Arithmetical properties of Multiple Ramanujan sums

0.23 MB·English

by  Yoshinori Yamasaki

Arithmetical properties of Multiple
7 Ramanujan sums
0
0
2
Yoshinori YAMASAKI
n
a
J
February 2, 2008

9
1

] Abstract
T
In the present paper, we introduce a multiple Ramanujan sum for arithmetic
N
functions, which gives a multivariable extension of the generalized Ramanujan sum
.
h studiedbyD.R.AndersonandT.M.Apostol. Wethenfindfundamentalarithmetic
t
a propertiesofthemultipleRamanujansumandstudyseveraltypesofDirichletseries
m
involving the multiple Ramanujan sum. As an application, we evaluate higher-
[ dimensional determinants of higher-dimensional matrices, the entries of which are

1 given by values of the multiple Ramanujan sum.
v
2000 Mathematics Subject Classification: Primary 11A25; Secondary 11C20.
8
2 Key words and phrases: Ramanujan sum, divisor function, Dirichlet convolu-
5 tion, Dirichlet series, Smith determinant, hyperdeterminant.
1
0
7
1 Introduction
0
/
h
t In 1918, S. Ramanujan [17] studied the sum
a
m
k
c(k,n) := e(k,nl) = µ d, (1.1)
:
v d
Xi glc(dmX(lo,dk)=k)1 d|gXcd(k,n) (cid:0) (cid:1)
r where, in the first sum, l runs over a reduced residue system modulo k with gcd(l,k) = 1,
a
e(r,n) := exp(2π√ 1n/r) and µ is the M¨obius function. The sum c(k,n) is called
−
the Ramanujan sum (or the Ramanujan trigonometric sum) and is widely investigated in
connection with, for example, even arithmetic functions [8, 9] and cyclotomic polynomials
[14, 15]. See [13] for the arithmetic theory of the Ramanujan sum. Among several
generalizations and variations of c(k,n), D. R. Anderson and T. M. Apostol [1] (see also
[3]) considered the sum
k
S (k,n) := f g(d), (1.2)
f,g
d
d|gcd(k,n)
X (cid:0) (cid:1)
where f and g are arithmetic functions. Clearly, S (k,n) extends the right-most expres-
f,g
sion in formula (1.1) and hence gives a generalization of the Ramanujan sum.
MotivatedbythestudyoftheabovegeneralizedRamanujansum, inthepresent paper,
we examine the following type of multiple sum for arithmetic functions f ,...,f ;
1 m+1
n d d
1 1 m−1
S (n ,...,n ) := f f f f d ,
f1,...,fm+1 1 m+1 1 d 2 d ··· m d m+1 m
1 2 m
dj|gcd(Xn1,...,nj+1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(j=1,...,m)
2 Yoshinori Yamasaki
where gcd(n ,...,n ) is the greatest common divisor of n ,...,n . We call this a
1 j+1 1 j+1
multiple Ramanujan sum for f ,...,f . Notice that the above expression gives the
1 m+1
generalized Ramanujan sum (1.2) when m = 1 and, moreover, the Dirichlet convolution
f f of f ,...,f in the “diagonal case”; n = = n .
1 m+1 1 m+1 1 m+1
∗···∗ ···
The present paper is organized as follows. In Section 2, we introduce a multiple Ra-
manujan sum S(γ1,...,γm) with positive integer parameters γ ,...,γ so that S =
f1,...,fm+1 1 m f1,...,fm+1
(1,...,1)
S and study its fundamental properties as a multivariable arithmetic function,
f1,...,fm+1
such as the degeneracies and the multiplicativity (see [21] for the theory of multivariable
arithmetic functions). Then, since S(γ1,...,γm) belongs to the class of even arithmetic func-
f1,...,fm+1
tions (mod n ) as a function of n ,...,n in the sense of Cohen [9], we calculate its
1 2 m+1
finite Fourier expansion, which any even arithmetic function possesses. This expression is
importantwithrespect toSection4. InSection3, westudyseveral typesofDirichletseries
having coefficients that are given by S(γ1,...,γm). We treat not only single-variable Dirichlet
f1,...,fm+1
series, but also multivariable Dirichlet series. For instance, an analogue of the formula of
J. M. Borwein and K. K. Choi [4], which contains the classical Ramanujan formula con-
cerning the divisor function σ , is obtained. Section 4 is devoted to a higher-dimensional
a
generalization of the so-called Smith determinant [19]. We evaluate higher-dimensional
determinants (hyperdeterminants) of higher-dimensional matrices (hypermatrices), the
entries of which are given by values of S(γ1,...,γm). In fact, we derive a hyperdeterminant
f1,...,fm+1
formula for even multivariable arithmetic functions. This includes the results of P. J.
