1
2
3
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6
Journal of Integer Sequences, Vol. 19 (2016),
Article 16.2.7
23 11
Sums of Products of Generalized Ramanujan Sums
Soichi Ikeda
Department of Mathematics
Shibaura Institute of Technology
307 Fukasaku, Minuma-ku, Saitama 337-8570
Japan
sikeda@sic.shibaura-it.ac.jp
Isao Kiuchi
Department of Mathematical Sciences
Faculty of Science
Yamaguchi University
Yoshida 1677-1 Yamaguchi 753-8512
Japan
kiuchi@yamaguchi-u.ac.jp
Kaneaki Matsuoka
Graduate School of Mathematics
Nagoya University
Furocho Chikusaku Nagoya 464-8602
Japan
m10041v@math.nagoya-u.ac.jp
Abstract
(1)
(n)
We consider weighted averages for the products tk1 (j) · · · tkn (j) of generalized RaP
(i)
manujan sums tki (j) =
d| gcd(ki ,j) fi (d)gi (ki /d)hi (j/d) with any arithmetical functions fi , gi and hi (i = 1, . . . , n), and derive formulas for several weighted averages
with weights concerning completely multiplicative functions, completely additive functions, and others.
1
1
Introduction
Let gcd(k, j) be the greatest common divisor of the positive integer k and the integer j, and
let K = lcm(k1 , . . . , kn ) be the least common multiple of n-tuple positive integers k1 , . . . , kn .
The function ck (j), as usual, denotes the Ramanujan sum defined as the sum of the m-th
powers of the primitive k-th roots of unity, namely,
X
mj
,
exp 2πi
ck (j) =
k
1≤m≤k
gcd(m,k)=1
which can be expressed as the well-known identity
X
k
dµ
d
d| gcd(k,j)
with the Möbius function µ. Anderson and Apostol [5] (also see [6, 7, 14, 17]) first introduced
the sum
X
k
sk (j) =
f (d)g
d
d| gcd(k,j)
with any arithmetical functions f and g, which is a generalization of Ramanujan’s sum. This
function is said to be the Anderson–Apostol sum. Now, let f be a completely multiplicative
function, and let g(k) = µ(k)u(k) with u a multiplicative function. Assume that f (p) 6= 0
and f (p) 6= u(p) for all primes p, and let F (k) = (f ∗ g)(k). Anderson and Apostol [5]
derived the identity sk (j) = F (k)µ (m) u (m)/F (m) with m = k/ gcd(k, j). This is said
to be Hölder’s identity for sk (j), which has been considered by many mathematicians and
physicists. For any fixed positive integer r, using the properties of the sum
k
X
j r ck (j),
(1)
j=1
Alkan [1, 3] derived exact formulas involving Euler’s and Jordan’s functions for averages of
the special values of L-functions. Tóth [18] showed a simpler proof of (1) and established
some identities for other weighted averages of Ramanujan’s sum with weights concerning
logarithms, values of arithmetic functions for gcd’s, the gamma function, the Bernoulli polynomials, and binomial coefficients. In a recent paper, Kiuchi, Minamide and Ueda [11] gave
a generalization of some identities due to Tóth [18]; they showed that
⌊r/2⌋ k
1
1 X r+1
1 X r
j sk (j) = (f ∗ g)(k) +
B2m (f ∗ id1−2m · g)(k)
k r j=1
2
r + 1 m=0 2m
2
(2)
for any fixed positive integer r,
k
X
j=1
sk (j) log j = (f · log ∗g · id)(k) + (f ∗ g · Log)(k),
√
1
j
= log 2π {(f ∗ g · id)(k) − (f ∗ g)(k)} − (f ∗ g · log)(k),
sk (j) log Γ
k
2
j=1
d
k X f (d) k X
lk
1 X k
lπ
sk (j) =
(−1) d cosk
g
k
2 j=0 j
d
d l=1
d
k
X
(3)
(4)
(5)
d|k
and
Bm
j
sk (j) = k−1 (idm−1 · f ∗ g)(k),
Bm
k
k
j=0
k−1
X
(6)
where the function Log d is given by log(d!) and Γ denotes the gamma function. Here, Bm (x)
[7] (also see [8, 9]) denotes the Bernoulli polynomials defined by the expansion
∞
X
zexz
zm
=
B
(x)
m
ez − 1 m=0
m!
where |z| < 2π. The number Bm is the Bernoulli number given by Bm (0). The expressions in
(2)–(6) give a generalization of some interesting identities by Tóth [12] and Alkan [1, 2, 3, 4].
The function
K
1 X
E(k1 , . . . , kn ) =
ck (j) · · · ckn (j)
K j=1 1
(7)
of the product ck1 (j) · · · ckn (j) of Ramanujan’s sums for n variables was investigated in the
studies of Liskovets [12] and Tóth [20], which has some interesting formulas for combinatorial
and topological applications. The expression (7) has been introduced by Mednykh and
Nedela [15] to handle certain problems of enumerative combinatorics. They showed that
all the values E(k1 , . . . , kn ) are nonnegative integers. Furthermore, Tóth derived that two
interesting representations [20, Propositions 3 and 9], [19, Corollary 4] for E hold:
X
d1 · · · dn
kn
k1
E(k1 , . . . , kn ) =
···µ
(8)
µ
lcm(d1 , . . . , dn )
d1
dn
d1 |k1 ,...,dn |kn
and
K
1 X
,
ck1 (d) · · · ckn (d)φ
E(k1 , . . . , kn ) =
K
d
d|K
3
(9)
where φ is the Euler totient function. He also noted that all values for E are nonnegative
e denote a
integers and that E is multiplicative as a function of several variables. Let E
generalization of E defined by
K
1 X
e
E(k1 , . . . , kn ) =
sk (j) · · · skn (j).
