On the Fourier transform of the greatest common divisor

0.42 MB·English

by  Peter H. van der Kamp

On the Fourier transform of
the greatest common divisor
2
1 Peter H. van der Kamp
0
2
Department of Mathematics and Statistics

n
La Trobe University
a
J Victoria 3086, Australia

6
1 January 17, 2012

]
T
N Abstract
The discrete Fourier transform of the greatest common divisor
.
h
m
at i(cid:98)d[a](m)=(cid:88)gcd(k,m)αmka,
m k=1
[ with αm a primitive m-th root of unity, is a multiplicative function that

generalisesboththegcd-sumfunctionandEuler’stotientfunction. Onthe
1
onehanditistheDirichletconvolutionoftheidentitywithRamanujan’s
v
9 sum, i(cid:98)d[a] = id∗c•(a), and on the other hand it can be written as a
3 generalised convolution product, i(cid:98)d[a]=id∗aφ.
1 We show that i(cid:98)d[a](m) counts the number of elements in the set of
3 orderedpairs(i,j)suchthati·j ≡a mod m. Furthermorewegeneralisea
. dozenknownidentitiesforthetotientfunction,toidentitieswhichinvolve
1
0 the discrete Fourier transform of the greatest common divisor, including
2 its partial sums, and its Lambert series.
1
:
v 1 Introduction
i
X
In [15] discrete Fourier transforms of functions of the greatest common divisor
r
a were studied, i.e.
m
(cid:88)
(cid:98)h[a](m)= h(gcd(k,m))αka,
m
k=1
where α is a primitive m-th root of unity. The main result in that paper is
m
the identity1 (cid:98)h[a]=h∗c•(a), where ∗ denoted Dirichlet convolution, i.e.
(cid:88) m
(cid:98)h[a](m)= h(d)cd(a), (1)
d|m
1Similarresultsinthecontextofr-evenfunctionwereobtainedearlier,see[10]fordetails.
1
and
m
(cid:88)
c (a)= αka (2)
m m
k=1
gcd(k,m)=1
denotes Ramanujan’s sum. Ramanujan’s sum generalises both Euler’s totient
function φ = c (0) and the M¨obius function µ = c (1). Thus, identity (1)
• •
generalizes the formula
m
(cid:88)
h(gcd(k,m))=(h∗φ)(m), (3)
k=1
already known to Ces`aro in 1885. The formula (1) shows that (cid:98)h[a] is mul-
tiplicative if h is multiplicative (because c (a) is multiplicative and Dirichlet
•
convolution preserves multiplicativity).
Here we will take h = id to be the identity function (of the greatest com-
mon divisor) and study its Fourier transform. Obviously, as id : id(n) = n is
multiplicative, the function i(cid:98)d[a] is multiplicative, for all a. Two special cases
are well-known. Taking a=0 we have i(cid:98)d[0]=P, where
m
(cid:88)
P(m)= gcd(k,m). (4)
k=1
isknownasPillai’sarithmeticalfunctionorthegcd-sumfunction. Secondly, by
taking a = 1 in (1), we find that i(cid:98)d[1] = id∗µ equals φ, by M¨obius inversion
of Euler’s identity φ∗u = id, where u = µ−1 is the unit function defined by
u(m)=1.
Let Fm denote the set of ordered pairs (i,j) such that i·j ≡a mod m, the
a
set of factorizations of a modulo m. We claim that i(cid:98)d[a](m) counts its number
of elements. Let us consider the mentioned special cases.
a=0 For given i∈{1,2,...,m} the congruence i·j ≡0 mod m yields
i m
j ≡0 mod ,
gcd(i,m) gcd(i,m)
whichhasauniquesolutionmodulom/gcd(i,m),andsotherearegcd(i,m)
solutionsmodulom. Hence, thetotalnumberofelementsinFm isP(m).
0
a=1 The totient function φ(m) counts the number of invertible congruence
classes modulo m. As for every invertible congruence class i modulo m
there is a unique j = i−1 mod m such that i·j ≡ 1 mod m, it counts
the number of elements in the set Fm.
