DOI: 10.2478/auom-2013-0020
An. Şt. Univ. Ovidius Constanţa
Vol. 21(2),2013, 5–19
Extremal orders of some functions connected
to regular integers modulo n
Brăduţ Apostol
Abstract
Let V (n) denote the number of positive regular integers (mod n) less
V (n)σ(n) V (n)ψ(n)
,
,
than or equal to n. We give extremal orders of
n2
n2
σ(n) ψ(n)
,
, where σ(n), ψ(n) are the sum-of-divisors function and
V (n) V (n)
the Dedekind function, respectively. We also give extremal orders for
σ ∗ (n)
φ∗ (n)
and
, where σ ∗ (n) and φ∗ (n) represent the sum of the
V (n)
V (n)
unitary divisors of n and the unitary function corresponding to φ(n), the
Euler’s function. Finally, we study some extremal orders of compositions
f (g(n)), involving the functions from above.
1
Introduction
Let n > 1 be a positive integer. An integer a is called regular (mod n) if there
exists an integer x such that a2 x ≡ a (mod n).
Properties of regular integers have been investigated by several authors. In
a recent paper O.Alkam and E.A. Osba [1], using ring theoretic considerations,
rediscovered some of the statements proved elementary by J.Morgado [3], [4].
It was proved in [3], [4] that a > 1 is regular (mod n) if and only if gcd (a, n)
is a unitary divisor of n. In [11] L.Tóth gives direct proofs of some properties,
Key Words: Arithmetical function, composition, regular integers (mod n), extremal
orders.
2010 Mathematics Subject Classification: 11A25, 11N37.
Received:February 2012
Revised:June 2012
Accepted:July 2012
5
6
Brăduţ Apostol
because the proofs of [3], [4] are lenghty and those of [1] are ring theoretical.
Let Regn = {a : 1 ≤ a ≤ n and a is regular (mod n)}, and V (n) = #Regn .
The function V is multiplicative and V (pα ) = φ(pαX
) + 1 = pα − pα−1 + 1,
where φ is the Euler function. Consequently, V (n) =
φ(d), for every n ≥ 1,
dkn
where d k n means unitary divisor (defined later). Also φ(n) < V (n) ≤ n, for
every n > 1, and V (n) = n if and only if n is a squarefree, see [4], [11], [1].
L.Tóth [11] proved results concerning the minimal and maximal orders of the
functions V (n) and V (n)/φ(n). The minimal order of V (n) was investigated
by O.Alkam and E.A.Osba in [1]. J. Sándor and L. Tóth [7] studied the
extremal orders of compositions of certain functions. In the present paper we
investigate the extremal orders of the function V (n) in connection with the
functions σ(n), ψ(n), σ ∗ (n), φ∗ (n). We also study extremal orders of certain
composite functions involving V (n), φ(n), σ(n), ψ(n), φ∗ (n), σ ∗ (n) and pose
some open problems.
For other arithmetic functions defined by regular integers modulo n we refer
to the papers [2] and [10].
αk
1
In what follows let n = pα
1 · · · pk > 1 be a positive integer. We will use
throughout the paper the following notation:
•
p1 , p2 , ... - the sequence of the primes;
n
•
d k n - d is a unitary divisor of n, that is d | n and (d, ) = 1;
d
•
σ(n) - the sum of the divisors of the natural number
n;
Y
1
•
ψ(n) - the Dedekind function, ψ(n) = n
1+
;
p
p|n
−1
Y 1
, s = σ +it ∈
•
ζ(n) - the Riemann zeta function, ζ(s) =
1− s
p
p prime
C and σ > 1;
•
φ(n) - the Euler function, φ(n) = n
Y
p|n
1−
1
;
p
1
1
+ ... + − log n);
2
n
•
γ - the Euler constant, γ = lim (1 +
•
φ∗ (n) - the unitary function corresponding to φ(n), φ∗ (n) =
•
n→∞
σ ∗ (n) - the unitary function corresponding to σ(n), σ ∗ (n) =
k
Y
i
(pα
i −1);
i=1
k
Y
i
(pα
i +1).
i=1
EXTREMAL ORDERS OF SOME FUNCTIONS CONNECTED TO REGULAR
INTEGERS MODULO N
2
7
Extremal orders concerning classical arithmetic functions
We know that φ(n) < n < σ(n) for every n > 1. It is easy to see that
6
φ(n)σ(n)
φ(n)σ(n)
6
φ(n)σ(n)
<
< 1, n > 1, lim inf
= 2 and lim sup
= 1.
n→∞
π2
n2
n2
π
n2
n→∞
φ(n)ψ(n)
6
φ(n)ψ(n)
In [5] it was proved that lim inf
= 2 and lim sup
= 1.
n→∞
n2
π
n2
n→∞
We recall that an integer n > 1 is called powerful if it is divisible by the square
of each of its prime factors. A powerful integer is also called a squarefull
integer.
