Some Remarks on Regular Integers Modulo n

Abstract

An integer $k$ is called regular (mod $n$) if there exists an integer $x$
such that $k^2x\equiv k$ (mod $n$). This holds true if and only if $k$
possesses a weak order (mod $n$), i.e., there is an integer $m\ge 1$ such that
$k^{m+1} \equiv k$ (mod $n$). Let $\varrho(n)$ denote the number of regular
integers (mod $n$) in the set $\{1,2,\ldots,n\}$. This is an analogue of
Euler's $\phi$ function. We introduce the multidimensional generalization of
$\varrho$, which is the analogue of Jordan's function. We establish identities
for the power sums of regular integers (mod $n$) and for some other finite sums
and products over regular integers (mod $n$), involving the Bernoulli
polynomials, the Gamma function and the cyclotomic polynomials, among others.
We also deduce an analogue of Menon's identity and investigate the maximal
orders of certain related functions.

Full text

arXiv:1304.2699v2 [math.NT] 13 May 2015
Some remarks on regular integers modulo n
Br˘adut¸ Apostol and L´aszl´o T´oth
Filomat 29 (2015), no 4, 687–701
Available at http://www.pmf.ni.ac.rs/filomat
Abstract
An integer kis called regular (mod n) if there exists an integer xsuch that k2x≡k
(mod n). This holds true if and only if kpossesses a weak order (mod n), i.e., there is an
integer m≥1 such that km+1 ≡k(mod n). Let ̺(n) denote the number of regular integers
(mod n) in the set {1,2,...,n}. This is an analogue of Euler’s φfunction. We introduce
the multidimensional generalization of ̺, which is the analogue of Jordan’s function. We
establish identities for the power sums of regular integers (mod n) and for some other finite
sums and products over regular integers (mod n), involving the Bernoulli polynomials, the
Gamma function and the cyclotomic polynomials, among others. We also deduce an analogue
of Menon’s identity and investigate the maximal orders of certain related functions.
2010 Mathematics Subject Classification: 11A25, 11B68, 11N37, 33B10, 33B15
Key Words and Phrases: regular integer (mod n), Euler’s totient function, Jordan’s function,
Ramanujan’s sum, unitary divisor, Bernoulli numbers and polynomials, Gamma function, finite
trigonometric sums and products, cyclotomic polynomial
1 Introduction
Throughout the paper we use the notations: N:= {1,2,...},N0:= {0,1,2,...},Zis the
set of integers, ⌊x⌋is the integer part of x,1is the function given by 1(n) = 1 (n∈N), id is
the function given by id(n) = n(n∈N), φis Euler’s totient function, τ(n) is the number of
divisors of n,µis the M¨obius function, ω(n) stands for the number of prime factors of n, Λ is
the von Mangoldt function, κ(n) := Qp|npis the largest squarefree divisor of n,cn(t) are the
Ramanujan sums defined by cn(t) := P1≤k≤n,gcd(k,n)=1 exp(2πikt/n) (n∈N, t ∈Z), ζis the
Riemann zeta function. Other notations will be fixed inside the paper.
Let n∈Nand k∈Z. Then kis called regular (mod n) if there exists x∈Zsuch that k2x≡k
(mod n). This holds true if and only if kpossesses a weak order (mod n), i.e., there is m∈N
such that km+1 ≡k(mod n). Every k∈Zis regular (mod 1). If n > 1 and its prime power
factorization is n=pν1
1···pνr
r, then kis regular (mod n) if and only if for every i∈ {1,...,r}
either pi∤kor pνi
i|k. Also, kis regular (mod n) if and only if gcd(k, n) is a unitary divisor of n.
We recall that dis said to be a unitary divisor of nif d|nand gcd(d, n/d) = 1, notation d||n.
Note that if nis squarefree, then every k∈Zis regular (mod n). See the papers [1,14,15,20]
for further discussion and properties of regular integers (mod n), and their connection with the
notion of regular elements of a ring in the sense of J. von Neumann.
1