1
To prove the general case we employ a Kluyver-like formula for i(cid:98)d[a], that is, a
formula similar to the formula for the Ramanujan sum function
(cid:88) k
c (a)= dµ( ). (5)
k d
d|gcd(a,k)
2
attributed to Kluyver. Together the identities (1) and (5) imply, cf. section 3,
(cid:88) m
i(cid:98)d[a](m)= dφ( ), (6)
d
d|gcd(a,m)
and we will show, in the next section, that the number of factorizations of a
mod m is given by the same sum.
The right hand sides of (5) and (6) are particular instances of the so called
generalized Ramanujan sums [1], and both formulas follow as consequence of a
generalformulafortheFouriercoefficientsofthesegeneralisedRamanujansums
[2, 3]. In section 3 we provide simple proofs for some of the nice properties of
these sums. In particular we interpret the sums as a generalization of Dirichlet
convolution. This interpretation lies at the heart of many of the generalised
totient identities we establish in section 4.
2 The number of factorizations of a mod m
For given i,m ∈ N, denote g = gcd(i,m). If the congruence i·j ≡ a mod m
has a solution j, then g |a and j ≡i−1a/g is unique mod m/g, so mod m there
are g solutions. This yields
m
(cid:88)
#Fm = gcd(i,m),
a
i=1
gcd(i,m)|a
which can be written as
m
(cid:88) (cid:88)
#Fm = d (7)
a
d|a i=1
gcd(i,m)=d
If d(cid:45)m then the sum
m
(cid:88)
1
i=1
gcd(i,m)=d
is empty. Now let d | m. The only integers i which contribute to the sum are
the multiples of d, kd, where gcd(k,m/d) = 1. There are exactly φ(m/d) of
them. Therefore the right hand sides of formulae (6) and (7) agree, and hence
#Fm =i(cid:98)d[a](m).
a
3 A historical remark, and generalised Ramanu-
jan sums
It is well known that Ramanujan was not the first who considered the sum
c (a). Kluyver proved his formula (5) in 1906, twelve years before Ramanujan
m
3
published the novel idea of expressing arithmetical functions in the form of a
(cid:80)
series a c (n) [11]. It is not well known that Kluyver actually showed that
s s s
c (a) equals Von Sterneck’s function, introduced in [13], i.e.
m
µ( m )φ(m)
gcd(a,m)
c (a)= . (8)
m φ( m )
gcd(a,m)
This relation is referred to in the literature as H¨older’s relation, cf. the remark
on page 213 in [1]. However, H¨older published this relation thirty years after
Kluyver[7]. Wereferto[1, Theorem2], or[2, Theorem8.8]forageneralisation
of (8). The so called generalized Ramanujan sums,
(cid:88) m
f ∗ g(m)= f(d)g( ), (9)
a d
d|gcd(a,m)
wereintroducedin[1]. Thenotation∗ isnew,thesumsaredenotedS(a;m)in
a
[1],s (a)in[2],andS (a,m)in[3]. Inthecontextofr-evenfunctions[10]the
m f,h
sumsaredenotedS (a),andconsideredassequencesofm-evenfunctions,with
f,g
argument a. We consider the sums as a sequence of functions with argument
m, labeled by a. We call f ∗ g the a-convolution of f and g.
a
The concept of a-convolution is a generalization of Dirichlet convolution as
f ∗ g = f ∗g. As we will see below, the function f ∗ g is multiplicative (for
0 a
all a) if f and g are, and the following inter-associative property holds, cf. [3,
Theorem 4].
(f ∗ g)∗h=f ∗ (g∗h). (10)
a a
We also adopt the notation f = id∗ f, and call this the Kluyver, or a-
a a
extension of f. Thus, we have f = id∗f, f = f, and formulas (5) and (6)
0 1
become cm(a)=µa(m), and i(cid:98)d[a]=φa, respectively.
The identity function I, defined by f ∗I = f, is given by I(k) = [k = 1],
where the Iverson bracket is, with P a logical statement,
(cid:26)
1 if P,
[P]=
0 if not P.
Let us consider the function f ∗ I. It is
a
(cid:88)
f ∗ I(k)= f(d)[d=k]=[k |a]f(k).
a
d|gcd(a,k)
Sincethefunctionk →[k |a]ismultiplicative, thefunctionf∗ I ismultiplica-
a
tive if f is multiplicative. Also, we may write, cf. [3, eq. (9)],
(cid:88) m
f ∗ g(m)= [d|a]f(d)g( )=(f ∗ I)∗g(m),
a d a
d|m
which shows that f ∗ g is multiplicative if f and g are. Also, the inter-
a
associativity property (10) now easily follows from the associativity of the
Dirichlet convolution,
(f ∗ g)∗h=((f ∗ I)∗g)∗h=(f ∗ I)∗(g∗h)=f ∗ (g∗h).
a a a a
4
Wenotethatthea-convolutionproductisneitherassociative,norcommutative.
