Some remarks on a paper of V. A. Liskovets

0.17 MB·English

by  László Tóth

Some remarks on a paper of V. A. Liskovets
La´szlo´ T´oth ∗
Department of Mathematics, University of P´ecs
1
1 Ifju´sa´g u. 6, H-7624 P´ecs, Hungary
0 and
2
Institute of Mathematics, Department of Integrative Biology

n
Universit¨at fu¨r Bodenkultur, Gregor Mendel-Straße 33, A-1180 Wien, Austria
a
J E-mail: [email protected]

5
2

]
Abstract
T
N WededucenewpropertiesoftheorbicyclicfunctionE ofseveralvariablesinvestigatedinarecent
. paper by V. A. Liskovets. We point out that the function E and its connection to the number
h
of solutions of certain linear congruences occur in the literature in a slightly different form. We
t
a investigate another similar function considered by Deitmar, Koyama and Kurokawa by studying
m analytic properties of some zeta functions of Igusa type. Simple number theoretic proofs for some
[ known properties are also given.

1
v 1 Introduction
3
2
In a recent paper Liskovets [10] investigated arithmetical properties of the function
8
4
M
. 1
1 E(m ,...,m ):= c (k)···c (k), (1)
0 1 r M m1 mr
k=1
1 X
1 where m ,...,m ,M ∈N:={1,2,...} (r ∈N), m:=lcm[m ,...,m ], m|M and c (k) is the Ramanu-
1 r 1 r n
:
v jan sum defined as the sum of k-th powers of the primitive n-th roots of unity (k,n∈N), i.e.,
i
X
c (k):= exp(2πijk/n). (2)
n
r
a 1≤j≤n
X
gcd(j,n)=1
The function E given by (1) has been introduced by Mednykh and Nedela [13] in order to handle
certain problems of enumerative combinatorics. The function (1) has also arithmetical and topological
applications and it is called in [10] as the “orbicyclic” arithmetic function.
For example, E(m ,...,m ) is the number of solutions (x ,...,x )∈Zr of the congruence
1 r 1 r M
x +...+x ≡0 (mod M)
1 r
satisfying gcd(x ,M) = M/m ,...,gcd(x ,M) = M/m , see [13, Lemma 4.1]. It follows from this
1 1 r r
interpretation that all the values E(m ,...,m ) are nonnegative integers.
1 r
Note that in case of one, respectively two variables,
M
1 1, m =1,
1
E(m ):= c (k)= (3)
1 M m1 (0, otherwise,
k=1
X
∗TheauthorgratefullyacknowledgessupportfromtheAustrianScienceFund(FWF)undertheprojectNr. P20847-N18.
1
M
1 φ(m), m =m =m,
1 2
E(m ,m ):= c (k)c (k)= (4)
1 2 M m1 m2 (0, otherwise,
k=1
X
where φ is Euler’s function. Formulae(3) and (4) are wellknownproperties of the Ramanujansums, (4)
being their orthogonality property leading to the Ramanujan expansions of arithmetical functions, see
for example [19].
Another function, similar to E, is
m
1
A(m ,...,m ):= gcd(k,m )···gcd(k,m ), (5)
1 r 1 r
m
k=1
X
where m ,...,m ∈N (r ∈N) and m:=lcm[m ,...,m ], as above.
1 r 1 r
Thefunction(5)wasmentionedbyLiskovets[10,section4]anditwasconsideredbyDeitmar,Koyama
andKurokawa[9]incasem |m (1≤j ≤r−1)bystudyinganalyticpropertiesofsomezetafunctions
j j+1
of Igusa type. The explicit formula for the values A(m ,...,m ) derived in [9, Section 3] was reproved
1 r
by Minami [14] for the general case m ,...,m ∈ N, using arguments of elementary probability theory.