K j=1 1
(10)
e hold:
Then Tóth [20, Proposition 19] deduced that two formulas for E
X
f
(d
)
·
·
·
f
(d
)
kn
k
1
n
1
e 1 , . . . , kn ) =
E(k
···g
g
lcm(d1 , . . . , dn )
d1
dn
(11)
d1 |k1 ,...,dn |kn
and
X
K
1
e 1 , . . . , kn ) =
.
sk1 (d) · · · skn (d)φ
E(k
K
d
(12)
d|K
He also showed that if f and g are multiplicative functions, then (10) is multiplicative as
a function of several variables. The weighted average of the products of Ramanujan’s sum
with weight concerning the function idr for any fixed positive integer r defined by
Sr (k1 , . . . , kn ) =
1
K r+1
K
X
j=1
j r ck1 (j) · · · ckn (j)
(13)
was considered by Tóth [18, Proposition 7], who derived the following identity
φ(k1 ) · · · φ(kn )
2K
⌊ 2r ⌋ X r + 1 B2m
X
Sr (k1 , . . . , kn ) =
1
+
r + 1 m=0
2m
K 2m
d1 |k1 ,...,dn |kn
(14)
d1 · · · dn
µ
lcm(d1 , . . . , dn )1−2m
k1
d1
···µ
kn
dn
.
He presented a multivariable generalization of (1) connected to the orbicyclic arithmetic
function, discussed by Liskovets [12] and Tóth [20]. Substituting r = 1 in (14), it follows
that
φ(k1 ) · · · φ(kn ) E(k1 , . . . , kn )
+
,
S1 (k1 , . . . , kn ) =
2K
2
which was given by Tóth [18, Corollary 2].
(i)
(i)
For any arithmetical functions fi , gi and hi (i = 1, 2, . . . , n), we define ski (j) and tki (j)
by
X
X
j
ki
ki
(i)
(i)
and tki (j) =
hi
,
ski (j) =
fi (d)gi
fi (d)gi
d
d
d
d| gcd(ki ,j)
d| gcd(ki ,j)
4
respectively.
The first aim of this study is to derive some identities for weighted averages of the product
(1)
(n)
tk1 (j) · · · tkn (j) with weight function wr , completely multiplicative, completely additive, and
others, for any fixed positive integer r, namely
Ur (k1 , . . . , kn ) =
K
X
j=1
(1)
(n)
wr (j)tk1 (j) · · · tkn (j).
(15)
This sum is a generalization of some of the identities for weighted averages mentioned by
Alkan [2], Liskovets [12], Tóth [18, 20] and Kiuchi, Minamide and Ueda [11]. To our knowledge, we derive some new and useful formulas in Theorems 1, 6 and 10. This study then
(1)
(n)
aims to establish some identities for the weighted averages of the product sk1 (j) · · · skn (j)
with weights concerning the gamma function, binomial coefficients, and Bernoulli polynomials, which provide a generalization of some interesting identities discussed by Tóth [18] and
Kiuchi, Minamide and Ueda [11].
Notations. We use the following notation throughout this paper. For any positive integer
n, the functions id and idq as well as the unit function 1 are given by id(n) = n, idq (n) = nq
for any real number q and 1(n) = 1, respectively. The symbols ∗ and · denote the Dirichlet
convolution and the ordinary product of arithmetical functions, respectively. The function
φs (n) defines the Jordan function by (ids ∗ µ)(n) for any real number s.
2
Some formulas for Ur (k1, . . . , kn)
We shall evaluate the function Ur (k1 , . . . , kn ). The weight function wr of Ur (k1 , . . . , kn ) only
deals with completely multiplicative functions and completely additive functions, and we
introduce a simple proof of (15) and some useful formulas in Theorems 1 and 6. Moreover,
(1)
(n)
we shall derive some identities of the weighted averages of the product sk1 (j) · · · skn (j) with
weights concerning the gamma function, binomial coefficients, and the Bernoulli polynomials.
Theorem 1. Let f1 , . . . , fn , g1 , . . . , gn , and h1 , . . . , hn denote any arithmetical functions,
and let
Ur (k1 , . . . , kn )
=
K
X
j=1
wr (j)
X
d1 | gcd(k1 ,j)
f1 (d1 )g1
k1
d1
h1
j
d1
5
···
X
dn | gcd(kn ,j)
fn (dn )gn
kn
dn
hn
j
dn
.
If wr is a completely multiplicative function, we have
Ur (k1 , . . . , kn )
X
kn
k1
· · · fn (dn )gn
wr (lcm(d1 , . . . , dn ))×
=
f1 (d1 )g1
d1
dn
(16)
d1 |k1 ,...,dn |kn
K
lcm(d1 ,...,dn )
X
×
wr (l)h1
l=1
lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
l · · · hn
l ,
d1
dn
and if in addition, h1 , . . . , hn are completely multiplicative functions, then
Ur (k1 , . . . , kn )
X
lcm(d1 , . . . , dn )
kn
k1
h1
· · · fn (dn )gn
×
=
f1 (d1 )g1
d1
d1
dn
(17)
d1 |k1 ,...,dn |kn
× hn
lcm(d1 , . . . , dn )
dn
K
lcm(d1 ,...,dn )
wr (lcm(d1 , . . . , dn ))
X
l=1
wr (l)h1 (l) · · · hn (l) .
