PORTUGALIAE MATHEMATICA
Vol. 53 Fasc. 3 – 1996
A GENERALIZATION OF MENON’S IDENTITY
WITH RESPECT TO A SET OF POLYNOMIALS
P. Haukkanen and J. Wang*
Abstract: P. Kesava Menon’s elegant identity states that
X
(a − 1, n) = φ(n) τ (n) ,
a (mod n)
(a,n)=1
where φ(n) is Euler’s totient function and τ (n) is the number of divisors of n. In this
paper we generalize this identity so that, among other things, a − 1 is replaced with a
set {fi (a)} of polynomials in Z[a1 , a2 , ..., au ].
1 – Introduction
For positive integers u and n, let Su (n) denote the set of all u-vectors {ai }
(mod n) such that ((ai ), n) = 1, where (ai ) is the g.c.d. of a1 , a2 , ..., au . It is
well-known that the cardinality of Su (n) is Jordan’s totient Ju (n). In particular,
J1 (n) is Euler’s totient φ(n). For the sake of brevity we write S1 (n) = S(n).
In [5], P. Kesava Menon established the elegant identity
X
(1.1)
(a − 1, n) = φ(n) τ (n) ,
a∈S(n)
where τ (n) is the number of divisors of n. Richards [10] mentioned among other
things the identity
(1.2)
X
a∈S(n)
(f (a), n) = φ(n)
o¯
X¯¯n
¯
¯ r ∈ S(d) : f (r) ≡ 0 (mod d) ¯ ,
d|n
Received : September 7, 1995.
AMS Subject Classification: 11A25.
Keywords and Phrases: Menon’s identity, Set of polynomials, Euler’s totient, Jordan’s totient, Regular arithmetical convolution.
* Supported by the Natural Science Foundations of China and Liaoning Province.
332
P. HAUKKANEN and J. WANG
where f (x) is any polynomial with integer coefficients. Nageswara Rao [8] generalized (1.1) to
(1.3)
X
(a1 − s1 , ..., au − su , n) = Ju (n) τ (n) ,
{ai }∈Su (n)
where {si } is a fixed element of Su (n). Sita Ramaiah [11, Section 9] considers
Menon’s identity with respect to regular convolutions.
In this paper we consider a combination of (1.2) and (1.3) with respect to
regular convolutions.
It should be noted that in the literature there are also other generalizations
of Menon’s identity (see e.g. [1, 3, 4, 7, 12–17]). The most general identity is
given in [3]. This identity arises from the theory of even functions and contains
as special cases most of the generalized Menon identities. However, it does not
deal with generalized Menon identities with respect to polynomials and therefore
does not contain (1.2). We do not consider the identity of [3] here.
2 – Preliminaries
We assume that the reader is familiar wit the notion of regular convolution.
This notion was introduced by Narkiewicz [9]. Further background material on
regular convolution can be found e.g. in [6, 11].
Let A be a regular convolution. For a positive integer k, denote
Ak (n) = {d : dk ∈ A(nk )} .
It is known [11] that Ak is a regular convolution. The Ak -convolution of two
arithmetical functions f and g is given as
(f ∗Ak g)(n) =
X
f (d) g(n/d) .
d∈Ak (n)
Let µAk denote the Ak -analogue of the Möbius function. We then have
(2.1)
f (n) =
X
(f ∗Ak µAk )(d) .
d∈Ak (n)
Let A be a regular convolution, and let k, u and n be positive integers. Let
(u)
SA,k (n) denote the set of all u-vectors {ai } (mod nk ) such that ((ai ), nk )A,k = 1,
where ((ai ), nk )A,k denotes the greatest k-th power divisor of (ai ) which belongs
(u)
(u)
to A(nk ). The number of elements in SA,k (n) is denoted by φA,k (n). If k = 1
333
A GENERALIZATION OF MENON’S IDENTITY
(u)
(u)
and A is the Dirichlet convolution, then SA,k (n) = Su (n) and φA,k is the Jordan
(u)
totient Ju . The function φA,k is thus a generalization of the Jordan totient Ju
and therefore a generalization of the Euler totient φ. It is known [3] that
(u)
(2.2)
dku µAk (n/d) .