1 r
We remark that the corresponding formulae of both papers [9, 14] contain misprints.
For r =1 (5) reduces to the function
m
1 φ(d)
A(m):= gcd(k,m)= (6)
m d
kX=1 Xd|m
giving the arithmetic mean of gcd(1,m),...,gcd(m,m). For arithmetic and analytic properties of (6)
and for a survey of other gcd-sum-type functions of one variable see T´oth [22].
In the presentpaper we deduce new properties ofthe functions E andA and use them to give simple
number theoretic proofs for some of their known properties. We derive convolution-type identities for E
and A (Propositions 3 and 12) to show that they are multiplicative as functions of severalvariables. We
giveother identities for E andA (Propositions9 and14) to obtainexplicit formulaefor their values. We
considercommongeneralizationsof these functions andpointout thatthe functionE andits connection
tothenumberofsolutionsofcertainlinearcongruencesoccurintheliteratureinaslightlydifferentform.
As an application of the identity of Proposition 3 we give a simple direct proof of the orthogonality
property (4) of the Ramanujan sums (Application 8).
2 Preliminaries
Wepresentinthissectionsomebasicnotionsandpropertiesneededthroughoutthepaper. Fortheprime
powerfactorizationofanintegern∈N we willuse the notationn= pνp(n), wherethe productis over
p
the primes p and all but a finite number of the exponents ν (n) are zero.
p
Q
We recall that an arithmetic function of r variables is a function f : Nr → C, notation f ∈ F . If
r
f,g ∈F , then their convolution is defined as
r
(f ∗g)(m ,...,m )= f(d ,...,d )g(m /d ,...,m /d ). (7)
1 r 1 r 1 1 r r
d1|m1X,...,dr|mr
ThesetF formsaunitalringwithordinaryadditionandconvolution(7),theunitybeingthefunction
r
ε(r) given by
1, m =...=m =1,
ε(r)(m ,...,m )= 1 r (8)
1 r
(0, otherwise.
A function f ∈F is invertible iff f(1,...,1)6=0. The inverse of the constant 1 function is given by
r
µ(r)(m ,...,m )=µ(m )...µ(m ), where µ is the Mo¨bius function.
1 r 1 r
A function f ∈F is said to be multiplicative if it is nonzero and
r
f(m n ,...,m n )=f(m ,...,m )f(n ,...,n )
1 1 r r 1 r 1 r
2
holds for any m ,...,m ,n ,...,n ∈N such that gcd(m ···m ,n ···n )=1.
1 r 1 r 1 r 1 r
If f is multiplicative, then it is determined by the values f(pa1,...,par), where p is a prime and
a ,...,a ∈N :={0,1,2,...}. More exactly, f(1,...,1)=1 and for any m ,...,m ∈N,
1 r 0 1 r
f(m ,...,m )= f(pνp(m1),...,pνp(mr)).
1 r
p
Y
For example, the functions (m ,...,m ) 7→ gcd(m ,...,m ) and (m ,...,m ) 7→ lcm[m ,...,m ]
1 r 1 r 1 r 1 r
are multiplicative. The function µ(r) is also multiplicative.
Theconvolution(7)preservesthemultiplicativityoffunctions. Moreover,themultiplicativefunctions
form a subgroup of the group of invertible functions with respect to convolution (7).
These properties, which are well known in the one variable case, follow easily from the definitions.
For further properties of arithmetic functions of several variables and of their ring we refer to [2], [20,
Ch. VII].