If wr is a completely additive function, we have
Ur (k1 , . . . , kn )
X
kn
k1
· · · fn (dn )gn
wr (lcm(d1 , . . . , dn ))×
=
f1 (d1 )g1
d1
dn
d1 |k1 ,...,dn |kn
K
lcm(d1 ,...,dn )
lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
×
l · · · hn
l
h1
d1
dn
l=1
X
k1
kn
f1 (d1 )g1
· · · fn (dn )gn
×
d1
dn
X
+
d1 |k1 ,...,dn |kn
K
lcm(d1 ,...,dn )
×
X
l=1
wr (l)h1
lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
l · · · hn
l ,
d1
dn
6
(18)
and if in addition, h1 , . . . , hn are completely multiplicative functions, then
Ur (k1 , . . . , kn )
(19)
X
lcm(d1 , . . . , dn )
kn
lcm(d1 , . . . , dn )
k1
h1
· · · fn (dn )gn
hn
×
=
f1 (d1 )g1
d1
d1
dn
dn
d1 |k1 ,...,dn |kn
K
lcm(d1 ,...,dn )
× wr (lcm(d1 , . . . , dn ))
+
X
f1 (d1 )g1
d1 |k1 ,...,dn |kn
k1
d1
X
l=1
h1
h1 (l) · · · hn (l)
lcm(d1 , . . . , dn )
d1
· · · fn (dn )gn
kn
dn
hn
lcm(d1 , . . . , dn )
dn
×
K
lcm(d1 ,...,dn )
×
X
l=1
wr (l)h1 (l) · · · hn (l) .
Remark 2. The formulas (16) and (17) immediately imply a generalization of the two formulas (8) and (11). We substitute wr = 1, f1 = · · · = fn = f , g1 = · · · = gn = g, and
h1 = · · · = hn = 1 in (16) to obtain the formula (11).
The formulas (16)–(19) give an analogue and a generalization of result of Tóth [22,
Proposition 1]. For any positive integer k, Kiuchi, Minamide and Ueda [11] recently showed
that if wr is a completely multiplicative function, then the identity
k
X
j=1
holds with W (d) =
identity
Pd
m=1
k
X
j=1
wr (j)tk (j) = (f · wr ∗ g · W )(k)
(20)
wr (m)h(m), and if wr is a completely additive function, then the
wr (j)tk (j) = (f · wr ∗ g · H)(k) + (f ∗ g · W )(k)
(21)
P
holds with H(d) = dm=1 h(m). Thus, the formulas (16) and (18) give a generalization of
(20) and (21), respectively. Substituting wr = idr and h1 = · · · = hn = 1 in (17), wr = idr
and g1 = · · · = gn = 1 in (16), and wr = idr and f1 = · · · = f1 = 1 in (16), we derive the
following formulas (22), (23), and (24), respectively.
7
Corollary 3. Let the notation be as above. Then we have
1
K r+1
=
+
K
X
j=1
(1)
(n)
j r sk1 (j) · · · skn (j)
(22)
1
(f1 ∗ g1 )(k1 ) · · · (fn ∗ gn )(kn )
2K
⌊r/2⌋ X
1 X r + 1 B2m
r+1
K
X
j
K 2m
2m
m=0
X
r
j=1
d1 |k1 ,...,dn |kn
f1 (d1 )h1
d1 | gcd(k1 ,j)
X
=
d1 |k1 ,...,dn |kn
j
d1
···
X
fn (dn )hn
dn | gcd(kn ,j)
k1
d1
j
dn
· · · gn
kn
dn
,
(23)
f1 (d1 ) · · · fn (dn ) lcm(d1 , . . . , dn )r ×
K
lcm(d1 ,...,dn )
l h1
g1
k1
d1
X
×
f1 (d1 ) · · · fn (dn )
g1
lcm(d1 , . . . , dn )1−2m
r
l=1
lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
l · · · hn
l
d1
dn
and
K
X
j
X
r
j=1
d1 | gcd(k1 ,j)
=
X
g1
d1 |k1 ,...,dn |kn
k1
d1
K
lcm(d1 ,...,dn )
×
X
l=1
r
l h1
X
j
j
kn
h1
···
hn
gn
d1
dn
dn
dn | gcd(kn ,j)
kn
· · · gn
lcm(d1 , . . . , dn )r ×
dn
(24)
lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
l · · · hn
l .
d1
dn
The formula (22) also gives a generalization of the formula (2). As an application of
Corollary 3, we give some formulas for weighted averages of the products of the arithmetical
functions of Anderson–Apostol sums.