McCarthy [12] and K. Bourque and S. Ligh [5] for the 2-dimensional case, that is, the
usual determinant case, and partially of P. Haukkanen [10], for the higher-dimensional
case.
We use the following notations in the present paper. The set of natural numbers,
the ring of rational integers, the field of real numbers and the field of complex numbers
are denoted respectively as N, Z, R, and C. For n ,...,n N, gcd(n ,...,n ) (resp.
1 k 1 k
∈
lcm(n ,...,n ))represents forthegreatest commondivisor (resp. least commonmultiple)
1 k
ofn ,...,n . Forx R, x isthegreatestintegernotexceedingx. WedenotetheM¨obius
1 k
∈ ⌊ ⌋
function as µ(n), the Euler totient function as ϕ(n), the power function as δx(n) := nx
for x C, and the identity element in the ring of arithmetic functions with respect to the
∈
Dirichlet convolution as ε(n) := 1 = 1 if n = 1, and 0 otherwise. Note that δ0 µ = ε.
∗ ⌊n⌋ ∗
Throughout the present paper, we consider a product (resp. a sum) over an empty set to
always be equal to 1 (resp. 0).
2 A multiple Ramanujan sum
2.1 Preliminary: γ-convolutions
Let be the set of all complex-valued arithmetic functions f : N C. We always
A →
understand that f(x) = 0 if x / N for f . The set of all multiplicative and completely
∈ ∈ A
multiplicative arithmetic functions are respectively denoted by and c. Namely,
M M
f (resp. f c) means that f(mn) = f(m)f(n) for gcd(m,n) = 1 (resp. for all
∈ M ∈ M
m,n N). As usual, the product fg of f,g is defined by (fg)(n) := f(n)g(n).
∈ ∈ A ∈ A
Arithmetical properties of Multiple Ramanujan sums 3
Let γ N. We define the γ-convolution of f,g by
∈ ∈ A
n
(g f)(n) := f g(dγ).
γ
∗ dγ
dγ|n
X (cid:0) (cid:1)
In particular, = stands for the usual Dirichlet convolution. Define the function
1
∗ ∗
a by a (n) = 1 if there exists d N such that n = dγ, and 0 otherwise. Then, it is
γ γ
∈ A ∈
clear that g f = g[γ] f, where g[γ] := a g. Note that the product does not satisfy the
γ γ γ
∗ ∗ ∗
commutativity or associativity properties unless γ = 1. We therefore inductively define
the γ = (γ ,...,γ )-convolution for γ ,...,γ N of f ,...,f as
1 m 1 m 1 m+1
∈ ∈ A
(f f )(n) : = (f f ) f (n)
m+1 ∗γm ···∗γ1 1 m+1 ∗γm ···∗γ2 2 ∗γ1 1
n dγ1 dγm−1
= (cid:0) f f 1 (cid:1)f m−1 f dγm .
1 dγ1 2 dγ2 ··· m dγm m+1 m
dγmmX|···|dγ11|n (cid:0) 1 (cid:1) (cid:0) 2 (cid:1) (cid:0) m (cid:1) (cid:0) (cid:1)
Let L(s;f) := ∞ f(n)n−s be the Dirichlet series attached to f and let σ(f)
n=1 ∈ A ∈
R be the abscissa of absolute convergence of L(s;f). Define fhγi for γ N by
∪{∞} P ∈ A ∈
fhγi(n) := f(nγ). Then, it is clear that L(γs;fhγi) = L(s;f[γ]) with σ(f[γ]) σ(f). For
≤
f ,...,f , we here calculate the Dirichlet series attached to f f .
1 m+1 ∈ A m+1 ∗γm ···∗γ1 1
Proposition 2.1. Suppose γ γ γ γ with γ = 1. Then, we have for Re(s) >
0 1 2 m 0
| | |···|
max σ(f ),...,σ(f )
1 m+1
{ }
m+1
L s;f f = L s;f[γj−1] . (2.1)
m+1 ∗γm ···∗γ1 1 j
j=1
(cid:0) (cid:1) Y (cid:0) (cid:1)
Proof. Assume dγm dγ1 n. Then, since γ γ γ , we can write dγj = dγj+1kγj for
m |···| 1 | 1| 2|···| m j j+1 j+1
1 j m 1 and n = k dγ1 = = k kγ1 kγm−1dγm with k ,k ,...,k N. We then
≤ ≤ − 1 1 ··· 1 2 ··· m m 1 2 m ∈
write d = k , and so
m m+1
∞
L s;f f = f (k )f (kγ1) f (kγm ) k kγ1 kγm −s
m+1 ∗γm ···∗γ1 1 1 1 2 2 ··· m+1 m+1 1 2 ··· m+1
(cid:0) (cid:1) k1,...X,km+1=1 (cid:0) (cid:1)
= L s;f L γ s;fhγ2i L γ s;fhγmi .