The investigation of the minimal and maximal order of V (n)σ(n) led us to
Proposition 1.
V (n)σ(n)
> 1,
n2
(i)
for every n > 1.
V (n)σ(n)
= 1,
n2
V (n)σ(n)
ζ(2)
≤
,
2
n
ζ(6)
lim inf
(ii)
n→∞
(iii)
for every powerful number n.
lim sup
n→∞
n powerful
ζ(2)
V (n)σ(n)
=
.
n2
ζ(6)
(iv)
Proof.
αk
1
Let n > 1 be an integer with the prime factorization n = pα
1 · · · pk .
1
p − pα
1
1
> 1, it follows that
Since 1 − + α ·
p p
p−1
(i)
p − α1
k i
V (n)σ(n) Y
1
1
pi i
=
1−
+ αi ·
> 1.
2
n
pi
pi
pi − 1
i=1
(ii)
V (p)σ(p)
p2 + p
=
lim
= 1, taking (i) into account, we
p→∞
p2
p2
p prime
p prime
Since p→∞
lim
obtain
lim inf
n→∞
V (n)σ(n)
= 1.
n2
8
Brăduţ Apostol
Let n = q1α1 · · · qkαk , q1 < q2 < ... < qk , αi ≥ 2, 1 ≤ i ≤ k and
k
p1 ,...,p
q α − q α−1 + 1 q α+1 − 1
1
1
the first k primes. We have
· α
· 1− 6
≤
qα
q (q − 1)
q
1 − q12
for α ≥ 2 and q prime, so
(iii)
k
k
Y
V (n)σ(n) Y qiαi − qiαi −1 + 1 qiαi +1 − 1
1
1
=
·
≤
·
1
−
.
n2
qiαi
qiαi (qi − 1) i=1 1 − q12
qi6
i=1
i
Since qi ≥
1 ≤ i ≤ k, it follows
that
pi for 1
1
1
1
·
1
−
≤
·
1
−
for 1 ≤ i ≤ k, so
qi6
p6i
1 − q12
1 − p12
i
i
k
k Y
V (n)σ(n) Y 1
1
≤
.
·
1
−
n2
p6i
1 − p12 i=1
i=1
i
Taking k → ∞, we obtain
V (n)σ(n)
ζ(2)
≤
.
2
n
ζ(6)
(iv)
Taking nk = p21 · · · p2k (p1 , ..., pk being the first k primes),
k
k Y
V (nk )σ(nk ) Y 1
1
=
·
1− 6 ,
n2k
pi
1 − p12 i=1
i=1
i
so
Y
Y 1
V (nk )σ(nk )
1
ζ(2)
=
·
.
lim
1− 6 =
1
2
k→∞
nk
p
ζ(6)
1 − p2 p prime
p prime
In view of (iii) , we obtain
lim sup
n→∞
n powerful
V (n)σ(n)
ζ(2)
=
.
n2
ζ(6)
Corollary 1. The minimal order of
V (n)σ(n)
ζ(2)
for n powerful is
.
n2
ζ(6)
V (n)σ(n)
is 1 and the maximal order of
n2
We now prove an analogous result for V (n)ψ(n):
EXTREMAL ORDERS OF SOME FUNCTIONS CONNECTED TO REGULAR
INTEGERS MODULO N
9
Proposition 2.
V (n)ψ(n)
= 1,
n2
n squarefree
(i)
V (n)ψ(n)
ζ(3)
≤
,
2
n
ζ(6)
(ii)
lim
inf
n→∞
for every powerful number n.
lim sup
n→∞
n powerful
ζ(3)
V (n)ψ(n)
=
.
n2
ζ(6)
(iii)
Proof.