Intheonevariablecase1,id (k ∈N),εandτ willdenotethefunctionsgivenby1(n)=1,id (n)=nk
k k
(n∈N), ε(1)=1, ε(n)=0 for n>1 and τ =1∗1 (divisor function), respectively.
The Ramanujan sums c (k) can be represented as
n
c (k)= dµ(n/d) (k,n∈N). (9)
n
d|gcXd(k,n)
Note that the function n 7→ c (k) is multiplicative for any fixed k ∈ N, and for any prime power pa
n
(a∈N),
pa−pa−1, if pa |k,
cpa(k)=−pa−1, if pa−1 |k,pa ∤k, (10)
0, if pa−1 ∤k.
Also, cn(k) is multiplicativeasa functionoftwovariables,i.e., consideredasthe functionc:N2 →Z,
c(k,n)=c (k). The inequality |c (k)|≤gcd(k,n) holds for any k,n∈N.
n n
These and other general accounts of Ramanujan sums can be found in the books by Apostol [6],
McCarthy [12], Schwarz and Spilker [19], Sivaramakrishnan [20].
3 The function E
The following results were proved by Liskovets [10].
Proposition 1. ([10, Lemmas 2, 5, Prop. 4])
(i) The function E is multiplicative (as a function of several variables).
(ii) Let pa1,...,par be any powers of a prime p (a1,...,ar ∈ N). Assume that a := a1 = a2 = ... =
a >a ≥a ≥...≥a ≥1 (r ≥s≥1). Then
s s+1 s+2 r
E(pa1,...,par)=pv(p−1)r−s+1h (p), (11)
s
where the integer v is defined by v = r a −r−a+1 and
j=1 j
P (x−1)s−1+(−1)s
h (x)= (12)
s
x
is a polynomial of degree s−2 (for s>1).
Note that Liskovets used the term semi-multiplicative function, but this is reserved for another con-
cept, see for example [20, Ch. XI].
Corollary 2. ([10, Th. 8, Cor. 11])
(i) For any integers m ,...,m ∈N,
1 r
E(m ,...,m )= pv(p)(p−1)r(p)−s(p)+1h (p), (13)
1 r s(p)
pY|m
3
where v(p) and s(p), depending now on p are the integers v and s, respectively, defined in Proposition 1.
(ii) For m =...=m =m,
1 r
(p−1)h (p)
f (m):=E(m,...,m)=mr−1 r . (14)
r pr−1
pY|m
We first give the following simple convolution representationfor the function E.
Proposition 3. For any m ,...,m ∈N,
1 r
d ···d
1 r
E(m ,...,m )= µ(m /d )···µ(m /d ). (15)
1 r 1 1 r r
lcm[d ,...,d ]
1 r
d1|m1X,...,dr|mr
Proof. Using formula (9),
M
1
E(m ,...,m )= d µ(m /d )··· d µ(m /d )
1 r 1 1 1 r r r
M
Xk=1d1|gcXd(k,m1) dr|gcXd(k,mr)
1
= d µ(m /d )···d µ(m /d ) 1,
1 1 1 r r r
M
d1|m1X,...,dr|mr 1≤k≤MX,d1|k,...dr|k
where the inner sum is 1=M/lcm[d ,...,d ], ending the proof.
1 r
1≤k≤M,lcXm[d1,...,dr]|k
By Mo¨bius inversion we obtain from (15),
Corollary 4. For any m ,...,m ∈N,
1 r
m ···m
1 r
E(d ,...,d )= . (16)
1 r
lcm[m ,...,m ]
1 r
d1|m1X,...,dr|mr
Formula (15) has also a number of other applications:
Application 5. Formula (15) shows that E is integral valued and that it does not depend on M, so in
(1) one can take M =m, the lcm of m ,...,m , remarked also by Liskovets [10, Section 1].
1 r
Application 6. If m ,...,m are pairwise relatively prime, then E(m ,...,m ) = ε(r)(m ,...,m ),
1 r 1 r 1 r
defined by (8). Indeed, in this case lcm[d ,...,d ] = d ···d for any d | m (1 ≤ i ≤ r) and the claim
1 r 1 r i i
follows from (15) using that µ(d)=ε(n).
d|n
Application 7. Also, (15)Pfurnishes a simple direct proof of the multiplicativity of E. Note that