1 1 2 ··· m m+1
This ends the proof. (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Remark 2.2. We cannot expect that L(s;f f ) is expressed as a product
m+1 ∗γm ···∗γ1 1
of a Dirichlet series such as (2.1) for general γ = (γ ,...,γ ) Nm.
1 m
∈
In the next subsection, we will introduce a multiple Ramanujan sum forf ,...,f
1 m+1
∈
, which gives the γ-convolution f f in the diagonal case (for more detail,
A m+1∗γm ···∗γ1 1
see Proposition 2.3 (iv)).
γ
2.2 Definition of S and its basic properties
f
Let γ = (γ ,...,γ ) Nm and f = (f ,...,f ) m+1. We define a multiple
1 m 1 m+1
∈ ∈ A
Ramanujan sum Sγ = S(γ1,...,γm) of f with the parameter γ by
f f1,...,fm+1
n dγ1 dγm−1
Sγ(n ,...,n ) := f 1 f 1 f m−1 f dγm ,
f 1 m+1 1 dγ1 2 dγ2 ··· m dγm m+1 m
γj 1 2 m
dj |gcdX(n1,...,nj+1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(j=1,...,m)
4 Yoshinori Yamasaki
wherethesumistakenoverallm-tuples(d ,...,d ) Nm satisfyingdγj gcd(n ,...,n )
1 m ∈ j | 1 j+1
for each 1 j m. Note that the summand vanishes unless dγm dγ1 n . We write
≤ ≤ m |···| 1 | 1
(1,...,1)
S := S and understand that S (n) = f(n) when m = 0. These are the fundamental
f f f
properties of Sγ, which are obtained from the elementary properties of the gcd-function.
f
Proposition 2.3. (i) For any 1 j m+1, we have
≤ ≤
Sγ(n ,...,n ) = S(γ1,...,γj−1) (n ,...,n ). (2.2)
f 1 m+1 f1,...,fj−1,Sf(γjj,.,....,.f,mγm+)1(·,nj+1,...,nm+1) 1 j
(ii) For any 1 j m+1, we have
≤ ≤
Sγ(n ,...,n ,1,n ,...,n ) = f (1) f (1) S(γ1,...,γj−2)(n ,...,n ).
f 1 j−1 j+1 m+1 j ··· m+1 · f1,...,fj−1 1 j−1
In particular, we have Sγ(1,n ,...,n ) = f (1) f (1).
f 2 m+1 1 ··· m+1
(iii) For any 1 j m, we have
≤ ≤
S(γ1,...,γm) (n ,...,n )
f1,...,fj−1,ε,fj+1,...,fm+1 1 m+1
= S(γ1,...,γj−1,γj+1,...,γm)(n ,...,n ,gcd(n ,n ),n ,...,n ) (2.3)
f1,...,fj−1,fj+1,...,fm+1 1 j−1 j j+1 j+2 m+1
and S(γ1,...,γm)(n ,...,n ) = S(γ1,...,γm−1)(n ,...,n ).
f1,...,fm,ε 1 m+1 f1,...,fm 1 m
(iv) Let n n for all 2 j m+1. Then, we have
1 j
| ≤ ≤
Sγ(n ,...,n ) = (f f )(n ). (2.4)
f 1 m+1 m+1 ∗γm ···∗γ1 1 1
Example 2.4. Let a = (a ,...,a ) Cm. Define a multiple divisor function σγ with
1 m ∈ a
the parameter γ by
σγ(n ,...,n ) := dγ1a1 dγmam. (2.5)
a 1 m+1 1 ··· m
dγmm|X···|dγ11|n1
dγjj|gcd(n1,...,nj+1) (j=1,...,m)
This is a generalization of the usual divisor function σ (n) := da; σ (n) = σ(1)(n,n).
a d|n a a
Then, it holds that σγ = Sγ with f = δa0+a1+···+aj−1 for 1 j m + 1 where
a f j P ≤ ≤
a = 0. Similarly, for b = (b ,...,b ) Cm+1, one can see that σγ(n ,...,n ) =
0 1 m+1 e 1 m+1
∈ b
n−b1Sγ (n ,...,n ) where b := (b b ,b b ,...,b b ) Cm. Note
1 δb1,...,δbm+1 1 m+1 2 − 1 3 − 2 m+1 − m ∈
that the sum Zγ(a ,...,a ) := σγ(n,...,n) is studied in [11] and called the multiple
n 1 m a
finite Riemann zeta function. From foermula (2.1), if γ γ γ γ with γ = 1, we have
0 1 2 m 0
| | |···|
L(s;Zγ(a ,...,a )) = m+1ζ(γ (s a a )) where ζ(s) = L(s;δ0) is the
· 1 m j=1 j−1 − 0 − ··· − j−1
Riemann zeta function.