An integer kis regular (mod n) if and only if k+nis regular (mod n). Therefore, it is
justified to consider the set
Regn:= {k∈N: 1 ≤k≤n, k is regular (mod n)}
and the quantity ̺(n) := # Regn. For example, Reg12 ={1,3,4,5,7,8,9,11,12}and ̺(12) = 9.
If nis squarefree, then Regn={1,2,...,n}and ̺(n) = n. Note that 1, n ∈Regnfor every
n∈N. The arithmetic function ̺is an analogue of Euler’s φfunction, it is multiplicative and
̺(pν) = φ(pν) + 1 = pν−pν−1+ 1 for every prime power pν(ν∈N). Consequently,
̺(n) = X
d|| n
φ(d) (n∈N).(1)
See, e.g., [13] for general properties of unitary divisors, in particular the unitary convolution
of the arithmetic functions fand gdefined by (f×g)(n) = Pd|| nf(d)g(n/d). Here f×g
preserves the multiplicativity of the functions fand g. We refer to [20] for asymptotic properties
of the function ̺.
The function
cn(t) := X
k∈Regn
exp(2πikt/n) (n∈N, t ∈Z),
representing an analogue of the Ramanujan sum cn(t) was investigated in the paper [8]. We
have
cn(t) = X
d|| n
cd(t) (n∈N, t ∈Z).
It turns out that for every fixed tthe function n7→ cn(t) is multiplicative, cn(0) = ̺(n) and
cn(1) = µ(n) is the characteristic function of the squarefull integers n.
The gcd-sum function is defined by P(n) := Pn
k=1 gcd(k, n) = Pd|nd φ(n/d), see [22]. The
following analogue of the gcd-sum function was introduced in the paper [21]:
e
P(n) := X
k∈Regn
gcd(k, n).
One has
e
P(n) = X
d|| n
d φ(n/d) = nY
p|n2−1
p(n∈N),
the asymptotic properties of e
P(n) being investigated in [10,22,26,27].
In the present paper we discuss some further properties of the regular integers (mod n). We
first introduce the multidimensional generalization ̺r(r∈N) of the function ̺, which is the
analogue of the Jordan function φr, where φr(n) is defined as the number of ordered r-tuples
(k1,...,kr)∈ {1,...,n}rsuch that gcd(k1,...,kr) is prime to n(see, e.g., [13,18]). Then we
consider the sum S[reg]r(n) of r-th powers of the regular integers (mod n) belonging to Regn. In
the case r∈Nwe deduce an exact formula for S[reg]r(n) involving the Bernoulli numbers Bm.
For a positive real number rwe derive an asymptotic formula for S[reg]r(n). We combine the
functions cn(t) and e
P(n) defined above and establish identities for sums, respectively products
over the integers in Regnconcerning the Bernoulli polynomials Bm(x), the Gamma function Γ,
the cyclotomic polynomials Φm(x) and certain trigonometric functions. We point out that for n
squarefree these identities reduce to the corresponding ones over {1,2,...,n}. We also deduce
an analogue of Menon’s identity and investigate the maximal orders of some related functions.
2

2 A generalization of the function ̺
For r∈Nlet ̺r(n) be the number of ordered r-tuples (k1,...,kr)∈ {1,...,n}rsuch that
gcd(k1,...,kr) is regular (mod n). If r= 1, then ̺1=̺. The arithmetic function ̺ris
the analogue of the Jordan function φr, defined in the Introduction and verifying φr(n) =
nrQp|n(1 −1/pr) (n∈N).
Proposition 2.1. i) For every r, n ∈N,
̺r(n) = X
d|| n
φr(d).
ii) The function ̺ris multiplicative and for every prime power pν(ν∈N),
̺r(pν) = prν −pr(ν−1) + 1.
Proof. i) The integer gcd(k1,...,kr) is regular (mod n) if and only if gcd(gcd(k1,...,kr), n)||n,
that is gcd(k1,...,kr, n)||nand grouping the r-tuples (k1,...,kr) according to the values
gcd(k1,...,kr, n) = dwe deduce that
̺r(n) = X
(k1,...,kr)∈{1,...,n}r
gcd(k1,...,kr) regular (mod n)
1 = X
d|| nX
(k1,...,kr)∈{1,...,n}r
gcd(k1,...,kr,n)=d
1
=X
d|| nX
(ℓ1,...,ℓr)∈{1,...,n/d}r
gcd(ℓ1,...,ℓr,n/d)=1
1,
where the inner sum is φr(n/d), according to its definition.
ii) Follows at once by i).
More generally, for a fixed real number slet φs(n) = Pd|ndsµ(n/d) be the generalized
Jordan function and define ̺sby
̺s(n) = X
d|| n
φs(d) (n∈N).(2)
The functions φsand ̺s(which will be used in the next results of the paper) are multiplicative
and for every prime power pν(ν∈N) one has φs(pν) = psν −ps(ν−1) and ̺s(pν) = psν −ps(ν−1) +1.
Note that φ−s(n) = n−sQpν|| n(1 −ps) and ̺−s(n) = n−sQpν|| n(psν −ps+ 1).
Proposition 2.2. If s > 1is a real number, then
X
n≤x
̺s(n) = xs+1
s+ 1 Y
p1−1
ps+1 +p−1
p(ps+1 −1)+O(xs).(3)
Proof. We need the following asymptotics. Let s > 0 be fixed real number. Then uniformly for
real x > 1 and t∈N,
φs(x, t) := X
n≤x
gcd(n,t)=1
φs(n) = xs+1
(s+ 1)ζ(s+ 1) ·tsφ(t)
φs+1(t)+O(xs2ω(t)).(4)
3

