Internat. J. Math. & Math. Sci.
VOL. 12 NO.
(1989) 123-130
123
ON INTEGERS n WITH Jt(n) < Jt(m) For m > n
J. CHIDAMBARASWAMY
and
P.V. KRISHNAIAH
Department of Mathematic-s
University of Toledo
Toledo, OH 43606
(Received April I0, 1987)
ABSTRACT.
Let F
all m > n,
Jt(n)
as
F
t
be the set of all positive integers n such that
<
Jt(m)
for
In this paper, it
being the Jordan totient function of order t.
been proved that (I) every postive integer d divides infinitely many members of
n
as n
in F
(3) every
(2) if n and n’ are consecutive members of F t,
t
prime p divides n for all sufficiently large n
Ft(x)
is the number of n
KEYS WORDS AND PHRASES.
g
F
t
F
g
t
and (4) Log
Ft(x)
t
<< log
x where
x.
that n
Euler totient function, Jordan totient function of order t.
IIA,N.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
Jr(n)
INTRODUCTION.
In [I] Masser and Shlu consider the set F of positive integers n such that
(n)
<
(m) for all m
(n) being the Euler totlent function.
> n,
Calling members
of F sparsely totient numbers, they prove, among other results, that (I) every
integer divides some member of F (2) every prime divides all sufficiently large
members of F (3) the ratio of consecutive members of F approaches
(4) log F(x)
<<
log1/2x where
and
F(x) is the counting function of F; that is, the number of
members of F which do not exceed x.
In this paper, using similar methods, we extend the above results to the set F
t
Jt(n)
of all positive integers n such that
<
Jt(m)
well known Jordan totlent function of order t [4].
as the number of incongruent t-vectors (a
it being understood that t-vectors (a I,
if a
i
bi(mod
g i g t and that
n) for
Jt(n)
Moreover this function
n
t
Jt(n)
(I
p
a
at)
Jt(n)
t
for all m > n,
Jr(n) being the
Jr(n) is defined
that ((a
at)n)
We recall that
mod n such
and (b
b
t)
I,
are congruent mod n
is given by the formula
-t)
coincides with Cohen’s [3] generalization
Euler totient function, defines as the number of positive integers a n
(I.I)
@t(n)
t
of the
such that
J. CHIDAMBARASWAMY AND P.V. KRISHNAIAH
124
t
(a,n)
where (x,y)
t
Clearly
denotes the largest t th power common divisor of x and y.
t
l(n))
(and hence
Jl(n)
is the same as (n).
Denoting by (n) the number of distinct prime factors of n, we order these prime
P1 > P2
factors as
>
P
thus
(n)"
Pr Pr(n)
is the r th largest prime factor of
Likewise we order the primes not dividing n as
n.
Pr
>
for the r th prime in the ascending
n, we write
n,
<
Q1
Q2
<
Further we write
sequence of all primes.
For positive integral
to denote the quotient of n by its largest square free divisor.
positive integer u, we write a
t x
t+u
+ (t+u)
x
For a
for the unique positive root of the equation
t
u
(1.2)
0.
It may be directly verified that
2u
t+u
Ft(x)
Finally we write
2.
=<
a
u
<
(1.3)
for the number of members of F
t
that do not exceed x.
MAIN RESULTS.
We prove the following results:
Let k
THEOREM 2.1.
d <
I,
2, d
0 be integers such that
(2.1)
Pk+l
and
d
Then d
t
t
.1.
tF
Every prime p divides n for all sufficiently large n
tLet u be a fixed positive integer. As n
in F we have
t
THEOREM 2.2.
THEOREM 2.3.
log-In
(a)
lim inf
Pu (n)
(b)
lim sup
Pl(n) log-ln
(c)
au
(d)
(2.2)
is a member of F
tEvery positive integer d divides infinitely many members of F t.
-I
If n and n’ are consecutive members of F t, then n n
as
COROLLARY 2.2.
in F
I).