Q
Example 2.5. Let f . Then, the composition function f gcd of f and the gcd-
∈ A ◦
function can be expressed in terms of the multiple Ramanujan sum. Actually, from
the degeneracy formula (2.3), we have (f gcd)(n ,...,n ) = S (n ,...,n ) with
1 m+1 f 1 m+1
◦
f = δ0, f = = f = ε and f = f µ.
1 2 m m+1
··· ∗
Arithmetical properties of Multiple Ramanujan sums 5
We next show the multiplicative property of the multiple Ramanujan sum Sγ. Recall
f
that an arithmetic function F(n ,...,n ) of k-variables is called multiplicative if
1 k
F(m n ,...,m n ) = F(m ,...,m ) F(n ,...,n )
1 1 k k 1 k 1 k
·
for relatively prime k-tuples (m ,...,m ) Nk and (n ,...,n ) Nk. Here, we say that
1 k 1 k
∈ ∈
(m ,...,m ) and (n ,...,n ) are relatively prime if gcd(m ,n ) = 1 for all i,j k, and
1 k 1 k i j
≤ ≤
equivalently, gcd(n n ,m m ) = 1 (see [21]). In this case, we have the following
1 k 1 k
··· ···
Euler product expression
F(n ,...,n ) = F(pαp,1,...,pαp,k),
1 k
p
Y
where α = ord n for 1 j k. Note that this is a finite product since F(1,...,1) = 1.
p,j p j
≤ ≤
Proposition 2.6. The function Sγ is multiplicative if f ,...,f .
f 1 m+1 ∈ M
Proof. Assume (n ,...,n ) and (k ,...,k ) are relatively prime. Then, for all 1
1 m+1 1 m+1
≤
j m+1, gcd(n k ,...,n k ) = gcd(n ,...,n ) gcd(k ,...,k ) and gcd(n ,...,n ) and
1 1 j j 1 j 1 j 1 j
≤ ·
gcd(k ,...,k ) are relatively prime. Hence, the first assertion follows in the same manner
1 j
as the that in the proof of Theorem 1 in [1].
2.3 Finite Fourier expansions
An arithmetic function F(r;n ,...,n ) of k-variables n ,...,n is called periodic (mod r)
1 k 1 k
if F(r;n ,...,n ) = F(r;n′,...,n′) whenever n n′ (mod r) for all 1 j k (see [13]
1 k 1 k j ≡ j ≤ ≤
for the case of k = 1). It is well-known that F is periodic (mod r) if and only if it has an
expression of the form of
r
F(r;n ,...,n ) = a (l ,...,l )e(r,n l ) e(r,n l ) (2.6)
1 k r 1 k 1 1 k k
···
l1,.X..,lk=1
and the coefficients a (l ,...,l ) are uniquely determined by
r 1 k
r
1
a (l ,...,l ) = F(r;n ,...,n )e(r, n l ) e(r, n l ). (2.7)
r 1 k rk 1 k − 1 1 ··· − k k
n1,.X..,nk=1
Moreover, F is called even (mod r) if F(r;n ,...,n ) = F(r;gcd(n ,r),...,gcd(n ,r)).
1 k 1 k
Note that F is periodic (mod r) if it is even (mod r). Then, as shown by E. Cohen [9], F
is even (mod r) if and only if it has an expression of the form of
F(r;n ,...,n ) = α (d ,...,d )c(d ,n ) c(d ,n ) (2.8)
1 k r 1 k 1 1 k k
···
d1,X...,dk|r
with
1 r r r r
α (d ,...,d ) = F(r;δ ,...,δ )c , c , . (2.9)
r 1 k rk 1 k δ d ··· δ d
1 1 k k
δ1,X...,δk|r (cid:0) (cid:1) (cid:0) (cid:1)
We call expressions (2.6) and (2.8) finite Fourier expansions and coefficients a and α
r r
finite Fourier coefficients of F.
6 Yoshinori Yamasaki
By definition, the multiple Ramanujan sum Sγ(n ,...,n ) is even (mod n ) as a
f 1 m+1 1
function of m-variables n ,...,n for any n N. Then, let us calculate the finite
2 m+1 1
∈
Fourier expansions of Sγ. To do so, we add one parameter (or, a weight) to Sγ. Let ξ be
f f
an arithmetic function of m-variables. Set
Sγ,ξ(n ,...,n ) = S(γ1,...,γm),ξ(n ,...,n )
f 1 m+1 f1,...,fm+1 1 m+1
n dγ1 dγm−1
:= ξ(dγ1,...,dγm)f 1 f 1 f m−1 f dγm .