Example 4. Let the notation be as above. Then we have
1
K r+1
K
X
j=1
j r gcd(k1 , j) · · · gcd(kn , j)
(25)
⌊r/2⌋ 1 X r + 1 B2m
k1 k2 · · · kn
+
=
2K
r + 1 m=0 2m K 2m
X
d1 |k1 ,...,dn |kn
φ(d1 ) · · · φ(dn )
,
lcm(d1 , . . . , dn )1−2m
(26)
8
K
1 X 2r
j
K 2r j=1
X
d1 | gcd(k1 ,j)
X φ(dr )
φ(d1 )
···
d1
dr
d1 |(kr ,j)
X
φ(d1 ) · · · φ(dr )
d1 · · · dr
d1 |k1 ,...,dr |kr
r K X 2r + 1 B2m
+
2r + 1 m=0
2m K 2m
=
1
2
X
d1 |k1 ,...,dr |kr
1
K r+n+1
(27)
K
X
j r+n
j=1
X
d1 | gcd(k1 ,j)
φ(d1 )
φ(dr )
···
lcm(d1 , . . . , dr )2m−1 ,
d1
dr
X fn (dn )
f1 (d1 )
···
d1
dn
(28)
dn |(kn ,j)
X fn (d1 )
1 X f1 (d1 )
=
···
2K
d1
dn
dn |kn
d1 |k1
⌊ 2 ⌋
X r + n + 1 B2m
1
+
2m
r + n + 1 m=0
K 2m
r+n
X
d1 |k1 ,...,dn |kn
f1 (d1 ) · · · fn (dn )
lcm(d1 , . . . , dn )2m−1
,
d1 · · · dn
and
X
1
kn 1
j
···
(29)
g1
gn
r+n+1
K
d1
dn dn
j=1
d1 | gcd(k1 ,j)
dn | gcd(kn ,j)
1
1
1
g1 ∗
(k1 ) · · · gn ∗
(kn )
=
2K
id
id
⌊ r+n
⌋
2
X
X
kn lcm(d1 , . . . , dn )2m−1
1
k1
r + n + 1 B2m
·
·
·
g
g
.
+
n
1
r + n + 1 m=0
K 2m
d1
dn
d1 · · · dn
2m
1
K
X
r+n
X
k1
d1
d1 |k1 ,...,dn |kn
We substitute f1 = · · · = fn = φ and g1 = · · · = gn = 1 in (22) to obtain (25) using
(φ ∗ 1)(gcd(ki , j)) = gcd(ki , j) (i = 1, . . . , n). The formula (25) is an analogue of the identity
K
1 X
gcd(k1 , j) · · · gcd(kn , j) =
K j=1
X
d1 |k1 ,...,dn |kn
φ(d1 ) · · · φ(dn )
lcm(d1 , . . . , dn )
(30)
[20, Proposition 12], [21, Corollary 2]. The formula (30) was mentioned by Liskovets [12],
and it was considered by Deitmar, Koyama and Kurokawa [10] in a special case, investigating analytic properties of Igusa-type zeta-functions. Using the arguments of elementary
probability theory, the explicit formula for the values of (30) derived in [10] was re-proved
by Minami [13] for the general case m1 , . . . , mn . We substitute n = r, f1 = · · · = fn = φ
9
and h1 = · · · = hn = id in (23) and use (51) below to obtain (27). The formulas (23) and
(24) imply a generalization of two formulas derived in [11], namely
k
1 X r
j
k r j=1
and
X
d| gcd(k,j)
k
1 X r
j
k r j=1
X X
d
k 1
j
=
h(l)lr
f
f (d)h
r
d
d d l=1
X
d| gcd(k,j)
d|k
X
d
g(d) X
j
k
h
=
h(l)lr ,
g
d
d
dr l=1
d|k
which follow from (20). Substituting h1 = · · · = hn = id in (23) and (24), respectively, and
using (51) below we easily obtain the two formulas (28) and (29).
(1)
(n)
We shall evaluate some identities of weighted averages for the product tk1 · · · tkn of
the weight wr with the completely additive function. We set wr = log and substitute
h1 = · · · = hn = 1 in (18), g1 = · · · = gn = 1 in (18), and f1 = · · · = fn = 1 in (18) to
obtain the following formulas (31), (32), and (33), respectively.
Corollary 5. Let the notation be as above. Then we have
K
X
(1)
(n)
sk1 (j) · · · skn (j) log j
j=1
(31)
k1
kn log lcm(d1 , . . . , dn )
=K
f1 (d1 )g1
· · · fn (dn )gn
d1
dn
lcm(d1 , . . . , dn )
d1 |k1 ,...,dn |kn
X
k1
K
kn
f1 (d1 )g1
+
!
· · · fn (dn )gn
log
d1
dn
lcm(d1 , . . . , dn )
X
d1 |k1 ,...,dn |kn
K
X
j=1
=
log j
X
f1 (d1 )h1
d1 | gcd(k1 ,j)
X
K
lcm(d1 ,...,dn )
+
X
X
l=1
d1 |k1 ,...,dn |kn
j
d1
···
X
fn (dn )hn
dn | gcd(kn ,j)
j
dn
(32)
f1 (d1 ) · · · fn (dn ) log lcm(d1 , . . . , dn )×
d1 |k1 ,...,dn |kn
×
h1
lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
l · · · hn
l
d1
dn
K
lcm(d1 ,...,dn )
f1 (d1 ) · · · fn (dn )
X
l=1
h1
lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
l · · · h1
l log l
d1
dn
10
and
X
j
j
kn
h1
···
hn
log j
gn
g1
d1
dn
dn
j=1
dn | gcd(kn ,j)
d1 | gcd(k1 ,j)
X
kn
k1
· · · gn
log lcm(d1 , . . . , dn )×
=
g1
d1
dn
K
X
X
k1
d1
(33)
d1 |k1 ,...,dn |kn
K
lcm(d1 ,...,dn )
lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
×
l · · · hn
l
h1
d1
dn
l=1
X
k1
kn
g1
· · · gn
×
d1
dn
X
+
d1 |k1 ,...,dn |kn
K
lcm(d1 ,...,dn )
×
X
h1
l=1
lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
l · · · hn
l log l.
d1
dn
(31) implies a generalization of (3). (32) and (33) give a generalization of the formulas
k
X
X
j
= (f · log ∗H)(k) + (f ∗ I)(k),
log j
f (d)h
d
j=1
d| gcd(k,j)
and
k
X
j=1
log j
X
d| gcd(k,j)
j
k
h
= (log ∗g · H)(k) + (1 ∗ g · W )(k)
g
d
d
Pd
Pd
Pd
with I(d) =
l=1 h(l)l, H(d) =
m=1 h(m) and W (d) =
m=1 wr (m)h(m). These two
identities immediately follow from (21).