To show (4) use the known estimate, valid for every fixed s > 0 and t∈N,
X
n≤x
gcd(n,t)=1
ns=xs+1
s+ 1 ·φ(t)
t+Oxs2ω(t).(5)
We obtain
φs(x, t) = X
de=n≤x
gcd(n,t)=1
µ(d)es=X
d≤x
gcd(d,t)=1
µ(d)X
e≤x/d
gcd(e,t)=1
es
=X
d≤x
gcd(d,t)=1
µ(d)(x/d)s+1
s+ 1 ·φ(t)
t+O(x/d)s2ω(t)
=xs+1
s+ 1 ·φ(t)
t
∞
X
d=1
gcd(d,t)=1
µ(d)
ds+1 +O xs+1 X
d>x
1
ds+1 !+Oxs2ω(t),
giving (4). Now from (2) and (4),
X
n≤x
̺s(n) = X
de=n≤x
gcd(d,e)=1
φs(e) = X
d≤xX
e≤x/d
gcd(e,d)=1
φs(e) = X
d≤x
φs(x/d, d)
=xs+1
(s+ 1)ζ(s+ 1)
∞
X
d=1
φ(d)
dφs+1(d)+O xs+1 X
d>x
φ(d)
dφs+1(d)!+O
xsX
d≤x
2ω(d)
ds
,
and for s > 1 this leads to (3).
Compare (3) to the corresponding formula for the Jordan function φs, i.e., to (4) with t= 1.
Remark 2.3. For the function ̺one has
X
n≤x
̺(n) = 1
2Y
p1−1
p2(p+ 1)x2+R(x),
where R(x) = O(xlog3x) can be obtained by the elementary arguments given above. This can
be improved into R(x) = O(xlog x) using analytic methods. See [20] for references.
3 A general scheme
In order to give exact formulas for certain sums and products over the regular integers (mod
n) we first present a simple result for a general sum over Regn, involving a weight function w
and an arithmetic function f. It would be possible to consider a more general sum, namely
over the ordered r-tuples (k1,...,kr)∈ {1,...,n}rsuch that gcd(k1,...,kr) is regular (mod n),
but we confine ourselves to the following result. See [24] for another similar scheme concerning
weighted gcd-sum functions.
4