1 m 1 dγ1 2 dγ2 ··· m dγm m+1 m
γj 1 2 m
dj |gcdX(n1,...,nj+1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(j=1,...,m)
Here, Sγ,ξ(n ,...,n ) is again even (mod n ) as a function of n ,...,n for any
f 1 m+1 1 2 m+1
n N and Sγ = Sγ,1m where 1 (d ,...,d ) 1. We also write Sξ := S(1,...,1),ξ. As
1 ∈ f f m 1 m ≡ f f
discussed in Proposition 2.6, Sγ,ξ can also be shown to be multiplicative if f ,...,f
f 1 m+1
and ξ are multiplicative.
For f = (f ,...,f ), we set tf = (δ0f ,δ1f ,...,δmf ). Further, for n N, we
1 m+1 m+1 m 1
∈
set
m
e n n n
tξγ(d ,...,d ) := a ξ ,..., .
n 1 m γj d · d d
m+1−j m 1
(cid:16)Yj=1 (cid:0) (cid:1)(cid:17) (cid:0) (cid:1)
Then, we obtain the following theorem. Roughly speaking, the finite Fourier coefficients
of Sγ,ξ can again be written as a multiple Ramanujan sum.
f
Theorem 2.7. Write the finite Fourier expansions of Sγ,ξ as
f
n1
Sγ,ξ(n ,...,n ) = aγ,ξ (l ,...,l )e(n ,n l ) e(n ,n l )
f 1 m+1 f,n1 2 m+1 1 2 2 ··· 1 m+1 m+1
l2,...X,lm+1=1
= αγ,ξ (d ,...,d )c(d ,n ) e(d ,n ).
f,n1 2 m+1 2 2 ··· m+1 m+1
d2,...X,dm+1|n1
Then, the finite Fourier coefficients aγ,ξ and αγ,ξ of Sγ,ξ are respectively given as
f,n1 f,n1 f
aγ,ξ (l ,...,l ) = n−mStξnγ1(n ,l ,...,l ), (2.10)
f,n1 2 m+1 1 tfe 1 m+1 2
αγ,ξ (d ,...,d ) = n−mStξnγ1 n , n1 ,..., n1 . (2.11)
f,n1 2 m+1 1 tfe 1 d d
m+1 2
(cid:0) (cid:1)
Proof. We only verify formula (2.11) (formula (2.10) can be similarly obtained). From
γ,ξ
formula (2.9), α (d ,...,d ) is given as
f,n1 2 m+1
n n n n
n−m Sγ,ξ(n ,δ ,...,δ )c 1, 1 c 1 , 1 .
1 f 1 2 m+1 δ d ··· δ d
2 2 m+1 m+1
δ2,...X,δm+1|n1 (cid:0) (cid:1) (cid:0) (cid:1)
Further, by changing the order of the summation, this expression is equivalent to
n eγ1 eγm−1
n−m ξ eγ1,...,eγm f 1 f 1 f m−1 f eγm
1 1 m 1 eγ1 2 eγ2 ··· m eγm m+1 m
eγmm|X···|eγ11|n1 (cid:0) (cid:1) (cid:0) 1 (cid:1) (cid:0) 2 (cid:1) (cid:0) m (cid:1) (cid:0) (cid:1)
n n n n
1 1 1 1
c , c , . (2.12)
× δ d ··· δ d
(cid:16)eγ11X|δ2|n1 (cid:0) 2 2(cid:1)(cid:17) (cid:16)eγmm|Xδm+1|n1 (cid:0) m+1 m+1(cid:1)(cid:17)
Arithmetical properties of Multiple Ramanujan sums 7
Here, we use the following identity. Let e n and d n. Then, we have
| |
n
n n if d e,
c , = e | (2.13)
δ d
(0 otherwise.
e|δ|n
X (cid:0) (cid:1)
Actually, one can obtainthis formula from the right-most expression (1.1) of the Ramanu-
jan sum c(k,n) and formula δ0 µ = ε. Then, applying formula (2.13), we see that (2.12)
∗
can be written as
n−m ξ eγ1,...,eγm
1 ··· 1 m
d2|Xeγ11|n1d3|Xeγ22|eγ11 dm+1|eXγmm|emγm−−11 (cid:0) (cid:1)
n eγ1 eγm−1 n n
f 1 f 1 f m−1 f eγm 1 1 .