X
φA,k (n) =
d∈Ak (n)
The Möbius inversion formula with respect to regular convolutions gives
nku =
(2.3)
(u)
X
φA,k (d) .
d∈Ak (n)
Further, it can be verified that
(u)
φA,k (n) = nku
(2.4)
Y
pt ∈A
(1 − p−kut ) ,
k (n)
where the product is over the Ak -primitive prime powers of n.
In what follows we denote u-vectors by boldface letters, i.e., {ai } = a.
(u)
Lemma. Let d ∈ Ak (n). Then for any b ∈ SA,k (d),
(2.5)
¯n
o¯
(u)
(u)
(u)
¯
¯
¯ a ∈ SA,k (n) : a ≡ b (mod dk ) ¯ = φA,k (n)/φA,k (d) .
t
t
Proof: Let pjj , j = 1, 2, ..., v, be the Ak -primitive prime powers such that
t
pjj ∈ Ak (n), pjj 6 | d. Denote
n
o
T = a (mod nk ) : a ≡ b (mod dk ) ,
n
t k
o
Tj = a ∈ T : pjj | (ai ) ,
j = 1, 2, ..., v .
(u)
It is clear that an element a ∈ T belongs to SA,k (n) if, and only if, ptk 6 | (ai ) for
all Ak -primitive prime powers pt ∈ Ak (n). If pt ∈ Ak (n) and pt | d, then ptk6 | (ai ).
(u)
Namely, otherwise ptk | (bi ), which contradicts the hypothesis b ∈ SA,k (d). Thus
(u)
t k
a ∈ T belongs to SA,k (n) if, and only if, pjj 6 | (ai ) for all j = 1, 2, ..., v. So we can
deduce that
v
¯
¯n
o¯ ¯
[
(u)
¯
¯ ¯
¯
Tj ¯ .
¯ a ∈ SA,k (n) : a ≡ b (mod dk ) ¯ = ¯T \
j=1
334
P. HAUKKANEN and J. WANG
By the inclusion-exclusion principle,
(2.6)
¯n
o¯
(u)
¯
¯
¯ a ∈ SA,k (n) : a ≡ b (mod dk ) ¯ =
= |T | +
v
X
(−1)j
j=1
X
|Te1 ∩ Te2 ∩ ... ∩ Tej | .
1≤e1 ≤e2 <...<ej ≤v
If a ∈ Te1 ∩ Te2 ∩ ... ∩ Tej , then for all i = 1, 2, ..., u there exists xi = 1, 2, ..., nk /dk
such that ai = bi + xi dk and
(2.7)
kte
kte
kte
bi + xi dk ≡ 0 (mod pe1 1 pe2 2 · · · pej j ) .
kte
kte
kte
t
t
Denote briefly mj = pe1 1 pe2 2 · · · pej j . Since d ∈ Ak (n) and pjj ∈ Ak (n), pjj 6 | d
for all j = 1, 2, ..., v, we have
(mj , dk ) = 1 .
Thus for all i = 1, 2, ..., u there exists a unique xi (mod mj ) satisfying (2.7).
Consequently, there exists exactly one xi satisfying (2.7) in each of the intervals
[1, mj ], [mj + 1, 2mj ], ..., [(c − 1)mj + 1, c mj ], where c mj = nk /dk . Therefore
|Te1 ∩ Te2 ∩ ... ∩ Tej | =
nku /dku
nku /dku
.
=
kute
kute
kute
muj
pe1 1 pe2 2 · · · p ej j
Now, by (2.6), we have
¯n
o¯
(u)
¯
¯
¯ a ∈ SA,k (n) : a ≡ b (mod dk ) ¯ =
=
v
nku X
+
(−1)j
dku j=1
1≤e
nku /dku
X
1 <e2 <...<ej ≤v
kute1
pe1
kute2
pe2
v
nku pt ∈Ak (n) (1 − p−kut )
nku Y
−kut
= ku
(1 − pj j ) = ku Q
−kut )
d j=1
d
pt ∈Ak (d) (1 − p
Q
(u)
=
φA,k (n)
(u)
φA,k (d)
.