Proposition 3.1. i) Let w:N2→Cand f:N→Cbe arbitrary functions and consider the
sum
Rw,f (n) := X
k∈Regn
w(k, n)f(gcd(k, n)).
Then
Rw,f (n) = X
d|| n
f(d)
n/d
X
j=1
gcd(j,n/d)=1
w(dj, n) (n∈N).(6)
ii) Assume that there is a function g: (0,1] →Csuch that w(k, n) = g(k/n)(1≤k≤n)
and let
G(n) =
n
X
k=1
gcd(k,n)=1
g(k/n) (n∈N).
Then
Rw,f (n) = X
d|| n
f(d)G(n/d) (n∈N).(7)
Proof. i) Using that kis regular (mod n) if and only if gcd(k, n)||nand grouping the terms
according to the values of gcd(k, n) = dand denoting k=dj we have at once
Rw,f (n) = X
d|| n
f(d)
n
X
k=1
gcd(k,n)=d
w(k, n) = X
d|| n
f(d)
n/d
X
j=1
gcd(j,n/d)=1
w(dj, n).
ii) Now
n/d
X
j=1
w(dj, n) =
n/d
X
j=1
g(j/(n/d)) = G(n/d).
Remark 3.2. For the function ggiven above let
G(n) :=
n
X
k=1
g(k/n).
Then we have
G(n) = X
d|n
µ(d)G(n/d) (n∈N).(8)
Indeed, as it is well known, G(n) = Pn
k=1 g(k/n)Pd|gcd(k,n)µ(d), giving (8).
5

4 Power sums of regular integers (mod n)
In this section we investigate the sum of r-th powers (r∈N) of the regular integers (mod
n). Let Bm(m∈N0) be the Bernoulli numbers defined by the exponential generating function
t
et−1=
∞
X
m=0
Bm
tm
m!.
Here B0= 1, B1=−1/2, B2= 1/6, B4=−1/30, Bm= 0 for every m≥3, modd and one
has the recurrence relation
Bm=
m
X
j=0 m
jBj(m≥2).(9)
It is well known that for every n, r ∈N,
Sr(n) :=
n
X
k=1
kr=1
r+ 1
r
X
m=0
(−1)mr+ 1
mBmnr+1−m
=nr
2+1
r+ 1
⌊r/2⌋
X
m=0 r+ 1
2mB2mnr+1−2m.(10)
From here one obtains, using the same device as that given in Remark 3.2 that for every
n, r ∈Nwith n≥2,
S[relpr]r(n) :=
n
X
k=1
gcd(k,n)=1
kr=nr
r+ 1
⌊r/2⌋
X
m=0 r+ 1
2mB2mφ1−2m(n),(11)
where φ1−2m(n) = n1−2mQp|n1−p2m−1. Formula (11) was given in [19]. Here we prove the
following result.
Proposition 4.1. For every n, r ∈N,
S[reg]r(n) := X
k∈Regn
kr=nr
2+nr
r+ 1
⌊r/2⌋
X
m=0 r+ 1
2mB2m̺1−2m(n),(12)
where
̺1−2m(n) = n1−2mY
pν|| np(2m−1)ν−p2m−1+ 1
is the generalized ̺function, discussed in Section 2.
Proof. Applying (6) for w(k, n) = krand f=1we have
S[reg]r(n) = X
d|| n
n/d
X
j=1
gcd(j,n/d)=1
(dj)r=X
d|| n
drS[relpr]r(n/d).
6

Now by (11) we deduce
S[reg]r(n) = nr+X
d|| n
d<n
dr
(n/d)r
r+ 1
⌊r/2⌋
X
m=0 r+ 1
2mB2mφ1−2m(n/d)

=nr+nr
r+ 1
⌊r/2⌋
X
m=0 r+ 1
2mB2mX
d|| n
d<n
φ1−2m(n/d)
=nr+nr
r+ 1
⌊r/2⌋
X
m=0 r+ 1
2mB2mX
d|| n
d>1
φ1−2m(d)
=nr−nr
r+ 1
⌊r/2⌋
X
m=0 r+ 1
2mB2m+nr
r+ 1
⌊r/2⌋
X
m=0 r+ 1
2mB2mX
d|| n
φ1−2m(d).
Here Pd|| nφ1−2m(n) = ̺1−2m(d) by (2). Also, by (9),
⌊r/2⌋
X
m=0 r+ 1
2mB2m=r+ 1
2
and this completes the proof.
For example, in the cases r= 1,2,3,4 we deduce that for every n∈N,
S[reg]1(n) = n(̺(n) + 1)
2,(13)
S[reg]2(n) = n2
2+n2̺(n)
3+n
6Y
pν|| n
(pν−p+ 1) ,(14)
S[reg]3(n) = n3
2+n3̺(n)
4+n2
4Y
pν|| n
(pν−p+ 1) ,
S[reg]4(n) = n4
2+n4̺(n)
5+n3
3Y
pν|| n
(pν−p+ 1) −n
30 Y
pν|| np3ν−p3+ 1.
The formula (13) was obtained in [20, Th. 3] and [3, Sec. 2], while (14) was given in a different
form in [3, Prop. 1]. Note that if nis squarefree, then (12) reduces to (10).
For a real number sconsider now the slightly more general sum
S[reg]s(n, x) := X
k≤x
kregular (mod n)
ks.
7