(i)
Let n = p1 · · · pk , where p1 ,...,pk are distinct prime numbers. We have
(p1 + 1) · · · (pk + 1)
V (p)ψ(p)
V (n)ψ(n)
=
> 1. Since p→∞
lim
= 1, we obtain
n2
p1 · · · pk
p2
p prime
V (n)ψ(n)
= 1.
n2
n squarefree
lim
inf
n→∞
(ii)
If n = q1α1 · · · qkαk , αi ≥ 2, and 1 ≤ i ≤ k, then we have
k
V (n)ψ(n) Y qiαi +1 − qiαi −1 + qi + 1
=
.
n2
qiαi +1
i=1
It is immediate that
q α+1 − q α−1 + q + 1
≤
q α+1
prime, so
1−
1
q2
1+
1
q2 − q
=1+
1
for α ≥ 2 and q
q3
Y
k k 1
V (n)ψ(n) Y
1
1
≤
1
−
1
+
=
1
+
.
n2
qi2
qi2 − qi
qi3
i=1
i=1
Let p1 , ..., pk the first k primes. Since qi ≥ pi for 1 ≤ i ≤ k, we get
k 1
1
V (n)ψ(n) Y
1
≤
1+
for
1
≤
i
≤
k,
hence
≤
1+
. Since the right
qi3
p3i
n2
p3i
i=1
ζ(3)
ζ(3)
V (n)ψ(n)
hand side tends increasingly to
as k → ∞, we get
, for
≤
ζ(6)
n2
ζ(6)
every powerful number n.
1+
10
Brăduţ Apostol
Take nk = p21 · · · p2k (p1 , ..., pk being the first k primes). Then
Y
Y
k k k ζ(3)
V (nk )ψ(nk ) Y
1
1
1
=
=
1− 2 ·
1+ 2
1+ 3 →
,
−
p
n2k
p
p
p
ζ(6)
i
i
i
i
i=1
i=1
i=1
(k → ∞) so, if we take into account (ii), we deduce that
ζ(3)
V (n)ψ(n)
V (n)ψ(n)
=
lim sup
, implying that the maximal order of
n→∞
n2
ζ(6)
n2
(iii)
n powerful
for n powerful is
ζ(3)
.
ζ(6)
In order to prove the properties below we apply the following result ([12],
Corollary 1) :
Lemma 1. If f is a nonnegative real-valued multiplicative arithmetic function
such that for each prime p,
−1
1
α
(i)
ρ(p) = sup(f (p )) ≤ 1 −
, and
p
α≥0
(ii)
1
there is an exponent ep = po(1) ∈ N satisfying f (pep ) ≥ 1 + ,
p
Y f (n)
1
γ
then lim sup
=e
ρ(p).
1−
p
n→∞ log log n
p prime
σ(n)
σ(n)
, we notice that
≥ 1 for every n ≥ 1.
V (n)
V (n)
σ(p)
σ(n)
Since p→∞
lim
= 1, we get lim inf
= 1, hence the minimal orn→∞ V (n)
V (p)
p prime
For the quotient
σ(n)
σ(n)
is 1. Proposition 3 shows that the maximal order of
is
V (n)
V (n)
e2γ (log log n)2 :
der of
Proposition 3.
lim sup
n→∞
s
Proof. Take f (n) =
s
α
f (p ) =
σ(n)
= e2γ .
V (n)(log log n)2
σ(n)
. Then
V (n)
pα+1 − 1
≤
(p − 1)(pα − pα−1 + 1)
1−
1
p
−1
= ρ(p),
EXTREMAL ORDERS OF SOME FUNCTIONS CONNECTED TO REGULAR
INTEGERS MODULO N
and
s
2
f (p ) =
p2 + p + 1
1
≥1+
p2 − p + 1
p
for every prime p, so (ii) in the above Lemma is satisfied. We obtain
p
σ(n)
lim sup p
= eγ ,
n→∞
V (n) log log n
so
lim sup
n→∞
σ(n)
= e2γ .
V (n)(log log n)2
ψ(n)
ψ(n)
. Since
≥ 1 for every n ≥ 1 and
V (n)
V (n)
p+1
ψ(n)
ψ(p)
=
for every prime p, it is immediate that lim inf
= 1.
n→∞
V (p)
p
V (n)
ψ(n)
Thus, the minimal order of
is 1.
V (n)
Consider now the quotient
Proposition 4.
lim sup
n→∞
s
Proof. Let f (n) =
s
α
f (p ) =
ψ(n)
6
= 2 e2γ .