Next, we shall evaluate the function Ur (k1 , . . . , kn ), which is another representation of
(16) and (18). Using the method of Tóth [20], we have the following formulas.
Theorem 6. Let k1 , . . . , kn be any positive integers and let K = lcm(k1 , . . . , kn ) be the least
common multiple of n-tuple integers k1 , . . . , kn . If wr is completely multiplicative function,
then
(1)
(n)
Ur (k1 , . . . , kn ) = (wr · tk1 · · · tkn ∗ Wr )(K)
with
d
X
Wr (d) =
(34)
wr (l),
l=1
(l,d)=1
and if wr is a completely additive function, then
(1)
(n)
(1)
(n)
Ur (k1 , . . . , kn ) = (wr · tk1 · · · tkn ∗ φ)(K) + (tk1 · · · tkn ∗ Wr )(K).
11
(35)
Remark 7. We substitute wr = 1, f1 = · · · = fn = id, g1 = · · · = gn = µ and h1 = · · · =
hn = 1 in (34) to obtain (9), and wr = 1, f1 = · · · = fn = f, g1 = · · · = gn = g and
h1 = · · · = hn = 1 in (34) to obtain the formula (12). The formulas (34) and (35) are an
analogue of (20) and (21), respectively.
We substitute wr = id and h1 = · · · = hn = 1 in (34) and use (54) below to obtain the
formula (36), which is a generalization of (14). We also substitute wr = log and h1 = · · · =
hn = 1 in (35) and use (52) below to obtain the formula (37).
Corollary 8. Let the notation be as above. Then we have
K
1 X r (1)
(n)
j sk1 (j) · · · skn (j)
r
K j=1
(36)
1 X (1)
1 (1)
(n)
= sk1 (K) · · · skn (K) +
s k1
2
r+1
d|K
K
d
(n)
· · · s kn
K
d
⌊r/2⌋
X
m=0
r+1
B2m φ1−2m (d)
2m
and
K
X
j=1
=
(1)
(n)
sk1 (j) · · · skn (j) log j
(1)
(sk1
−
(n)
· · · s kn
X
(1)
s k1
d|K
(37)
· log ∗φ)(K) +
K
d
(n)
· · · s kn
X
d|K
K
d
(1)
s k1
K
d
φ(d)
(n)
· · · s kn
K
d
X
e|d
µ(e) Log
d
e
X log p
p−1
p|d
where Log d is given by log(d!).
As an application of Corollary 8, we provide two formulas for the weighted averages of
the product gcd(k1 , j) gcd(k2 , j) · · · gcd(kn , j) of the gcd’s.
Example 9. Let the notation be as above. Then we have
K
1 X r
j gcd(k1 , j) · · · gcd(kn , j)
K r j=1
(38)
1
gcd(k1 , K) · · · gcd(kn , K)
2
⌊r/2⌋ 1 X
K X r+1
K
+
B2m φ1−2m (d),
· · · gcd kn ,
gcd k1 ,
r+1
d
d m=0 2m
=
d|K
12
and
K
X
j=1
gcd(k1 , j) · · · gcd(kn , j) log j
X
=
d|K
gcd (k1 , d) · · · gcd (kn , d) (log d)φ
(39)
K
d
K X
d
K
· · · gcd kn ,
+
µ(e)Log
gcd k1 ,
d
d
e
e|d
d|K
X log p
X
K
K
.
· · · gcd kn ,
φ(d)
−
gcd k1 ,
d
d
p−1
X
p|d
d|K
We substitute wr = idr , f1 = · · · = fn = φ and g1 = · · · = gn = 1 in (36) to obtain (38),
which is a generalization of Tóth’s result [20, Proposition 14]:
K
1 X
1 X
K
gcd(k1 , j) · · · gcd(kn , j) =
.
gcd(d, k1 ) · · · gcd(d, kn )φ
K j=1
K
d
d|K
We also substitute wr = log, f1 = · · · = fn = φ and g1 = · · · = gn = 1 in (37) to obtain (39).
(1)
(n)
Lastly, we shall consider some weighted averages of the product sk1 (j) · · · skn (j) with
weights concerning the gamma function, binomial coefficients, and Bernoulli polynomials.
To state Theorem 10, we use the well-known multiplication formula of Gauss–Legendre [8,
Proposition 9.6.33] for the gamma function
n−1
n
Y
j
(2π) 2
Γ
= √
n
n
j=1
(40)
for any positive integer n, and the binomial formula [22, (27)]
[n/r] X
k=0
n
kr
n
r r
j
2n X n jπ
1X
njπ
1 + exp 2πi
=
cos
=
cos
r j=1
r
r j=1
r
r
(41)
for any positive integers n and r. Furthermore, we use the well-known formula for the
Bernoulli polynomial [8, Proposition 9.1.3]
Bm
j
= m−1
Bm
k
k
j=0
k−1
X
for any positive integer k.