Proposition 4.2. Let s≥0be a fixed real number. Then uniformly for real x > 1and n∈N,
S[reg]s(n, x) = xs+1
s+ 1 ·̺(n)
n+Oxs3ω(n).
Proof. Similar to the proof of Proposition 3.1,
S[reg]s(n, x) = X
k≤x
gcd(k,n)|| n
ks=X
d|| n
dsX
j≤x/d
gcd(j,n/d)=1
js.
Now using the estimate (5) we deduce
S[reg]s(n, x) = X
d|| n
ds(x/d)s+1φ(n/d)
(s+ 1)(n/d)+O(x/d)s2ω(n/d)
=xs+1
(s+ 1)nX
d|| n
φ(n/d) + O
xsX
d|| n
2ω(n/d)
,
and using (1) the proof is complete.
5 Identities for other sums and products over regular integers
(mod n)
5.1 Sums involving Bernoulli polynomials
Let Bm(x) (m∈N0) be the Bernoulli polynomials defined by
text
et−1=
∞
X
m=0
Bm(x)tm
m!.
Here B0(x) = 1, B1(x) = x−1/2, B2(x) = x2−x+ 1/6, B3(x) = x3−3x2/2 + x/2,
Bm(0) = Bm(m∈N0) are the Bernoulli numbers already defined in Section 4and one has the
recurrence relation
Bm(x) =
m
X
j=0 m
jBjxm−j(m∈N0).
It is well known (see, e.g., [5, Sect. 9.1]) that for every n, m ∈N,m≥2,
Tm(n) :=
n
X
k=1
Bm(k/n) = Bm
nm−1.(15)
Furthermore, applying (8) one obtains from (15) that for every n, m ∈N,m≥2,
T[relpr]m(n) :=
n
X
k=1
gcd(k,n)=1
Bm(k/n) = Bmφ1−m(n),(16)
8

where φ1−m(n) = n1−mQp|n1−pm−1. See [5, Sect. 9.9, Ex. 7]. We now show the validity of
the next formula:
Proposition 5.1.1 For every n, m ∈N,m≥2,
T[reg]m(n) := X
k∈Regn
Bm(k/n) = Bm̺1−m(n),(17)
where ̺1−m(n) = n1−mQpν|| np(m−1)ν−pm−1+ 1.
Proof. Choosing g(x) = Bm(x) and f=1we deduce from (7) by using (16) that
T[reg]m(n) = X
d|| n
T[relpr]m(d)
=BmX
d|| n
φ1−m(d) = Bm̺1−m(n),
according to (2).
Remark 5.1.2 In the case m= 1 a direct computation and (13) show that T[reg]1(n) = 1/2.
Also, (17) can be put in the form
n−1
X
k=0
kregular (mod n)
Bm(k/n) = Bm̺1−m(n),
which holds true for every n, m ∈N, also for m= 1.
5.2 Sums involving gcd’s and the exp function
Consider in what follows the function
P[reg]f,t(n) := X
k∈Regn
f(gcd(k, n)) exp(2πikt/n) (n∈N, t ∈Z),
where fis an arbitrary arithmetic function. For t= 0 and f(n) = n(n∈N) we reobtain the
function e
P(n) and for f=1we have cn(t), the analogue of the Ramanujan sums, both given in
the Introduction. We have
Proposition 5.2.1 For every fand every n∈Nand t∈Z,
P[reg]f,t(n) = X
d|| n
f(d)cn/d(t).
If fis integer valued and multiplicative (in particular, if f= id), then n7→ P[reg]f ,t(n)also
has these properties.
9