2
V (n)(log log n)
π
ψ(n)
in Lemma 1. Here
V (n)
pα + pα−1
≤
α
p − pα−1 + 1
and
s
f (p4 ) =
r
p+1
= ρ(p) <
p−1
1−
1
p
−1
,
1
p4 + p3
≥1+ ,
4
3
p −p +1
p
so (ii) is fulfilled in the cited Lemma, for every prime p. We obtain
r
p
r
Y
ψ(n)
1
6
γ
γ
lim sup p
=e
1− 2 =e
,
p
π2
n→∞
V (n) log log n
p prime
so
lim sup
n→∞
ψ(n)
6
= 2 e2γ .
V (n)(log log n)2
π
11
12
3
Brăduţ Apostol
Extremal orders concerning unitary analogues
In what follows we consider the functions σ ∗ (n) and φ∗ (n), representing the
sum of the unitary divisors of n and the unitary Euler function, respectively.
αk
1
The functions σ ∗ (n) and φ∗ (n) are multiplicative. If n = pα
1 · · · pk is the
prime factorisation of n > 1, then
αk
1
φ∗ (n) = (pα
1 − 1) · · · (pk − 1),
αk
1
σ ∗ (n) = (pα
1 + 1) · · · (pk + 1)
Note that σ ∗ (n) = σ(n), φ∗ (n) = φ(n) for all squarefree n, and for every n ≥ 1
φ(n) ≤ φ∗ (n) ≤ n ≤ σ ∗ (n) ≤ σ(n).
σ ∗ (n)
φ∗ (n)
and
, the minimal order
V (n)
V (n)
σ ∗ (n)
φ∗ (n)
being studied for powerful numbers. Since
≥ 1 and for prime
of
V (n)
V (n)
σ ∗ (p)
p+1
σ ∗ (n)
numbers p, lim
= lim
= 1, it follows that lim inf
= 1.
p→∞ V (p)
p→∞
n→∞ V (n)
p
φ∗ (n)
≥ 1, taking into account that
If n is powerful, it is easy to see that
V (n)
∗ α
φ (p )
φ∗ (p2 )
≥
1
for
α
≥
2.
For
prime
numbers
p,
we
notice
that
lim
=
p→∞ V (p2 )
V (pα )
φ∗ (n)
p2 − 1
= 1, which implies that lim inf
= 1, so the minimal
lim 2
n→∞ V (n)
p→∞ p − p + 1
φ∗ (n)
order of
is 1. For the maximal orders of these quotients we give:
V (n)
We give extremal orders for the quotients
Proposition 5.
lim sup
n→∞
lim sup
n→∞
σ ∗ (n)
= eγ ,
V (n) log log n
(i)
φ∗ (n)
= eγ .
V (n) log log n
(ii)
Proof.
σ ∗ (n)
, which is a nonnegative real-valued multiplicative
V (n)
−1
pα + 1
1
arithmetic function. We have f (pα ) = α
≤
1
−
= ρ(p),
p − pα−1 + 1
p
and
(i)
Take f (n) =
EXTREMAL ORDERS OF SOME FUNCTIONS CONNECTED TO REGULAR
INTEGERS MODULO N
f (p) = 1 +
13
1
1
≥ 1 + for every prime p. Applying Lemma 1, we get
p
p
lim sup
n→∞
σ ∗ (n)
= eγ .
V (n) log log n
φ∗ (n)
. Here
V (n)
−1
1
pα − 1
= ρ(p), and
f (pα ) = α
≤
1
−
p − pα−1 + 1
p
p4 − 1
1
f (p4 ) = 4
≥ 1 + , for every prime p. According to Lemma 1,
p − p3 + 1
p
(ii)
Now let f (n) =
lim sup
n→∞
φ∗ (n)
= eγ .
V (n) log log n
Corollary 2. The maximal order of both
4
φ∗ (n)
σ ∗ (n)
and
is eγ log log n.
V (n)
V (n)
Extremal orders regarding compositions of functions
We now move to the study of extremal orders of some composite arithmetic
functions. We start with V (V (n)) and φ(V (n)).