13
(42)
Theorem 10. Let k1 , . . . , kn be any positive integers and let K = lcm(k1 , . . . , kn ) be the least
common multiple of n-tuple integers k1 , . . . , kn . Then we have
K
X
j=1
(1)
(n)
sk1 (j) · · · skn (j) log Γ
j
K
(43)
f1 (d1 ) · · · fn (dn )
kn
k1
= K log 2π
· · · gn
g1
lcm(d1 , . . . , dn )
d1
dn
d1 |k1 ,...,dn |kn
√
− (f1 ∗ g1 )(k1 ) · · · (fn ∗ gn )(kn ) log 2πK
X
p
k1
kn
f1 (d1 )g1
+
· · · fn (dn )gn
log lcm(d1 , . . . , dn ),
d1
dn
X
√
d1 |k1 ,...,dn |kn
K X
K (1)
(n)
sk1 (j) · · · skn (j)
j
j=0
=2
X
K
d1 |k1 ,...,dn |kn
f1 (d1 ) · · · fn (dn )
g1
lcm(d1 , . . . , dn )
lcm(d1 ,...,dn )
X
×
(44)
Kl
(−1) lcm(d1 ,...,dn ) cosK
l=1
k1
d1
· · · gn
kn
dn
×
lπ
,
lcm(d1 , . . . , dn )
and
K−1
X
Bm
j=0
Bm
= m−1
K
j
K
(1)
(n)
sk1 (j) · · · skn (j)
X
d1 |k1 ,...,dn |kn
(45)
f1 (d1 ) · · · fn (dn )
g1
lcm(d1 , . . . , dn )1−m
k1
d1
· · · gn
kn
dn
.
As an application of Theorem 10, we give three formulas for weighted averages of the
product gcd(k1 , j) gcd(k2 , j) · · · gcd(kn , j) of the gcd’s.
Example 11. Let the notation be as above. Then we have
K
X
j=1
gcd(k1 , j) · · · gcd(kn , j) log Γ
= K log
√
2π
X
d1 |k1 ,...,dn |kn
− k1 k2 · · · kn log
√
j
K
(46)
φ(d1 ) · · · φ(dn )
1
+
lcm(d1 , . . . , dn ) 2
X
d1 |k1 ,...,dn |kn
2πK,
14
φ(d1 ) · · · φ(dn ) log lcm(d1 , . . . , dn )
K 1 X K
gcd(k1 , j) · · · gcd(kn , j)
2K j=0 j
X
=
d1 |k1 ,...,dn |kn
φ(d1 ) · · · φ(dn )
lcm(d1 , . . . , dn )
(47)
lcm(d1 ,...,dn )
X
Kl
(−1) lcm(d1 ,...,dn ) cosK
l=1
lπ
lcm(d1 , . . . , dn )
and
K−1
X
Bm
j=0
j
K
Bm
K m−1
gcd(k1 , j) · · · gcd(kn , j) =
X
d1 |k1 ,...,dn |kn
φ(d1 ) · · · φ(dn )
.
lcm(d1 , . . . , dn )1−m
(48)
We substitute f1 = · · · = fn = φ and g1 = · · · = gn = 1 in (43) and f1 = · · · = fn = φ
and g1 = · · · = gn = 1 in (44) to obtain (46) and (47), respectively. We also substitute
f1 = · · · = fn = φ and g1 = · · · = gn = 1 in (45) to obtain (48), which is an analogue of (30).
3
Proofs of Theorems 1, 6, 10 and Corollaries 3, 5, 8
Proof of Theorem 1. Since
(i)
tki (j)
X
=
di | gcd(ki ,j)
j
ki
hi
fi (di )gi
di
di
(i = 1, 2, . . . , n),
we have
Ur (k1 , . . . , kn )
=
X
(49)
f1 (d1 )g1
d1 |k1 ,...,dn |kn
=
X
f1 (d1 )g1
d1 |k1 ,...,dn |kn
k1
d1
k1
d1
· · · fn (dn )gn
kn
dn
· · · fn (dn )gn
kn
dn
K
lcm(d1 ,...,dn )
×
X
wr (l lcm(d1 , . . . , dn )) h1
l=1
K
X
j=1
d1 |j,...,dn |j
wr (j)h1
j
d1
· · · hn
j
dn
×
lcm(d1 , . . . , dn )
lcm(dn , . . . , dn )
l · · · hn
l .
d1
dn
We use the completely multiplicative function wr in (49) to obtain
Ur (k1 , . . . , kn )
X
k1
kn
=
f1 (d1 )g1
· · · fn (dn )gn
wr (lcm(d1 , . . . , dn )) ×
d1
dn
d1 |k1 ,...,dn |kn
K
lcm(d1 ,...,dn )
×
X
l=1
wr (l) h1
lcm(d1 , . . . , dn )
lcm(dn , . . . , dn )
l · · · hn
l ,
d1
dn
15
(50)
and if h1 , . . . , hn are completely multiplicative functions, (50) gives
Ur (k1 , . . . , kn )
X
lcm(d1 , . . . , dn )
kn
k1
h1
· · · fn (dn )gn
×
=
f1 (d1 )g1
d1
d1
dn
d1 |k1 ,...,dn |kn
× hn
lcm(d1 , . . . , dn )
dn
K
lcm(d1 ,...,dn )
wr (lcm(d1 , . . . , dn ))
X
l=1
wr (l)h1 (l) · · · hn (l) .
Similarly, as in the proof of the above, using the completely additive function wr in (49), we
have the identity (18), and if h1 , . . . , hn are completely multiplicative functions, we establish
the identity (19). This completes the proof of Theorem 1.