The inter-associativity and the commutativity of Dirichlet convolution imply
that
f ∗g =(f ∗g) =f ∗g . (11)
a a a
Formula(6)statesthattheFouriertransformofthegreatestcommondivisor
is the Kluyver extension of the totient function. We provide a simple proof.
Proof[of(6)]Employing(1),(5)and(11)wehavei(cid:98)d[a]=id∗c•(a)=id∗µa =
(id∗µ) =φ . (cid:3)
a a
The formula (6) also follows as a special case of the following formula for the
Fourier coefficients of a-convolutions,
m
(cid:88) f
f ∗ g(m)= h (m)αka, h =g∗ , (12)
a k m k k id
k=1
given in [1, 2]. The formula (12) combines with (1) and (5) to yield a formula
for functions of the greatest common divisor, h¯[k]:m→h(gcd(k,m)), namely
h¯[k]=(h∗µ)∗ u. (13)
k
Proof [of (13)] The Fourier coefficients of (cid:98)h[a](m) are h¯[k](m). But (cid:98)h[a] =
h∗c (a)=(id∗ µ)∗h=id∗ (µ∗h),andso,using(12),theFouriercoefficients
• a a
are also given by (h∗µ)∗ u(m). (cid:3)
k
ForaDirichletconvolutionwithaFouriertransformofafunctionofthegreatest
common divisor we have
(cid:91)
f ∗g[a]=f ∗g[a]. (14)
(cid:98)
Proof [of (14)] f ∗g[a]=f ∗(g∗µ )=(f ∗g)∗µ =f(cid:91)∗g[a] (cid:3)
(cid:98) a a
Similarly, for an a-convolution with a Fourier transform of a function of the
greatest common divisor,
(cid:92)
f ∗ g[b]=f ∗ g[b]. (15)
a(cid:98) a
Proof [of (15)] f ∗ g[b]=f ∗ (g∗µ )=(f ∗ g)∗µ =f(cid:92)∗ g[b] (cid:3)
a(cid:98) a b a b a
4 Generalised totient identities
The totient function is an important function in number theory, and related
fieldsofmathematics. Itisextensivelystudied,connectedtomanyothernotions
and functions, and there exist numerous generalisation and extensions, cf. the
chapter ”The many facets of Euler’s totient” in [12]. The Kluyver extension of
the totient function is a very natural extension, and it is most surprising it has
not been studied before. In this section we generalise a dozen known identities
for the totient function φ, to identities which involve its Kluyver extension φ ,
a
a.k.a. the discrete Fourier transform of the greatest common divisor. This
includes a generalisation of Euler’s identity, the partial sums of φ , and its
a
Lambert series.
5
4.1 The value of φ at powers of primes
a
We start by providing a formula for the value of φ at powers of primes. This
a
dependsonlyonthemultiplicityoftheprimeina. Theformulae, withpprime,
P(pk)=(k+1)pk−kpk−1, φ(pk)=pk−pk−1,
of which the first one is Theorem 2.2 in [4], generalise to
(cid:26) (pk−pk−1)(l+1) if l<k,
φ (pk)= (16)
a (k+1)pk−kpk−1 if l≥k,
where l is the largest integer, or infinity, such that pl |a.
Proof [of (16)] We have
(cid:88) pk
φ (pk)= dφ( )
a d
d|gcd(pl,pk)
min(l,k)
(cid:88)
= prφ(pk−r)
r=0
(cid:40)
(cid:80)l pk−pk−1 if l<k,
= r=0
((cid:80)k−1pk−pk−1)+pk if l≥k,
r=0
which equals (16).