V (V (n))
V (n)
We know that V (n) ≤ n for every n ≥ 1, so
≤
≤ 1 and
n
n
V (V (p))
V (p)
lim
= p→∞
lim
= 1, so the maximal order of V (V (n)) is n.
p→∞
p
p
p prime
p prime
Since φ(n) ≤ n and V (n) ≤ n for any n ≥ 1, we have
φ(V (n))
V (n)
≤
≤ 1.
n
n
φ(V (p))
p−1
= lim
= 1, so the maximal order of φ(V (n)) is n.
p→∞
p
p
p prime
But p→∞
lim
In [7] was investigated the maximal order of φ∗ (φ(n)). Using the general idea
of that proof, we show:
Proposition 6. The maximal order of V (φ(n)) is n.
Proof. We will use Linnik’s theorem which states that if (k, `) = 1, then
there exists a prime p such that p ≡ ` (mod k) and p k c , where c is a
constant (one
Y can take c ≤ 11).
Let A =
p. Since (A2 , A + 1) = 1, by Linnik’s theorem there is a prime
p≤x
p prime
number q such that q ≡ A + 1 (mod A2 ) and q (A2 )c = A2c , where c
14
Brăduţ Apostol
satisfies c ≤ 11. Let q be the least prime satisfying the above condition. So,
q − A − 1 = kA2 , for some k. We have
φ(q) = q − 1 = A + kA2 = A(1 + kA) = AB, where B = 1 + kA. Thus
(A, B) = 1, so B is free of prime factors ≤ x. We have q − 1 = AB, so
q = AB + 1.
Since V (n) is multiplicative, we have
V (AB)
V (A) V (B)
AB
V (φ(q))
=
=
·
·
.
q
AB + 1
A
B
AB + 1
(1)
AB
V (A)
V (B)
→ 1 as x → ∞, so it is sufficient to study
and
.
AB + 1
A
B
Clearly,
Y
Y V (p)
V
p
V (A)
p≤x
p≤x
= Y
= 1.
(2)
= Y
A
p
p
Here
p≤x
It is well-known that A =
B A10 we have B e
Y
p=e
p≤x
O(x) 10
p≤x
O(x)
. Since q A2c and A = eO(x) , from
= eO(x) , so
log B x.
If B =
k
Y
qibi is the prime factorization of B, we obtain
i=1
k
X
log B =
bi log qi > (log x)
k
X
bi , as qi > x for all i ∈ {1, 2, ..., k}. But
i=1
i=1
k
X
(3)
bi ≥ k, so log B > k log x, implying that k <
i=1
log B
x
(by(3)). We
log x
log x
get:
V
V (B)
=
B
Y
k
qibi
i=1
k
Y
i=1
qibi
k
Y
=
k
Y
(qibi − qibi −1 + 1)
i=1
k
Y
>
qibi
i=1
(qibi − qibi −1 )
i=1
k
Y
=
qibi
i=1
x
k O( log x )
k Y
1
1
1
1
> 1−
≥ 1−
>1+O
,
=
1−
qi
x
x
log x
i=1
EXTREMAL ORDERS OF SOME FUNCTIONS CONNECTED TO REGULAR
INTEGERS MODULO N
because 1 −
15
1
1
> 1 − . So,
qi
x
V (B)
1
>1+O
.
B
log x
By (1), (2), (4) and
AB
→ 1 as x → ∞, we obtain
AB + 1
V (φ(q))
1
>1+O
.
q
log x
By relation (5), and since
(4)
(5)
V (φ(n))
φ(n)
≤
≤ 1, it follows that
n
n
lim sup
n→∞
V (φ(n))
= 1.
n
Proposition 7. The maximal order of V (φ∗ (n)) is n.
Proof. We apply the following result:
If a is an integer, a > 1, p is a prime number and f (n) is an arithmetical
function satisfying φ(n) ≤ f (n) ≤ σ(n), one has
lim
p→∞
f (N (a, p))
= 1,
N (a, p)
(6)
ap − 1
(see e.g. D.Suryanarayana [9]).
a−1
∗
Since φ (n) ≤ n, it follows that V (φ∗ (n)) ≤ φ∗ (n) ≤ n, so
where N (a, p) =
V (φ∗ (n))
≤ 1.
n
Let n = 2p , p prime number. Then we have
V (2p − 1) 2p − 1
V (φ∗ (2p ))
=
·
.
p
2
2p − 1
2p
(7)
(8)
Since φ(n) ≤ V (n) ≤ σ(n) and N (2, p) = 2p − 1, it follows that
V (2p − 1)
= 1,
p→∞
2p − 1
lim
taking into account (6). By (8), taking p → ∞, we obtain
V (φ∗ (2p ))
= 1.
p→∞
2p
lim
(9)
16
Brăduţ Apostol
V (φ∗ (n))
= 1.
n
n→∞
σ(φ∗ (n)) ψ(φ∗ (n))
,
we give
For the maximal orders of
V (φ∗ (n)) V (φ∗ (n))
Now (7) and (9) imply lim sup
Proposition 8.