Proof of Corollary 3. We substitute wr = idr and h1 = · · · = hn = 1 in (17) and use
⌊2⌋ r+1
1 X
Nr
r
B2m N r+1−2m
+
m =
2m
2
r
+
1
m=0
m=1
r
N
X
(51)
for any positive integer N > 1 [8, Proposition 9.2.12], [9, Section 3.9] to obtain
K
X
j=1
(1)
(n)
j r sk1 (j) · · · skn (j)
=
k1
d1
k1
d1
K
lcm(d1 , . . . , dn )
r
X
f1 (d1 )g1
X
f1 (d1 )g1
d1 |k1 ,...,dn |kn
=
d1 |k1 ,...,dn |kn
×
=
+

1 2
· · · fn (dn )gn
· · · fn (dn )gn
m=0
2m
K 2m
kn
dn
K
lcm(d1 ,...,dn )
+
d1 |k1 ,...,dn |kn
2m
lr
l=1
lcm(d1 , . . . , dn )r ×
⌊2⌋ X
r+1
1
r + 1 m=0
X
lcm(d1 , . . . , dn )r
r
(f1 ∗ g1 )(k1 ) · · · (fn ∗ gn )(kn ) r
K
2
⌊r/2⌋ X
K r+1 X r + 1 B2m
r+1
kn
dn
B2m
K
lcm(d1 , . . . , dn )
f1 (d1 ) · · · fn (dn )
g1
lcm(d1 , . . . , dn )1−2m
k1
d1

r+1−2m 
· · · gn

kn
dn
,
which proves (22). We substitute g1 = · · · = gn = 1 and wr = idr in (16) and f1 = · · · =
fn = 1 and wr = idr in (16) to obtain the formulas for (23) and (24), respectively.
16
Proof of Corollary 5. We substitute wr = log and h1 = · · · = hn = 1 in (18) to obtain
K
X
j=1
(1)
(n)
sk1 (j) · · · skn (j) log j
=K
X
f1 (d1 )g1
d1 |k1 ,...,dn |kn
k1
d1
· · · fn (dn )gn
kn
dn
log lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
K
lcm(dX
1 ,...,dn )
kn
k1
· · · fn (dn )gn
log l
+
f1 (d1 )g1
d1
dn
l=1
d1 |k1 ,...,dn |kn
X
kn log lcm(d1 , . . . , dn )
k1
· · · fn (dn )gn
=K
f1 (d1 )g1
d1
dn
lcm(d1 , . . . , dn )
d1 |k1 ,...,dn |kn
X
k1
K
kn
f1 (d1 )g1
+
!.
· · · fn (dn )gn
log
d1
dn
lcm(d1 , . . . , dn )
X
d1 |k1 ,...,dn |kn
Furthermore, we substitute g1 = · · · = gn = 1 and wr = log in (18) and f1 = · · · = fn = 1
and wr = log in (18) to get the formulas (32) and (33), respectively.
Proof of Theorem 6. Let K = lcm(k1 , . . . , kn ) be the least common multiple of n-tuple positive integers k1 , . . . , kn . Note that tk (j) = tk (gcd(k, j)) for any positive integers k and j.
Since gcd(gcd(j, K), ki ) = gcd(j, gcd(K, ki )) = gcd(j, ki ) for any i ∈ {1, 2, . . . , n}, we observe
(i)
(i)
that tki (j) is equal to tki (gcd(j, K)). If wr is a completely multiplicative function, we have
Ur (k1 , . . . , kn ) =
K
X
wr (j)tk1 (gcd(j, K)) · · · tkn (gcd(j, K))
X
(1)
(n)
wr (d)tk1 (d) · · · tkn (d)
j=1
=
d|K
(n)
(1)
K
d
X
wr (l)
l=1
gcd l, K
d
)=1
X
d
X K (1) K K
(n)
t k1
· · · t kn
wr (l),
=
wr
d
d
d
l=1
(
d|K
gcd(l,d)=1
which completes the proof of the formula (34). Similarly, as in the proof of (34), we have
(35).
To prove Corollary 8, we need the following formula.
Lemma 12. For any positive integers N > 1 and r, we have
N
X log p
X
X N log(d!) − φ(N )
log l =
µ
d
p−1
l=1
gcd(l,N )=1
p|N
d|N
17
(52)
and
⌊r/2⌋ X
m=0
r+1
r+1
B2m =
.
2m
2
(53)
Proof. Using the well-known identity
X µ(d)
d
d|N
log d = −
φ(N ) X log p
N
p−1
p|N
[8, Exercise 10.8.45 (c)], we have
N
X
log l =
l=1
gcd(l,N )=1
X
µ(d)
d|N
N/d
X
log (dj)
j=1
X µ(d)
N
!+N
log d
=
µ(d) log
d
d
d|N
d|N
X
X log p
N
=
µ(d) log
! − φ(N )
,
d
p−1
X
d|N
p|N
which completes the proof of (52). From Theorem 12.15 in [7], we have
r X
r+1
m=0
It follows that
m
Bm = 0.
r+1
r+1
r+1
r+1
B0 +
B1 +
B2 + . . . +
B2[ r2 ] = 0.
0
1
2
2[ 2r ]
Since B0 = 1, B1 = − 12 and B2m+1 = 0 for any positive integer m, we obtain the formula
(53).