× 1 eγ1 2 eγ2 ··· m eγm m+1 m eγ1 ··· eγm
1 2 m 1 m
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Changing variables e′ = n /eγk for all 1 k m, we have a (n /e′) = 1. Moreover,
k 1 k ≤ ≤ γk 1 k
it holds that e′ n1 for 1 k m and e′ e′ n , because eγm eγ1 n . Hence, we
k|dk+1 ≤ ≤ 1|···| m| 1 m |···| 1 | 1
can rewrite the above expression as
n n n n
n−m a 1 a 1 ξ 1,..., 1
1 ··· γ1 e′ ··· γm e′ · e′ e′
e′meX′m|d|mnn11+1 ee′m′mX−−11||den′mm1 eeX′1′1||nde′221 (cid:0) 1(cid:1) (cid:0) m(cid:1) (cid:0) 1 m(cid:1)
e′ e′ n
f e′ f 2 f m f 1 e′ e′ . (2.14)
× 1 1 2 e′ ··· m e′ m+1 e′ 1··· m
1 m−1 m
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
It is easy to see that (2.14) coincides with the right-hand side of formula (2.11). This
completes the proof of theorem.
Remark 2.8. Suppose that γ γ γ with γ = 1. Then, we have Stξnγ1 = Stξn1 where
0| 1|···| m 0 tfe tfe[γ]
tξ := tξ(1,...,1) and tf[γ] = (δ0f[γm],δ1f[γm−1],...,δmf[γ0]). In fact, since the condition
n1 n1 m+1 m 1
above means that a (n /e′) = a (e′ /e′) for all 1 k m 1, the summand in (2.14)
γk 1 k γk k+1 k ≤ ≤ −
is written as e
n n e′ e′ n
ξ 1,..., 1 f e′ f[γ1] 2 f[γm−1] m f[γm] 1 e′ e′ .
e′ e′ 1 1 2 e′ ··· m e′ m+1 e′ 1··· m
1 m 1 m−1 m
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Hence, the claim follows.
Example 2.9. Retaining the notation in Example 2.4, one can easily see that tf =
(δa1+···+am,δa1+···+am−1+1,...,δa1+m−1,δm). Hence, the finite Fourier coefficient α :=
f,n1
α(1,...,1),1m of the multiple divisor function σ := σ(1,...,1) is given by e
f,n1 a a
n n
αf,n1(d2,...,dm+1) = na11+···+am−mσ1−ta n1, d 1 ,..., d1 ,
m+1 2
(cid:0) (cid:1)
where 1 := (1,...,1) Cm and ta := (a ,...,a ) Cm
m 1
∈ ∈
Example 2.10. Retaining the notation in Example 2.5, we have S = f gcd. Again, by
f
◦
the degeneracy formula of (2.3), the finite Fourier coefficient α of f gcd is given by
f,n1 ◦
n n
α (d ,...,d ) = n−mS n ,gcd( 1,..., 1 ) .
f,n1 2 m+1 1 f∗µ,δm 1 d d
2 m+1
(cid:0) (cid:1)
8 Yoshinori Yamasaki
γ
3 Dirichlet series attached to S
f
In this section, we examine both single-variable and multivariable Dirichlet series, the
coefficients of which are given by the multiple Ramanujan sum Sγ.
f
3.1 Single-variable Dirichlet series
Wefirstexamineasingle-variableDirichletseries. Recallthefollowingwell-knownformula
concerning the Ramanujan sum c(k,n) (see, e.g., [20]):
∞
σ (n)
c(k,n)k−s = 1−s (Re(s) > 1). (3.1)
ζ(s)
k=1
X
In this subsection, we give a generalization of this formula. For j = 1,2,...,m+1, let
∞
Φγ(s;nˇ ) := Sγ(n ,...,n )n−s,
f j f 1 m+1 j
nXj=1
where nˇ := (n ,...,n ,n ,...,n ) Nm. The following proposition says that the
j 1 j−1 j+1 m+1
∈
series Φγ(s;nˇ ) is written as the product of a Dirichlet series and a finite sum, which is
f j
again given by the multiple Ramanujan sum.
Proposition 3.1. (i) For j = 1, we have
Φγ(s;nˇ ) = L(s;f )S (n ) = L(s;f )F (n ) (Re(s) > σ(f )), (3.2)
f 1 1 F1 2 1 1 2 1
where F = Fγ,s := δ−s S(γ2,...,γm)( ,n ,...,n ) δs .