PI’’" Pk-lPk+
COROLLARY
n
(d+l)t(p
1) <
(Pk+
llm inf
n
llm sup
n
2
Qu(n) log-In
llm sup
n
(n) log-In
(2.3)
au-I
Pt+u(n) log-ln
(2.4)
and
(e)
lim sup
n
Pl(n) log-t-In
THEOREM 2.4.
3.
log
(2.5)
Ft(x) << logx.
FOR THE PROOFS OF THE THEOREMS WE NEED THE FOLLOWING LEMMAS.
LEMMA 3.1.
Let r be a positive integer and
numbers satisfying (i)
(ii)
S t
Xl’’’Xr
x <
yl...yr
x
y"
i
Yi
Then
and y
(Xl-l)
x
i
Xl,...,Xr,
for i
(x r-l)(x-l) <
YI’’’’’Yr’
x and y be real
r and
(Yl-l)...(Yr-l)(y-l).
ON INTEGERS n WITH
Jr(n)
Jr(m)
FOR m > n
125
This is lemma 3.1 of [6].
For positive integers a and b we write f(a,b) to denote the smallest multiple of b
Clearly
that exceeds a.
a < f(a,b) _-< a
LEMMA 3.2.
If n e F
then k-2t < re(n)
PROOF.
n <
(3.2)
Pk Pk+l
Pl
k.
The second inequality in (3.2) implies that m(n)
possible, that 0 < m(n)
f(n,
(I
p
i-t)
< m <_-I
n
On the other hand, since
Pl
Suppose if
k.
-t
(I
(3.3)
Pk_2t
Pk-t )’ we have, by
+
+ Pl" n "Pk-t
Pl
-<
It follows from (I.I) that
k- 2t.
Jt(n)m-t
Choosing m
(3.1)
b.
and
t
Pk
Pl
+
(3.1) and (3.2),
+
<
Pk-t+l" "Pk
(3.4)
Pk-t
divide m, we have, in virtue of (3.3),
Pk-t
-t
-t
-t
Jt(m)m -<- (l-Pl
(l-pk-t)
-<-
Jt(n)n-t (l-Pk-t2t+l)
=<
t
Jt (n)n-t (l-pk-tt)
(l-Pk-t)
so that, (3.4) now yields
Jt (m)
Jt(n)
+
< (I
-t
Pk_t)
t
-t
(I
Pk_t)
<
Hence the lemma follows.
contrary to the hypothesis that n e F
t
REMARK I. Since for each n there is a unique k such that (3.2) holds, lemma 3.2
implies that m(n)
in F
as n
t
eu Pt+u<
t + u then
LEMMA 3.3. If u is a positive integer, n e F t and m(n)
-I f(a,b) so that
and
na
m
PROOF. We write a
b
QI
Qu
PI
+
n
Since
Qu
Q1
PI
Pt+u
so that
Pt+u’
b
+
u
Qu p-t-u
t+u
(3.5)
are prime factors of m but not of n where as the prime factors
of n may or may not divide m, we have
Jt(m)m
-i
(l-Qut)( l-p-t1
(l-Qlt)
(3.5) and the hypothesis that n
<
Jt(m)(Jt(n))-I
Qu"
<
F
t
uP-t-u.t+u u
(l+QU
(l+Qu p-t-u,
t+u
u
t
I--t
-I
t(n)n
t+n)
J
(l-p?t)
-I
together imply
i=l
<
-I
t-
(l-Qi
(I-
-t u
Qu
t+u
i=l
(I
-t
-Pt+u
-t-u
-t
126
J. CHIDAMBARASWAMY AND P.V. KRISNAIAB
Taking (t
+ u) th
roots and employing the well known inequality x
r
<-
+ r(x-l)
for x > 0, 0 < r < I, we obtain
I-Q:
<
l-uQ t (t+u)-I
< l-uQ
Cancelling
1+Q.U
<
-t
-I
(t+u)
u
+
u
-t-u
Pt+u (tdm)-I
tQu
(t+u)Q
in the above and multiplying by
t(Qu
t+u
-I
Pt+u
+ (t+u)
(Qu
When t
-I
t
Qu
Pt+u
Pt+u
which, in virtue of (1.2), implies that
REMARK 2.