Proof of Corollary 8. We substitute wr = idr and h1 = · · · = hn = 1 in (34) and use the
formula [16, Corollary 4]
N
X
m=1
gcd(m,N )=1
⌊2⌋ r+1
N r+1 X
r
B2m φ1−2m (N )
m =
r + 1 m=0 2m
r
18
(54)
for any positive integers N > 1, and r to obtain
K
X
j=1
=
(1)
X
d|K
=
(n)
j r sk1 (j) · · · skn (j)
(1)
(n)
dr sk1 (d) · · · skn (d)
K/d
X
l=1
gcd l, K
d
(
(1)
(n)
K r sk1 (K) · · · skn (K)
lr
)=1
⌊r/2⌋ X r + 1
K
K r X (1)
(n)
B2m φ1−2m
sk1 (d) · · · skn (d)
+
2m
r + 1 d|K
d
m=0
d<K
=
(1)
(n)
K r sk1 (K) · · · skn (K)
K r X (1)
s
+
r + 1 d|K k1
K
d
(n)
· · · s kn
K
d
d>1
⌊r/2⌋
X
m=0
r+1
B2m φ1−2m (d)
2m
⌊r/2⌋ X r + 1
K r (1)
(n)
−
=
B2m
s (K) · · · skn (K)
2m
r + 1 k1
m=0
⌊r/2⌋
K r X (1) K
K X r+1
(n)
+
· · · s kn
B2m φ1−2m (d) .
s k1
r+1
d
d m=0 2m
(1)
(n)
K r sk1 (K) · · · skn (K)
d|K
Using the well-known identity (53), we obtain the formula (36). We also substitute wr = log
and h1 = · · · = hn = 1 in (35), and use (52) to obtain
K
X
(1)
(n)
sk1 (j) · · · skn (j) log j
j=1
=
(1)
(sk1
−
(n)
· · · s kn
X
d|K
(1)
s k1
· log ∗φ)(K) +
K
d
(n)
· · · s kn
X
(1)
s k1
d|K
K
d
K
d
(n)
· · · s kn
K
d
X
e|d
d
!
µ(e) log
e
X log p
φ(d)
.
p−1
p|d
Proof of Theorem 10. Using (40) and substituting wr (d) = log Γ
19
d
K
and h1 = · · · = h1 = 1
in (49), we have
K
X
j=1
(1)
(n)
sk1 (j) · · · skn (j) log Γ
X
=
f1 (d1 )g1
d1 |k1 ,...,dn |kn
=
X
f1 (d1 )g1
d1 |k1 ,...,dn |kn
j
K
k1
d1
k1
d1
· · · fn (dn )gn
kn
dn
· · · fn (dn )gn
kn
dn
K
lcm(dX
1 ,...,dn )
log Γ
l=1
lcm(d1 , . . . , dn )
l
K
×
p
√
√
K
log 2π − log 2πK + log lcm(d1 , . . . , dn )
×
lcm(d1 , . . . , dn )
X
√
f1 (d1 ) · · · fn (dn )
kn
k1
· · · gn
g1
= K log 2π
lcm(d1 , . . . , dn )
d1
dn
d1 |k1 ,...,dn |kn
X
X
√
k1
kn
f1 (d1 )g1
···
fn (dn )gn
− log 2πK
d1
dn
d1 |k1
dn |kn
X
p
k1
kn
+
f1 (d1 )g1
· · · fn (dn )gn
log lcm(d1 , . . . , dn ),
d1
dn
d1 |k1 ,...,dn |kn
which completes the proof of (43). We set wr (j) =
we have
K X
K (1)
(n)
sk1 (j) · · · skn (j)
j
j=0
X
=
f1 (d1 )g1
d1 |k1 ,...,dn |kn
X
= 2K
d1 |k1 ,...,dn |kn
× g1
= 2K
k1
d1
X
× g1
k1
d1
k1
d1
· · · fn (dn )gn
kn
dn
and h1 = · · · = h1 = 1. Using (41),
K
lcm(dX
1 ,··· ,dn ) m=0
K
lcm(d1 , . . . , dn )m
f1 (d1 ) · · · fn (dn )
×
lcm(d1 , . . . , dn )
· · · gn
d1 |k1 ,...,dn |kn
K
j
kn
dn
lcm(dX
1 ,...,dn )
cosK
j=1
jπ
Kjπ
cos
lcm(d1 , . . . , dn )
lcm(d1 , . . . , dn )
f1 (d1 ) · · · fn (dn )
×
lcm(d1 , . . . , dn )
· · · gn
kn
dn
lcm(dX
1 ,...,dn )
Kj
(−1) lcm(d1 ,...,dn ) cosK
j=1
20
jπ
,
lcm(d1 , . . . , dn )
hence, we obtain (44). Setting wr (j) = Bm
have
K−1
X
j
(1)
(n)
sk1 (j) · · · skn (j)
Bm
K
j=0
j
K
and h1 = · · · = h1 = 1 and using (42), we
−1
lcm(d1K
,...,dn )
X
kn
lcm(d1 , . . . , dn )
=
f1 (d1 )g1
· · · fn (dn )gn
Bm
l
dn
K
l=0
d1 |k1 ,...,dn |kn
X
Bm
kn
f1 (d1 ) · · · fn (dn )
k1
= m−1
· · · gn
.
g1
1−m
K
lcm(d1 , . . . , dn )
d1
dn
X
k1
d1
d1 |k1 ,...,dn |kn
Hence, we prove (45).
4
Acknowledgments
The authors deeply thank the referee for carefully reading this paper and useful comments.
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2010 Mathematics Subject Classification: Primary 11A25; Secondary 11B68.
Keywords: arithmetical function, Ramanujan’s sum, greatest common divisor.
(Concerned with sequences A018804, A051193, A056188, and A159068.)
Received August 18 2015; revised versions received November 21 2015; March 17 2016.
Published in Journal of Integer Sequences, March 18 2016.
Return to Journal of Integer Sequences home page.
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