4.2 Partial sums of φ /id
a
To generalise the totient identity
n n
(cid:88)φ(k) (cid:88)µ(k) n
= (cid:98) (cid:99). (17)
k k k
k=1 k=1
to an identity for φ we first establish
a
n n
(cid:88)f0(k) =(cid:88)f(k)(cid:98)n(cid:99). (18)
k k k
k=1 k=1
Proof [of (18)] Since there are (cid:98)n/d(cid:99) multiples of d in the range [1,n] it follows
that
n n n
(cid:88)f ∗id(k) (cid:88)(cid:88)f(d) (cid:88)f(d) n
= = (cid:98) (cid:99).
k d d d
k=1 k=1 d|k d=1
(cid:3)
As a corollary we obtain
n n
(cid:88)f ∗ag0(k) =(cid:88)f ∗ag(k)(cid:98)n(cid:99). (19)
k k k
k=1 k=1
6
Proof [of (19)] Employing (10) we find
n n n
(cid:88)f ∗a(g∗id)(k) =(cid:88)(f ∗ag)∗id(k) =(cid:88)f ∗ag(k)(cid:98)n(cid:99).
k k k k
k=1 k=1 k=1
(cid:3)
Now taking f =id and g =µ in (19) we find
n n
(cid:88)φa(k) =(cid:88)ck(a)(cid:98)n(cid:99). (20)
k k k
k=1 k=1
4.3 Partial sums of P /id expressed in terms of φ
a a
Taking f =id and g =φ in (19) we find
n n
(cid:88)Pa(k) =(cid:88)φa(k)(cid:98)n(cid:99). (21)
k k k
k=1 k=1
Notethatbytakingeithera=0in(20),ora=1in(21),wefindanidentity
involving the totient function and the gcd-sum function,
n n
(cid:88)P(k) (cid:88)φ(k) n
= (cid:98) (cid:99). (22)
k k k
k=1 k=1
4.4 Partial sums of φ
a
To generalise the totient identity, with n>0,
n (cid:32) n (cid:33)
(cid:88) 1 (cid:88) n
φ(k)= 1+ µ(k)(cid:98) (cid:99)2 , (23)
2 k
k=1 k=1
we first establish
n (cid:32) n n (cid:33)
(cid:88) 1 (cid:88) n (cid:88)
f (k)= f(k)(cid:98) (cid:99)2+ f ∗u(k) . (24)
0 2 k
k=1 k=1 k=1
Proof [of (24)] We have, by changing variable k =dl,
n n
(cid:88) (cid:88)(cid:88) 2k
(2f ∗id−f ∗u)(k)= f(d)( −1)
d
k=1 k=1 d|k
n (cid:98)n/d(cid:99)
(cid:88) (cid:88)
= f(d)(2l−1)
d=1 l=1
n
(cid:88) n
= f(d)(cid:98) (cid:99)2.
d
d=1
(cid:3)
7
Note that this gives a nice proof of (23), taking f =µ, as (cid:80)n I(k)=[k >0].
k=1
When f =µ , then (11) implies f ∗id=φ , and f ∗u=I , and therefore as a
a a a
special case of (24) we obtain
 
n n
(cid:88) 1 (cid:88) (cid:88) n
φa(k)= 2 k[k ≤n]+ ck(a)(cid:98)k(cid:99)2. (25)
k=1 k|a k=1
(cid:80)
We remark that when n≥a we have k[k ≤n]=σ(a), where σ =id∗u is
k|a
the sum of divisors function.
4.5 Generalisation of Euler’s identity
Euler’s identity, φ∗u=id, generalises to
(cid:88)
φ (d)=τ(gcd(a,m))m, (26)
a
d|m
where τ =u∗u is the number of divisors function.
