central Missouri State University
Warrensburg, MO 64093
(Received February 7, 1989 and in revised form March 9, 1989)
{n: (n) is a factor of n} is called a Tau
number, where T(n) denotes the number of divisors of the integer n. We determine the
natural density of this set by use of Hardy and Wright’s Theorem 437 (4th ed.).
ABSTRACT.
I.
An
element
of
the
set T
INTRODUCTION.
Hardy and Wrlght’s, An Introduction to the Theory of Numbers [I] is one of the
classic
references for number theorists.
If one has a number theoretic question,
chances are that it is either answered or will be a corollary to a result contained in
this book.
This note relates how the latter possibility can occur.
Recently,
a
colleague
of
ours
(Alvin
Tinsley)
questions that concerned the divisors of an integer.
was
searching
for
suitable
One such question was, "Find an
Integer n such that (n) divides n" where (n) denotes the number of divisors of n.
He, of course, found many such integers that met this condition. However, for his own
curiosity, he investigated whether or not such integers could be characterized.
a characterization is still an open problem.
Such
However, since we are often interested
in the concept of natural density, we also investigated the existence of the natural
density of this set of integers.
In what follows, we outline the steps that show that
the natural density of this set is 0.
2.
NOTATION AND TERMINOLOGY.
For a set, A, of integers, A(x) denotes the number of members of A not exceeding
x.
If
llm
x
A(x)
x
exists, we call this limit the natural density of A, and is symbolized by (A).
we consider the set of positive integers
Here
R.E. KENNEDY AND C.N. COOPER
384
{n: T(n) is a factor of n}
T
In what follows, an element of T will be called a Tau number.
and determine (T).
Thus, we will determine (T) in order to show that the natural density of T is zero.
Any undefined notation or terminology can be found in the basic references Hardy and
Wright [I] and Apostol [2].
THE NATURAL DENSITY OF K N T.
3.
Then
Let k be a fixed positive integer, and K be the collection of k-free integers.
n E K N T implies that n is both k-free and T(n) is a factor of n. If the prime
factorizatlon of n is
n
t
Pi
i--I
then
i
t
n (n i + I).
(n)
ill
t
Le
(n)-i=l
hi,
and note that since n is k-free, (n) can also be written in the form
2 ml 3m2 4m3
z(n)
where
mj
#{ni:
n
j}.
i
.
Also since n
k-I
(n)
k-I
j.
Jl
This
gives
an
upper
k
bound
mj
ink_
T,
k2
k.
mj
k
It follows that
3.
jl
for (n) which,
though seemingly
too
large,
will
be
sufficient for our purposes.
Stating this result as a lemma, we have
LEMMA 3.1.
integers.
4.
Let k be a positive integer,
3
T.
k for each n
K
and K be the collection of k-free
Then (n)
A COROLLARY TO HARDY AND WRIGHT’S THEOREM 437.
In [I; page 368], the following theorem is presented:
(Theorem 437) Let T
{n: (n)
t}.
Then,
THEOREM 4.2.
t
Tt(x
Since by Lemma 3.
t-I
x(log log x)
(t
I)! log x
I,
k
K
3
Tc__tffiUl
Tt’
TAU N791BERS, NATURAL DENSITY AND HARDY AND WRIGHT’S THEOREM
385
it follows that
k
3
[ Tt(x).
t=l
(K0 T)(x)
Hence by Theorem 4.2,
llm
.
k
(KOT)(x)
lira
x/
k3
t=l
3
T (x)
t
x
(log log x)
t-I
x/
0.
Thus, (K 0 T)ffi 0, and we have, as a corollary to Hardy and Wright’s Theorem 437,
that the natural density of the intersection of the set of k-free integers and the Tau
numbers Is 0.
This result is the key to finding (T).
By observing that since Tc__K’ U(KOT),
we have that
K’(x) + (KOT)(x)
X
X
T(x)
X
Here, K’ is the complement of K with respect to the positive integers.
Thus,
K(x)
T(x)
X
X
and we see that knowing the natural density of the k-free integers will give some
In fact, knowing the value of (K)will help to determine
information about (T).
that the value of (T) is 0.
5.
THE NATURAL DENSITY OF THE TAU NUMBERS.
Using the above results, and the well-known fact that
(K)
(k)
we have that
T(x)
llm
for any k
)
2.
;(k)
x
X
But since
lira
(k)
I,
k/
we conclude that
is
O.
(T)
O, and have proven that the natural density of the Tau numbers
386
R.E. KENNEDY AND C.N. COOPER
REFERENCES
1.
HARDY, G.H. and WRIGHT, E.M., An Introduction to the Theory of Numbers, 4th ed.
Oxford: Clarendon Press (1960).
2.
APOSTOL, T.M., Introduction to Analytic Number Theory, Sprlnger-Verlag, 1976.
3.
KNUTH, D.E., The Art of Computer Programmi. ng, Vol. I, Addlson-Wesley, 1969.