(i) lim sup
n→∞
(ii) lim sup
n→∞
σ(φ∗ (n))
σ(φ∗ (n))
=
lim
sup
= e2γ ,
∗
∗
2
V (φ∗ (n))(log log n)2
n→∞ V (φ (n))(log log φ (n))
ψ(φ∗ (n))
V
(φ∗ (n))(log log n)2
= lim sup
n→∞
ψ(φ∗ (n))
V
(φ∗ (n))(log log φ∗ (n))2
=
6 γ
e .
π2
Proof.
(i) Let
l1 := lim sup
n→∞
and
l2 := lim sup
n→∞
σ(φ∗ (n))
V (φ∗ (n))(log log n)2
σ(φ∗ (n))
V
(φ∗ (n))(log log φ∗ (n))2
.
Since φ∗ (n) ≤ n for every n ≥ 1,
σ(φ∗ (n))
σ(φ∗ (n))
≤
≤
l
=
lim
sup
l1 = lim sup
2
∗
2
∗
∗
2
n→∞ V (φ (n))(log log φ (n))
n→∞ V (φ (n))(log log n)
σ(m)
lim sup
= e2γ , by Proposition 3.
2
m→∞ V (m)(log log m)
Since (n, 1) = 1, by Linnik’s theorem, there exists a prime number p such
that p ≡ 1 (mod n) and p nc . Let pn be the least prime such that pn ≡ 1
(mod n), for every n. Then n | pn − 1 and pn nc , so log log pn ∼ log log n.
σ(a)
σ(b)
Observe that a | b implies
≤
. If pβ | pα (β ≤ α), it is easy to
V (a)
V (b)
σ(pα )
σ(pβ )
≤
. The general case follows, taking into account that
see that
β
V (p )
V (pα )
σ(n)
is multiplicative. So,
V (n)
σ(φ∗ (pn ))
=
∗
V (φ (pn ))(log log pn )2
σ(pn − 1)
σ(pn − 1)
σ(n)
∼
≥
.
V (pn − 1)(log log pn )2
V (pn − 1)(log log n)2
V (n)(log log n)2
But
lim sup
n→∞
σ(φ∗ (n))
σ(φ∗ (pn ))
≥
lim
sup
≥
∗
2
V (φ∗ (n))(log log n)2
n→∞ V (φ (pn ))(log log pn )
EXTREMAL ORDERS OF SOME FUNCTIONS CONNECTED TO REGULAR
INTEGERS MODULO N
lim sup
n→∞
17
σ(n)
= e2γ .
V (n)(log log n)2
We obtain e2γ ≤ l1 ≤ l2 ≤ e2γ , hence l1 = l2 = e2γ .
(ii) The proof is similar to the proof of (i), taking into account that a | b
ψ(a)
6
ψ(b)
ψ(n)
impies
= 2 e2γ , by Proposition 4.
≤
and lim sup
2
V (a)
V (b)
π
n→∞ V (n)(log log n)
So, the maximal orders of
σ(φ∗ (n)) ψ(φ∗ (n))
,
are e2γ (log log n)2 and
V (φ∗ (n)) V (φ∗ (n))
6 2γ
e (log log n)2 ,
respectively.
In
a
similar
manner,
since
π2
∗
∗
σ (n)
φ (n)
lim sup
= lim sup
= eγ (Proposition 5), a | b
V
(n)
log
log
n
V
(n)
log log n
n→∞
n→∞
σ ∗ (b)
φ∗ (a)
φ∗ (b)
σ ∗ (a)
≤
and
≤
, respectively, it can be shown that
implies
V (a)
V (b)
V (a)
V (b)
lim sup
n→∞
lim sup
n→∞
5
σ ∗ (φ∗ (n))
σ ∗ (φ∗ (n))
=
lim
sup
= eγ and
∗
∗
V (φ∗ (n)) log log n
n→∞ V (φ (n)) log log φ (n)
φ∗ (φ∗ (n))
φ∗ (φ∗ (n))
=
lim
sup
= eγ .