Proof. Choosing g(x) = exp(2πitx) from (7) we deduce at once that
P[reg]f,t (n) = X
d|| n
f(d)
n/d
X
j=1
gcd(j,n/d)=1
exp(2πijt/(n/d)) = X
d|| n
f(d)cn/d(t).
For t= 1 and f= id this gives the multiplicative function
P[reg]id,1(n) = X
d|| n
dµ(n/d),
not investigated in the literature, as far as we know. Here P[reg]id,1(pν) = p−1 for every prime
pand P[reg]id,1(pν) = pνfor every prime power pνwith ν≥2.
Proposition 5.2.2 We have
X
n≤x
P[reg]id,1(n) = x2
2Y
p1−1
p2+1
p3+O(xlog2x).
Proof. Using (5) for s= 1 we deduce
X
n≤x
P[reg]id,1(n) = X
d≤x
µ(d)X
δ≤x/d
gcd(δ,d)=1
δ
=X
d≤x
µ(d)φ(d)(x/d)2
2d+O((x/d)2ω(d))
=x2
2
∞
X
d=1
µ(d)φ(d)
d3+O x2X
d>x
1
d2!+O
xX
d≤x
2ω(d)
d
,
giving the result.
5.3 An analogue of Menon’s identity
Our next result is the analogue of Menon’s identity ([12], see also [23])
n
X
k=1
gcd(k,n)=1
gcd(k−1, n) = φ(n)τ(n) (n∈N).(18)
Proposition 5.3.1 For every n∈N,
X
k∈Regn
gcd(k−1, n) = X
d|| n
φ(d)τ(d) = Y
pν|| npν−1(p−1)(ν+ 1) + 1.
10

Proof. Applying (6) for w(k, n) = gcd(k−1, n) and f=1we deduce
Sn:= X
k∈Regn
gcd(k−1, n) = X
d|| n
n/d
X
j=1
gcd(j,n/d)=1
gcd(dj −1, n)
=X
d|| n
n/d
X
j=1
gcd(j,n/d)=1
gcd(dj −1, n/d),
since gcd(dj −1, d) = 1 for every dand j. Now we use the identity
n
X
k=1
gcd(k,n)=1
gcd(ak −1, n) = φ(n)τ(n) (n∈N),
valid for every fixed a∈Nwith gcd(a, n) = 1, see [23, Cor. 14] (for a= 1 this reduces to (18)).
Choose a=d. Since d||nwe have gcd(d, n/d) = 1 and obtain
Sn=X
d|| n
φ(n/d)τ(n/d) = X
d|| n
φ(d)τ(d).
5.4 Trigonometric sums
Further identities for sums over Regncan be derived. As examples, consider the following
known trigonometric identities. For every n∈N,n≥2,
n
X
k=1
cos2kπ
n=n
2;
furthermore, for every n∈Nodd number,
n
X
k=1
tan2kπ
n=n2−n;
and also for every n∈Nodd,
n
X
k=1
tan4kπ
n=1
3(n4−4n2+ 3n).
See, for example, [4] for a discussion and proofs of these identities. See [16, Appendix 3] for
other similar identities. By the approach given in Remark 3.2 we deduce that for every n∈N,
n
X
k=1
gcd(k,n)=1
cos2kπ
n=φ(n) + µ(n)
2;
11

for every n∈Nodd number,
n
X
k=1
gcd(k,n)=1
tan2kπ
n=φ2(n)−φ(n);
and for every n∈Nodd,
n
X
k=1
gcd(k,n)=1
tan4kπ
n=1
3(φ4(n)−4φ2(n) + 3φ(n)).
This gives the next results. The proof is similar to the proofs given above.
Proposition 5.4.1 For every n∈N,
X
k∈Regn
cos2kπ
n=̺(n) + µ(n)
2,
where µ(n) = Pd|| nµ(d)is the characteristic function of the squarefull integers n, given in the
Introduction.
Proposition 5.4.2 For every n∈Nodd number,
X
k∈Regn
tan2kπ
n=̺2(n)−̺(n),
X
k∈Regn
tan4kπ
n=1
3(̺4(n)−4̺2(n) + 3̺(n)).
5.5 The product of numbers in Regn
It is known (see, e.g., [16, p. 197, Ex. 24]) that for every n∈N,
Q[relpr](n) :=
n
Y
k=1
gcd(k,n)=1
k=nφ(n)A(n),(19)
where
A(n) = Y
d|n
(d!/dd)µ(n/d).
We show that
Proposition 5.5.1 For every n∈N,
Q[reg](n) := Y
k∈Regn
k=n̺(n)Y
d|| n
A(d).
12