Pt+u-t-u (t+u)-I)
u
l+tQu
u > 0
-I
this lemma 4 of
u
we arrive at
>
u
[I] and when
this yields a
t
slight improvement of lemma 7 of [I].
m
t(Ql(n))t+l.
LEMMA 3.4. For n e F t we have Pl(n) <
PROOF. We write P for Pl(n) and Q for
-I we see
n f(P,Q)P
that
Ql(n)"
Suppose P
->
t
Q
t+-+
choosing
+_=_<o +__l
n
(3.6)
tQt
Qt+l
Arguing as in lemma 3,3, we have, since P
Jt (m) m-t
<
by our assumption,
l-p-t)-I Jt (n)n-t
(1-Q-t)
(l_Q-t)(l_Q-t(t+l))
+ Q_t
(l+Q-t +
Jt(n)n
-I
2
-t
Jt(n)n
-t
so that by (3.6)
Jt (m) (Jt (n) )-I
contrary to the hypothesis.
(l+Q-t +
+
Q-t 2)-I (l+t-IQ-t)t
.< I,
This establishes the lemma.
For n
F t,
n,
Writing m
n f
(n,,Q)n,
LEMMA 3.5.
PROOF.
.<
<
t(Ql(n))
t+l
we note that
+Q
<m<=
n
n,
so that, since n e
exp(O)
Ft,
Jt(m)(Jt(n))-I
<=
t
(l-Q-t)(l+Qn, I)
< exp
(-Q-t)exp(tQn, I).
The lemma follows on comparing the exponents.
LEMMA 3.6.
A < p
Let A > 0, M > 3 and (A;M) be the number of primes p with
A+M ((A;M)
(A+M)
w(A)).
Jr(n)
ON INTEGERS n WITH
<
Jr(m)
127
FOR m> n
Then
<-
(A;M)
2M(log M)
-I {i
+ 0 (log log M
log-IM)}
and the 0-constant is independent of A.
This is Theorem 4.5, Chapter 19 of [2].
REMARK 3.
Lemma 3.6 implies that (A;M)
3M log
M
for all A
0 and
sufficiently large M.
4.
PROOFS
OF THEOREMS.
PROOF OF THEOREM 2.1.
(2.2).
Let n
d pl...Pk_l
From (I.I) and (2.2) we have
t
Jt (n) <= dt (Pl -I)
t
(d+t)t(pl-I)
<
Let m > n.
t
(t
Pk_l)(Pk+
Jt(m)
-> m t
t
s->
Case (I)
k+l.
(plt_l
>
(Pl-l)
">
Jt(n)
<=
s
(l-ps
(Ps
& m <
Pl
Ps+l
and the last
-t
t-I
(4.2)
t
(pkt-l)(Pk+l-l)
t
t
(Pk-l)(d+l)
t
by (2.1)
by (4.1)
k, (m) .< k-l.
Jt(m)m-t
Ps
We have, by (4.2),
Jr(m) ->
Case (2)
-t
(Pl t_l
>-
Pl
(4 I)
Hence
s.
(l-Pl
-I)
(Pk-l- I) (pkt-I
There is a unique s such that
inequality implies that m(m)
where d,k, satisfy (2.1) and
Pk+
In this case
(l-Pl -t)
-t
(I
Pk-I
where as
-t
Jt (n)n-S (l-p t)
(1-plt)
so that
Jt(m)
Case (3)
>
-t
(l-Pk_ I) (l-Pk+)
--t
(l-Pk_ I)
Jr(n)"
s
m(m) and
k
t
Jt(m)
t
t
(Pl-l)
__> (d+l) t
>
d+l.