Proof [of (26)] We have φ ∗u=(φ∗u) =id where
a a a
(cid:88) m
id (m)= d =mτ(gcd(a,m)). (27)
a d
d|gcd(a,m)
(cid:3)
4.6 Partial sums of P expressed in terms of φ (and τ)
a a
If f =φ , then f ∗id=P , and (24) becomes, using (26),
a a
n (cid:32) n n (cid:33)
(cid:88) 1 (cid:88) (cid:88) n
P (k)= τ(gcd(a,k))k+ φ (k)(cid:98) (cid:99)2 . (28)
a 2 a k
k=1 k=1 k=1
4.7 Four identities of C´esaro
AccordingtoDickson[5]thefollowingthreeidentitieswereobtainedbyC´esaro:
(cid:88) n
dφ( )=P(n), (29)
d
d|n
n
(cid:88) d (cid:88) 1
φ(d)= , (30)
n gcd(j,n)
d|n j=1
n
(cid:88) n (cid:88)
φ(d)φ( )= φ(gcd(j,n)). (31)
d
d|n j=1
8
Identity (29), which is Theorem 2.3 in [4], is obtained by taking a=0 in (6), or
h=id in (3). It generalises to
(cid:88) n
dφ ( )=P (n). (32)
a d a
d|n
Proof [of (32)] By taking f =φ and g =id in (11). (cid:3)
Identity (30) is obtained by taking h=1/id in (3) and generalises to
n
(cid:88) d (cid:88) (cid:88) 1
φ (d)= , (33)
n a gcd(j,n)
d|n j=1d|gcd(a,n) d
Proof [of (33)] By taking f =φ and g =1/id in (11). (cid:3)
Identity (31) is also a special case of (3), with h=φ. It generalises to
n
(cid:88) n (cid:88) (cid:88) n
φ (d)φ ( )= φ (gcd(j, )). (34)
a b d b d
d|n j=1d|gcd(a,n)
Proof [of (34)] We have
(id∗aφ)∗(id∗bφ)=id∗a(id∗b(φ∗φ))=id∗a(id∗bφ(cid:98)[0])=id∗aφ(cid:98)b[0],
and evaluation at m yields
m/d m
(cid:88) (cid:88) m (cid:88) (cid:88) m
d φ (gcd(j, ))= φ (gcd(j, )).
b d b d
d|gcd(a,m) j=1 d|gcd(a,m)j=1
(cid:3)
The more general identity (3) generalises to
m
(cid:88)
h (gcd(k,m))=h∗φ (m). (35)
a a
k=1
4.8 Three identities of Liouville
Dickson [5, p.285-286] states, amongst many others identities that were pre-
sented by Liouville in the series [9], the following
(cid:88) m
φ(d)τ( )=σ(m), (36)
d
d|m
(cid:88) m
φ(d)σ[n+1]( )=mσ[n](m), (37)
d
d|m
(cid:88) m2 (cid:88) m
φ(d)τ( )= dθ( ), (38)
d2 d
d|m d|m
9
whereσ[n]=id[n]∗u,id[n](m)=mn,andθ(m)isthenumberofdecompositions
of m into two relatively prime factors. All three are of the form φ∗f = g and
therefore they gain significance due to (3), thought Liouville might not have
been aware of this. For example, (3) and (36) combine to yield
m
(cid:88)
τ(gcd(k,m))=σ(m).
k=1
The three identities are easily proven by substituting τ =u∗u, σ[n]=id[n]∗u,
τ ◦id[2]=θ∗u, φ=µ∗id, and using µ∗u=I. They generalise to
(cid:88) m
φ (d)τ( )=σ (m), (39)
a d a
d|m
(cid:88) m
φ (d)σ[n+1]( )=mu∗ σ[n](m), (40)
a d a
d|m
(cid:88) m2 (cid:88) m
φ (d)τ( )= dτ(gcd(a,d))θ( ). (41)
a d2 d
d|m d|m
These generalisation are proven using the same substitutions, together with
(11), or for the latter identity, (10) and (27).
4.9 One identity of Dirichlet
Dickson [5] writes that Dirichlet [6], by taking partial sums on both sides of
Euler’s identity, obtained
n (cid:18) (cid:19)
(cid:88) n n+1
(cid:98) (cid:99)φ(k)= .
k 2
k=1
By taking partial sums on both sides of equation (26) we obtain
(cid:88)n n (cid:88) (cid:18)(cid:98)n(cid:99)+1(cid:19)
(cid:98) (cid:99)φ (k)= d d . (42)
k a 2
k=1 d|a
Proof [of (42)] Summing the left hand side of (26) over m yields
n n
(cid:88) (cid:88) (cid:88) n
φ (d)= (cid:98) (cid:99)φ (d)
a d a
m=1d|m d=1
and summing the right hand side of (26) over m yields
n n (cid:98)n/d(cid:99)
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) n (cid:16) n (cid:17)
τ(gcd(a,m))m= m= dk = d(cid:98) (cid:99) (cid:98) (cid:99)+1 /2.
d d
m=1 m=1d|gcd(a,m) d|a k=1 d|a
(cid:3)
10