1 1,f,nˇ1 f2,...,fm+1 · 3 m+1 ∗γ1
(ii) For 2 j m+1, we have
≤ ≤ (cid:0) (cid:1)
Φγ(s;nˇ ) = ζ(s)S(γ1,...,γj−2) (n ,...,n ) (Re(s) > 1), (3.3)
f j f1,...,fj−2,Fj 1 j−1
where F = Fγ,s := δ−s S(γj,...,γm)( ,n ,...,n ) (δsf ) .
j j,f,nˇj fj,...,fm+1 · j+1 m+1 ∗γj−1 j−1
Proof. Suppose that dγk (cid:0)gcd(n ,...,n ) for all k = 1,...,m. The(cid:1)n, n is a multiple of
lcm(dγ1,...,dγm) = dγk1 |if j = 11 and lckm+1(dγj−1,...,dγm) = dγj−1 if 2 jj m+ 1 (note
1 m 1 j−1 m j−1 ≤ ≤
that we only consider m-tuples (d ,...,d ) such that dγm dγ1). Hence, for j = 1 and
1 m m |···| 1
Re(s) > σ(f ), Φγ(s;nˇ ) is expressed as
1 f j
∞ dγ1l dγ1 dγm−1
f 1 f 1 f m−1 f dγm dγ1l −s
1 dγ1 2 dγ2 ··· m dγm m+1 m 1
dγkk|gcdX(n2,...,nk+1) Xl=1 (cid:0) 1 (cid:1) (cid:0) 2 (cid:1) (cid:0) m (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
(k=1,...,m)
and, for 2 j m+1 and Re(s) > 1,
≤ ≤
n dγ1 dγj−3 dγj−2
f 1 f 1 f j−3 f j−2
1 dγ1 2 dγ2 ··· j−2 dγj−2 j−1 dγj−1
dγkk|gcdX(n1,...,nk+1) (cid:0) 1 (cid:1) (cid:0) 2 (cid:1) (cid:0) j−2 (cid:1)djγ−j−1X1|djγ−j−22 (cid:0) j−1 (cid:1)
(k=1,2,...,j−2)
∞ dγj−1 dγm−1
f j−1 f m−1 f dγm dγj−1l −s.
× j dγj ··· m dγm m+1 m j−1
dγkk|gcd(dγj−j−1X1,nj+1,...,nk+1) Xl=1 (cid:0) j (cid:1) (cid:0) m (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
(k=j,...,m)
Arithmetical properties of Multiple Ramanujan sums 9
Thus, it is easy to see that these coincide, respectively, with (3.2) and (3.3). This com-
pletes the proof.
Example 3.2. For small m, the series Φγ(s;nˇ ) is explicitly given as follows. For m = 1,
f j
we have
Φ(γ1) (s;n ) = L(s;f ) f (dγ1)d−γ1s, (3.4)
f1,f2 2 1 2
dXγ1|n2
n
Φ(γ1) (s;n ) = ζ(s) f 1 f (dγ1)d−γ1s (3.5)
f1,f2 1 1 dγ1 2
dXγ1|n1 (cid:0) (cid:1)
and for m = 2, we have
dγ1
Φ(γ1,γ2) (s;n ,n ) = L(s;f ) f 1 f (dγ2)d−γ1s,
f1,f2,f3 2 3 1 2 dγ2 3 2 1
dXγ11|n2dγ22|gcXd(dγ11,n3) (cid:0) 2 (cid:1)
n dγ1
Φ(γ1,γ2) (s;n ,n ) = ζ(s) f 1 f 1 f (dγ2)d−γ1s,
f1,f2,f3 1 3 1 dγ1 2 dγ2 3 2 1
dXγ11|n1dγ22|gcXd(dγ11,n3) (cid:0) 1 (cid:1) (cid:0) 2 (cid:1)
n dγ1
Φ(γ1,γ2) (s;n ,n ) = ζ(s) f 1 f 1 f (dγ2)d−γ2s.
f1,f2,f3 1 2 1 dγ1 2 dγ2 3 2 2
dγ11|gXcd(n1,n2)dγ2X2|dγ11 (cid:0) 1 (cid:1) (cid:0) 2 (cid:1)
Formulas (3.4) and (3.5) were obtained in [1] for the case of γ = 1.
1
Example 3.3. For 0 k m + 1 and a = (a ,...,a ) Cm+1−k, we define
1 m+1−k
≤ ≤ ∈
generalizations of the classical Ramanujan sum c(n ,n ) as
1 2
cam+1,k(n1,...,nm+1) := Sµ,...,µ,δa1,...,δam+1−k(n1,...,nm+1).
k
(1)
It is clear that c(n ,n ) = c (n ,n ). |Th{ezn,}for 1 k m, using formulas (3.2) and
1 2 2,1 1 2 ≤ ≤
L(s;µ) = 1/ζ(s), we can verify by induction on m and k that
∞
ca (n ,...,n )n−s1 n−sk
m+1,k 1 m+1 1 ··· k
n1,.X..,nk=1
k
ζ(s )
= k j=2 j da1−(s1+···+sk)σae(d,nk+2,...,nm+1), (3.6)
ζ(s + +s )
j=1Q 1 ··· j dX|nk+1
where a := (a aQ,a a ,...,a a ) Cm−k (see Example 2.4). Setting
2 1 3 2 m+1−k m−k
− − − ∈
m = 1, k = 1 (in this case, σae 1) and a1 = 1, we obtain formula (3.1). See [6] for a
≡
multivaeriable analogue of formula (3.1).