In this case s
(pt-l)
m,
m,
m, =>
Jt (n)
t
(Pl-l)
t
(Pk-l)
t
(Pk-l)
k
(m) and
128
J. CHIDAMBARASWAMY AND P.V. KRISHNAIAH
in virtue of
(4.1).
Case (4)
-<
s
(m) and
k
m, =<
q
k-l, Y
l
Taking r
>
qk-I qk
ql
t
Pl
Pk-I
q
y
x
t
t
p
i
(qk_l)(qk_l)
(ql_l)
>
ql ">
Then
m,
(d Pk+g
and
qk’ ql ’s
m, ql
Let m
d.
distinct prime factors of m in ascending order.
x
d
-I
t t
-t
Pk+gm*
(P-I
(p-l)
Pl
being the
and since m > n we have
I) (d
in lemma 3
we obtain
-t
Pk+ m,-I)
t t
so that
(Pk_l-l) (d
> dt
">
by (4.1).
t
(Pk_l-l)(Pk+ -I)
Jt (n)
This completes the proof of Theorem 2.1.
PROOF OF COROLLARY 2.1.
Let d be any positive integer.
that (2.1) holds, we can take
each k
t
t
(Pl-l)
pk+-m,
2 such that
0 so that (2.2) holds.
For each k
Thus d
2 such
Pk
Pl
Ft
for
> d+l.
Pk+l
As the proof of Corollary 2.2 is essentially the same as that of the Corollary
given in section 3 of [I], we omit it.
PROOF OF THEOREM 2.2.
Pr
>
pal
-I
Let p be a given prime. Choosing r such that
we see that (n) E r + t +
for all n in F such that n
Pl
t
Pr+3t
For such n we have
in virtue of lemma 2.
> P
Pt+l Pr
al
-I
so that
>
QI
> p
Ul Pt+l
in virtue of lemma 3.3 (u
REMARK I.
I), yielding that
pln.
Though this theorem follows immediately from Theorem 2.3 a direct
proof seems desirable.
(a) For any n e
PROOF OF THEOREM 2.3.
Pl
n <
Pk
(n) >- k
By lemma 3.2,
Pu(n)
P-u+l
(e(x)
x),
Pk
ffi>
Pk-2t-u+2
log
(Pl
Pl
2t
Ft,
there is a unique integer k satisfying
(4.3)
Pk+l
+
so that, for a fixed integer u,
in F
log n as n
t
slnce, by the prime numbers theorem
log n
Pk
and
Pk-2t-u+2
Pk
as
k
(hence as n
in
Ft).
ON INTEGERS n WIT
p (n)
u
>-
Thus lim inf
n
log n
P
0 in theorem 2.1), we have
P (n)
P u (n)
log n
--Pk"
Hence
through such members of F t
P (n)
u
I.
lim inf
log n
as n
->
(b) For k
>-
2,
t
Pk+
I) says that
0 theorem 2.1 (with d
t
< 2
129
FOR m > n
On the other hand, considering members n of F t of the form
1.
l,
(take d
Pk
Jr(n) < Jr(m)
(p-l)
+
=>
Ft"
Pk-I Pk+
Pl
to be the largest subject to the above condition we have
choosing
t
Pk+
+I
>
2
t
t
+
(Pk-l)
t
>
(4.4)
Pk+
so that
Pk++l
where n-
Now
Pl
Pl(n)
<
2(Pk-1)
2 log n
Pk-lPk+ since n satisfies
Pk+ Pk++l and this yields
p1(n)
2
lim sup
lira inf
n
>-
Io= n
(c) Since, by lemma 3.3,
Qu
Qu (n)
(4.3) in virtue of (4.4).
>
au Pt+u
-> a
log n
(a)
we have, from
u
On the other hand, choosing k as in (4.3) we see that QI <=
for all n in F t, where as, for members of F t of the form P
Since
log n we have
Pk+u
Pk+l
Qu < Pk+u
have Qu
Pk+u*
and hence
Pk
we
Pk
lira sup
n
Qu (n)
log n
i.