This completes the proof.
Remark. Let D(b, d, n) denote the arithmetic progression
D(b, d, n) = {b, b + d, ..., b + (n − 1)d} ,
kutej
· · · p ej
335
A GENERALIZATION OF MENON’S IDENTITY
where (b, d) = 1. Let φ(b, d, n) denote the number of elements in D(b, d, n) that
are relatively prime to n (see [2]). A direct consequence of the above lemma is
that
φ(b, d, n) = d φ(n)/φ(d) .
This result could be generalized to the general case of the lemma, that is, with
respect to k, u and A.
3 – The identity
Let F = {f1 , f2 , ..., fs } be a set of polynomials in Z[x1 , x2 , ..., xu ]. Let NF (n)
(u)
denote the number of elements a ∈ SA,k (n) such that fi (a) ≡ 0 (mod nk ) for
every i = 1, 2, ..., s and let TF be the Ak -convolution of NF and the ζ-function,
i.e.,
X
TF (n) =
(3.1)
NF (d) ζ(n/d) =
X
NF (d) .
d∈Ak (n)
d∈Ak (n)
Theorem. Let g be any arithmetical function. Then
(3.2)
µ
¶
X
g
(u)
a∈SA,k
(n)
³
(fi (a)), nk
´1/k
(u)
= φA,k (n)
A,k
X
(u)
(g ∗Ak µAk )(d) NF (d)/φA,k (d) .
d∈Ak (n)
Proof: By (2.1),
X
(u)
a∈SA,k
(n)
g
µ³
(fi (a)), n
=
k
´1/k ¶
A,k
X
=
X
(g ∗Ak µAk )(d)
d∈A (n)
(u)
a∈SA,k
(n) k k
d |(fi (a))
=
X
(g ∗Ak µAk )(d)
X
(g ∗Ak µAk )(d)
d∈Ak (n)
1
X
¯n
o¯
(u)
¯
¯
¯ a ∈ SA,k (n) : a ≡ b (mod dk ) ¯ .
(u)
a∈SA,k
(n)
dk |(fi (a))
d∈Ak (n)
=
X
(u)
b∈SA,k
(d)
k
d |(fi (b))
336
P. HAUKKANEN and J. WANG
By the lemma,
X
(u)
a∈SA,k
(n)
g
µ³
(fi (a)), nk
´1/k ¶
A,k
=
X
(u)
(u)
(g ∗Ak µAk )(d) NF (d) φA,k (n)/φA,k (d)
d∈Ak (n)
(u)
= φA,k (n)
X
(u)
(g ∗Ak µAk )(d) NF (d)/φA,k (d) .
d∈Ak (n)
This completes the proof.
Corollary. We have
(3.3)
X
(u)
a∈SA,k
(n)
³
(fi (a)), nk
´u
A,k
(u)
= φA,k (n) TF (n) .
Proof: Take g(n) = nku in the theorem and apply (2.2) and (3.1).
Example 1: If F = {f }, k = u = 1 and A is the Dirichlet convolution, then
(3.3) reduces to (1.2). If f (x) = x − 1, then (1.2) reduces to (1.1).
Example 2: If F = {x1 − s1 , x2 − s2 , ..., xu − su }, s ∈ Su (n), k = 1 and A
is the Dirichlet convolution, then NF (n) = 1, TF (n) = τ (n) and therefore (3.3)
reduces to (1.3).
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Pentti Haukkanen,
Department of Mathematical Sciences, University of Tampere,
P.O. Box 607, FIN-33101 Tampere – FINLAND
and
Jun Wang,
Institute of Mathematical Sciences, Dalian University of Technology,
Dalian 116024 – PEOPLE’S REPUBLIC OF CHINA