Proof. Choosing w(k, n) = log kand f=1in Proposition 3.1 we have
log Q[reg](n) = X
k∈Regn
log k=X
d|| n
n/d
X
j=1
gcd(j,n/d)=1
log(dj)
=X
d|| n
(φ(n/d) log d+ log Q[relpr](n/d))
=X
d|| n
(φ(d) log(n/d) + log Q[relpr](d))
= (log n)X
d|| n
φ(d)−X
d|| n
φ(d) log d+X
d|| n
log Q[relpr](d).
Hence,
Q[reg](n) = n̺(n)Y
d|| n
Q[relpr](d)
dφ(d).
Now the result follows from the identity (19).
5.6 Products involving the Gamma function
Let Γ be the Gamma function defined for x > 0 by
Γ(x) = Z∞
0
e−ttx−1dt.
It is well known that for every n∈N,
R(n) :=
n
Y
k=1
Γ(k/n) = (2π)(n−1)/2
√n,(20)
which is a consequence of Gauss’ multiplication formula. For the q-analogs of the Gamma and
Beta functions and the multiplication formula see the recent papers [6,7] published in this
journal. Furthermore, for every n∈N,n≥2,
R[relpr](n) :=
n
Y
k=1
gcd(k,n)=1
Γ(k/n) = (2π)φ(n)/2
exp(Λ(n)/2) ,(21)
see [17,11].
Proposition 5.6.1 For every n∈N,
R[reg](n) := Y
k∈Regn
Γ(k/n) = (2π)(̺(n)−1)/2
pκ(n).(22)
13

Proof. Choosing g= log Γ and f=1in (7) and using (21) we deduce
log R[reg](n) = X
k∈Regn
log Γ(k/n) = X
d|| n
log R[relpr](d)
=X
d|| n
d>1log 2π
2φ(d)−1
2Λ(d)
=X
d|| nlog 2π
2φ(d)−1
2Λ(d)−log 2π
2=log 2π
2(̺(n)−1) −1
2X
d|| n
Λ(d),
where the last sum is log κ(n).
For squarefree n(22) reduces to (20).
5.7 Identities involving cyclotomic polynomials
Let Φn(x) (n∈N) stand for the cyclotomic polynomials (see, e.g., [9, Ch. 13]) defined by
Φn(x) =
n
Y
k=1
gcd(k,n)=1
(x−exp(2πik/n)) .
Consider now the following analogue of the cyclotomic polynomials Φn(x):
Φ[reg]n(x) = Y
k∈Regn
(x−exp(2πik/n)) .
The application of Proposition 3.1 gives the following result.
Proposition 5.7.1 For every n∈N,
Φ[reg]n(x) = Y
d|| n
Φd(x).
Here the degree of Φ[reg]n(x) is ̺(n). If nis squarefree, then Φ[reg]n(x) = xn−1 and for
example, Φ[reg]12(x) = Φ1(x)Φ3(x)Φ4(x)Φ12 (x) = x9−x6+x3−1.
It is well known that for every n∈N,n≥2,
U(n) :=
n
Y
k=1
gcd(k,n)=1
sin kπ
n=Φn(1)
2φ(n),(23)
where
Φn(1) = (p, n =pν, ν ≥1,
1,otherwise, i.e., if ω(n)≥2,
and for n≥3,
V(n) :=
n
Y
k=1
gcd(k,n)=1
cos kπ
n=Φn(−1)
(−4)φ(n)/2,(24)
14