∗
∗
V (φ∗ (n)) log log n
n→∞ V (φ (n)) log log φ (n)
Open Problems
Problem 1. Note that
lim sup
n→∞
V (n)σ(n)
V (n)ψ(n)
= lim sup
= ∞,
n2
n2
n→∞
since for nk = p1 · · · pk (the product of the first k primes),
k V (nk )σ(nk )
(p1 + 1) · · · (pk + 1) Y
1
=
=
1
+
→ ∞, k → ∞;
n2k
p1 · · · pk
pi
i=1
the other relation follows in a similar manner. What are the maximal orders
V (n)ψ(n)
V (n)σ(n)
for
and
?
2
n
n2
Problem 2. Note that
lim inf
n→∞
V (φ(n))
V (φ∗ (n))
φ∗ (V (n))
= lim inf
= lim inf
= 0.
n→∞
n→∞
n
n
n
18
Brăduţ Apostol
For nk = p1 · · · pk (the product of the first k primes),
V (φ(nk ))
V ((p1 − 1) · · · (pk − 1))
(p1 − 1) · · · (pk − 1)
=
≤
nk
p1 · · · pk
p1 · · · pk
1
1
= 1−
··· 1 −
,
p1
pk
so
V (φ(nk ))
1
1
lim
= lim 1 −
··· 1 −
= 0,
k→∞
k→∞
nk
p1
pk
similarly the other relations. What are the minimal orders for the V (φ(n)),
V (φ∗ (n)), φ∗ (V (n)) ?
Problem 3. Taking nk = p1 · · · pk (the product of the first k primes),
σ ∗ (p1 · · · pk )
(p1 + 1) · · · (pk + 1)
σ ∗ (V (nk ))
=
=
nk
p1 · · · pk
p1 · · · pk
1
1
= 1+
··· 1 +
→∞
p1
pk
as k → ∞, so lim sup
∗
n→∞
σ ∗ (V (n))
= ∞.
n
What is the maximal order for
σ (V (n)) ?
ACKNOWLEDGEMENT. The author would like to thank Professor
L.Tóth and Professor G. Mincu for the valuable suggestions which improved
the presentation.
The author also thanks the referees for helpful suggestions.
References
[1] O. Alkam, E. Osba, On the regular elements in Zn , Turk.J.Math.,
32(2008).
[2] P. Haukkanen, L. Tóth, An analogue of Ramanujan’s sum with respect
to regular integers (mod r), Ramanujan J., 27(2012) 71-88.
[3] J. Morgado, A property of the Euler φ-function concerning the integers
which are regular modulo n, Portugal. Math., 33 (1974), 185–191.
EXTREMAL ORDERS OF SOME FUNCTIONS CONNECTED TO REGULAR
INTEGERS MODULO N
19
[4] J. Morgado, Inteiros regulares módulo n, Gazeta de Matematica (Lisboa),
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[6] J. Sándor, B. Crstici, Handbook of number theory II, Kluwer Academic
Publishers, Dordrecht, 2004.
[7] J. Sándor, L. Tóth, Extremal orders of compositions of certain arithmetical functions, Integers: Electronic Journal of Combinatorial Number
Theory, 8 (2008), # A34.
[8] R. Sivaramakrishnan, Classical theory of arithmetic functions, Monographs and Textbooks in Pure and Applied Mathematics, 126 . Marcel
Dekker, Inc., New York, 1989.
[9] D. Suryanarayana, On a class of sequences of integers, Amer. Math.
Monthly, 84(1977), 728–730.
[10] L. Tóth, A gcd-sum function over regular integers modulo n, J. Integer
Seq., 12(2009), no.2, Article 09.2.5, 8 pp.
[11] L. Tóth, Regular integers (mod n), Annales Univ. Sci. Budapest., Sect
Comp., 29(2008), 264–275, see http://front.math.ucdavis.edu/0710.1936.
[12] L. Tóth, E. Wirsing, The maximal order of a class of multiplicative
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22(2003), 353–364.
Brăduţ APOSTOL,
”Spiru Haret” Pedagogical High School,
1 Timotei Cipariu St. , RO - 620004 Focşani, ROMANIA,
Email:apo brad@yahoo.com
20
Brăduţ Apostol