Example 3.4. Retaining the notation in Example 2.5, we have S = f gcd. Let
f
◦
gcd(nˇ ) := gcd(n ,...,n ,n ,...,n ). Then,sinceitholdsthatgcd(n ,...,n ) =
j 1 j−1 j+1 m+1 1 m+1
gcd(n ,gcd(nˇ )), we have the following well-known formula
j j
∞
(f gcd)(n ,...,n )n−s = Φ (s;gcd(nˇ )) = ζ(s) (f µ)(d)d−s.
◦ 1 m+1 j δ0,f∗µ j ∗
nXj=1 d|gXcd(nˇj)
10 Yoshinori Yamasaki
3.2 Multivariable Dirichlet series
For an arithmetic function F(n ,...,n ) of k-variables, we denote L(s;F) by the multi-
1 k
variable Dirichlet series attached to F, that is,
∞
L(s;F) := F(n ,...,n )n−s1 n−sk
1 k 1 ··· k
n1,.X..,nk=1
for s = (s ,...,s ) Σ(F), where Σ(F) Ck is the region of absolute convergence for
1 k
∈ ⊂
L(s;F). The first goal of this subsection is to calculate L(s;Sγ) explicitly.
f
Theorem 3.5. Let γ γ γ γ with γ N. Define the function a Sγ of (m+1)-
0| 1| 2|···| m 0 ∈ γ0 f
variables as (a Sγ)(n ,...,n ) := a (n )Sγ(n ,...,n ). Then, we have
γ0 f 1 m+1 γ0 1 f 1 m+1
m+1 m+1
L(s;a Sγ) = ζ(s ) L(s + +s ;f[γj−1]) (3.7)
γ0 f j · 1 ··· j j
j=2 j=1
Y Y
in the region
Re(s ) > 1 (2 j m+1),
s = (s ,...,s ) Cm+1 j ≤ ≤ .
1 m+1
( ∈ (cid:12) Re(s1 + +sj) > σ(fj) (1 j m+1) )
(cid:12) ··· ≤ ≤
In particular, setting γ = 1, we h(cid:12)ave
0 (cid:12)
(cid:12)
m+1 m+1
L(s;Sγ) = ζ(s ) L(s ;f ) L s + +s ;f[γj−1] . (3.8)
f j · 1 1 1 ··· j j
j=2 j=2
Y Y (cid:0) (cid:1)
Proof. This is proven by induction on m. Suppose that m = 1. Then, it holds from
formula (3.5) that
∞
L (s ,s );a S(γ1) = Φ(γ1) (s ;nγ0)n−γ0s
1 2 γ0 f1,f2 f1,f2 2 1 1
(cid:0) (cid:1) nX1=1
∞
n
= ζ(s ) f 1 f dγ1 d−γ1s2n−γ0s1.
2 1 dγ1 2 1
nX1=1dγX1|nγ10 (cid:0) (cid:1) (cid:0) (cid:1)
Here, since γ γ , nγ0 is expressed as nγ0 = dγ1lγ0 with l N. Hence, this expression is
0| 1 1 1 ∈
written as
ζ(s ) ∞ ∞ f dγ1lγ0 f dγ1 d−γ1s2 dγ1lγ0 −s1,
2 1 dγ1 2
d=1 l=1
XX (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
γ
thuscompletingtheproofform = 1. Next,supposethatm 1. NotethatS (n ,...,n )
− f 1 m+1
= S(γ1) (n ,n ) from formula (2.2). Then, from (3.7) for m = 1,
f1,Sf(γ22,.,....,.f,mγm+)1(·,n3,...,nm+1) 1 2
L(s;a Sγ) is equal to
γ0 f
∞
L (s ,s );a S(γ1) n−s1 n−sm+1
n3,...X,nm+1=1 (cid:0) 1 2 γ0 f1,Sf(γ22,.,....,.f,mγm+)1(·,n3,...,nm+1)(cid:1) 3 ··· m+1
∞
= ζ(s )L(s ;f[γ0]) L s +s ;a S(γ2,...,γm)( ,n ,...,n ) n−s1 n−sm+1
2 1 1 1 2 γ1 f2,...,fm+1 · 3 m+1 3 ··· m+1
n3,...X,nm+1=1 (cid:0) (cid:1)
= ζ(s )L(s ;f[γ0])L (s +s ,s ,...,s );a S(γ2,...,γm) .
2 1 1 1 2 3 m+1 γ1 f2,...,fm+1
(cid:0) (cid:1)