where
Φn(−1) = 




2, n = 2 ν,
p, n = 2pν, p > 2 prime, ν ≥1,
1,otherwise.
For every n∈N,Qk∈Regnsin(kπ/n) = 0, since n∈Regn. This suggests to consider also the
modified products
U[regmod](n) :=
n−1
Y
k=1
kregular (mod n)
sin kπ
n,
V[regmod](n) :=
n−1
Y
k=1
kregular (mod n)
cos kπ
n.
We show that U[regmod](n) is nonzero for every n≥2. More precisely, define the modified
polynomials
Φ[regmod]n(x) = (x−1)−1Φ[reg]n(x) = Y
d|| n
d>1
Φd(x).
Here, for example, Φ[regmod]12(x) = Φ3(x)Φ4(x)Φ12(x) = x8+x7+x6+x2+x+ 1. All
of the polynomials Φ[regmod]n(x) have symmetric coefficients. By arguments similar to those
leading to the formulas (23) and (24) we obtain the following identities.
Proposition 5.7.2 For every n∈N,n≥2,
U[regmod](n) = Φ[regmod]n(1)
2̺(n)−1=κ(n)
2̺(n)−1,
and for every n∈N,n≥3odd,
V[regmod](n) = Φ[regmod]n(−1)
(−4)(̺(n)−1)/2= (−1/4)(̺(n)−1)/2.
Note that ̺(n) is odd for every n∈Nodd.
6 Maximal orders of certain functions
Let σ(n) be the sum of divisors of nand let ψ(n) = nQp|n(1 + 1/p) be the Dedekind
function. The following open problems were formulated in [2]: What are the maximal orders of
the functions ̺(n)σ(n) and ̺(n)ψ(n)?
The answer is the following:
Proposition 6.1.
lim sup
n→∞
̺(n)σ(n)
n2log log n= lim sup
n→∞
̺(n)ψ(n)
n2log log n=6
π2eγ,
where γis the Euler-Mascheroni constant.
15

Proof. Apply the following general result, see [25, Cor. 1]: If fis a nonnegative real-valued
multiplicative arithmetic function such that for each prime p,
i) ρ(p) := supν≥0f(pν)≤(1 −1/p)−1, and
ii) there is an exponent ep=po(1) ∈Nsatisfying f(pep)≥1 + 1/p,
then
lim sup
n→∞
f(n)
log log n=eγY
p1−1
pρ(p).
Take f(n) = ̺(n)σ(n)/n2. Here f(p) = 1 + 1/p and f(pν) = 1 + 1/pν+ 1/pν+2 + 1/pν+3 +
...+ 1/p2ν<1 + 1/p for every prime pand every ν≥2. This shows that ρ(p) = 1 + 1/p and
obtain that
lim sup
n→∞
f(n)
log log n=eγY
p1−1
p2=6
π2eγ.
The proof is similar for the function g(n) = ̺(n)ψ(n)/n2. In fact, g(p) = f(p) = 1 + 1/p and
g(pν)≤f(pν) for every prime pand every ν≥2, therefore the result for g(n) follows from the
previous one.
Remark 6.2. Let σs(n) = Pd|nds. Then for every real s > 1,
lim sup
n→∞
̺s(n)σs(n)
n2s=ζ(s)
ζ(2s).
This follows by observing that for fs(n) = ̺s(n)σs(n)/n2s,fs(p) = 1 + 1/psand fs(pν) =
1 + 1/psν + 1/ps(ν+2) + 1/ps(ν+3) +...+ 1/p2sν <1 + 1/psfor every prime pand every ν≥2.
Hence, for every n∈N,
fs(n)≤Y
p|n1 + 1
ps<Y
p1 + 1
ps=ζ(s)
ζ(2s),
and the lim sup is attained for n=nk=Q1≤j≤kpjwith k→ ∞, where pjis the j-th prime.
7 Acknowledgement
L. T´oth gratefully acknowledges support from the Austrian Science Fund (FWF) under the
project Nr. M1376-N18.
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Br˘adut¸ Apostol
Pedagogic High School ”Spiru Haret”
Str. Timotei Cipariu, RO-620004 Foc¸sani, Romania
E-mail: apo brad@yahoo.com
L´aszl´o T´oth
Institute of Mathematics, Universit¨at f¨ur Bodenkultur
Gregor Mendel-Straße 33, A-1180 Vienna, Austria
and
Department of Mathematics, University of P´ecs
Ifj´us´ag u. 6, H-7624 P´ecs, Hungary
E-mail: ltoth@gamma.ttk.pte.hu
18