(d) and (e) now follow by applying lemmas 3 and 4 respectively.
number of members n of
PROOF. OF THEOREM 2.4. We write G(x)
Ft(x)
x <
the theorem follows
from
which
F satisfying
n -< x and show that log G(x)
Ft()
log1/2x
t
easily.
Throughout the proof we assume that x is a sufficiently large positive real
number.
We write
u
[logx
t
and note that t+u _-> 2
Putting
Q1
P+I
we conclude that
+
(log log x)
2t.
-I]
For n e
and noting that
+
(4.5)
t
Ft,
x
n >
-> _34
we have
ZQl(n)
log
Q1 (n)
Ql(n)
4
i
log n by (2.3)
by the prime number theorem
130
J. CHIDAMBARASWAMY AND P.V. KRISHNAIAH
flog
log}x
n
>
log log n
>t+u
log log x
Each n is specified uniquely when
n,
2
t
+2t.
and the prime factors of n are given.
We
estimate an upperbound for G(x) by estimating upper bounds for the number of choices,
x
F
consistent with n
x], for each of (a) n, (b) the largest t+u prime factors
t
of n, namely
and
(c) the prime factors of n that are less that
(,
au Pt+u
Pl’’’’’Pt+u
Pt+u’ Pt+u
(d) the prime factors of n that lie in [a u
)"
(a) By lemma 3.5 and (2.3) the number of cholcesfor n, does not exceed
t+l
t+l
t (2 log x)
and hence does not exceed (2 t log x)
t+l
-< i <- t+u in virtue of (1.8) so that the
"<
<" 2t (log x)
for
(b)
Pi PI
number of choices for
Pt+u does not
P1
(c) Each choice of
QI’
factors of n that are < a
u
to at least one choice for
Qu
P
exceed (2t log
x)(t+l)(t+u).
gives rise to exactly one choice of the prime
and each choice of these prime factors gives rise
t+u
au
Pt+u < Qu and all primes in
(I, au Pt+u
{QI
Qu divide n. If P1 Pr then r Pr P" "< 2t logt+Ix,
"<
"<
and
(3t lolgt+Ix) z so that the
Qu Pr+u" Hence Qi <- Pr+u < (r+)
Q1 Pr+l
QI’""Qu
since
<
2
number of choices for
does not exceed (3t
(d) Let M
M
’Qu
Q1
and hence for the prime factors of n in
logt+Ix) 2u.
Pt+u
(I
u Pt+u and
(l,auPt+u)
note that
u Pt+u <" 2t(t+u) -I Pt+u
-< 3t(t) -I a -I log x
u
< 4t a -1
U
log1/2x
log log x
in virtue of (1.3), (2.4) and (4.5) respectively.
lemma 3.6, the number of primes in
[au Pt+u’Pt+u
Hence, by remark 3 following
does not exceed 24t
au-I log1/2x
and
consequently the number of choices for the prime factors of n that lle in this
interval does not exceed 2
24tu-llog
x.
3 (t+u) (t+l)
(3t log x)
Combining the estimates (a) through (d) we obtain G(x)
-1
exp (24 t aU
log 2) from which the desired order estimate follows in virtue
logx
of (4.5).
It would be interesting to investigate whether results of this nature are
available for more general totients; e.g. totients with respect to a polynomial [3]
and with respect to a set of polynomials [4], introduced by the first author.
REFERENCES
I.
MASSER, D.W. and SHIU, P.
(1986), 407-426.
2.
HUA, L.K.
On sparsely totient numbers, Pacific J. of Math. 121
Introduction to Number
Theory, Springer-Verlag, Berlin-Heidelberg-New
York, 1982.
3.
CHIDAMBARASAMY, J.
Totients with respect to a polynomial, Indian J. of Pure &
Applied Math. _5 (1974), 601-608.
4.
Totients and unitary totlents with respect to a set of polynomials,
Ibid, I0 (1979), 287-302.