Liskovets [10] used other arguments to show the multiplicativity. Observe that, according to (15), E
is the convolution of the functions F and µ(r), where F is given by
m ···m
1 r
F(m ,...,m )= .
1 r
lcm[m ,...,m ]
1 r
Since F and µ(r) are multiplicative, E is multiplicative too.
Application 8. Formula(15)leadstoasimpledirectproofoftheorthogonalityproperty (4). Usingthe
Gauss formula φ(d)=n,
d|n
P M
1 d d µ(m /d )µ(m /d )
1 2 1 1 2 2
E(m ,m ):= c (k)c (k)=
1 2 M m1 m2 lcm[d ,d ]
1 2
Xk=1 d1|mX1,d2|m2
= µ(m /d )µ(m /d )gcd(d ,d )= µ(m /d )µ(m /d ) φ(δ)
1 1 2 2 1 2 1 1 2 2
d1|mX1,d2|m2 d1|mX1,d2|m2 δ|gcdX(d1,d2)
= µ(k)µ(ℓ)φ(δ)= φ(δ) µ(k) µ(ℓ),
δak=mX1,δbℓ=m2 δu=mX1,δv=m2 aXk=u bXℓ=v
where one of the inner sums are zero, unless u = v = 1 and obtain that E(m ,m ) = φ(m) for m =
1 2 1
m =m and E(m ,m )=0 otherwise.
2 1 2
4
NowwederiveanotheridentityforEwhichfurnishesanalternativeproofofformula(ii)inProposition
1.
Proposition 9. For any m ,...,m ∈N,
1 r
1
E(m ,...,m )= c (d)···c (d)φ(m/d). (17)
1 r m m1 mr
Xd|m
Proof. It follows from (9) that c (k) = c (gcd(k,n)) (k,n ∈ N). Observe that for any i ∈ {1,...,r},
n n
gcd(gcd(k,m),m ) = gcd(k,gcd(m,m )) = gcd(k,m ), since m | m, hence c (k) = c (gcd(k,m )) =
i i i i mi mi i
c (gcd(gcd(k,m),m ))=c (gcd(k,m)). We obtain
mi i mi
m
1
E(m ,...,m )= c (gcd(k,m))···c (gcd(k,m)),
1 r m m1 mr
k=1
X
and by grouping the terms according to the values gcd(k,m) = d, where d | m, k = dj, 1 ≤ j ≤ m/d,
gcd(j,m/d)=1, we obtain (17).
Application 10. By (17), with the notation of Proposition 1,
1
E(pa1,...,par)= pa cpa1(d)···cpar(d)φ(pa/d). (18)
dX|pa
Using (10) we see that only two terms of (18) are nonzero, namely those for d = pa and d = pa−1.
Hence,
1
E(pa1,...,par)= pa cpa1(pa)···cpar(pa)φ(1)+cpa1(pa−1)···cpar(pa−1)φ(p)
1 (cid:0) (cid:1)
= (p−1)pa1−1···(p−1)par−1+(−pa−1)s(p−1)pas+1−1···(p−1)par−1(p−1) ,
pa
and a short com(cid:0)putation gives formula (ii) in Proposition 1. (cid:1)
Consider the function f (m) defined in Corollary 2 (case m =...=m =m). Here f =ε, f =φ.
r 1 r 1 2
Proposition 11. Let r ≥3. The average order of the function f (m) is C mr−1, where
r r
(p−1)h (p)−pr−1
r
C := 1+ . (19)
r pr
p (cid:18) (cid:19)
Y
More exactly, for any 0<ε<1,
C
f (m)= rxr+O(xr−1+ε). (20)
r
r
m≤x
X
Proof. The function f is multiplicative and by (14),
r
∞ f (m) (p−1)h (p)−pr−1
r r
=ζ(s−r+1) 1+
ms ps
m=1 p (cid:18) (cid:19)
X Y
for s ∈ C, Res > r, where the infinite product is absolutely convergent for Res > r − 1. Hence
f =g ∗id in terms of the Dirichlet convolution, where g is multiplicative and for any prime power
r r r−1 r
pa (a∈N),
(p−1)h (p)−pr−1, a=1,
g (pa)= r
r
(0, a≥2.
We obtain
xr g (d) |g (d)|
f (m)= g (d) er−1 = r +O xr−1 r
r r r dr  dr−1 
m≤x d≤x e≤x/d d≤x d≤x
X X X X X
 
and (20) follows by usual estimates.
5
4 The function A
Consider now the function A given by (5).
The next formulae are similar to (15) and (17). The following one was already given in [22, Section
3].
Proposition 12. For any m ,...,m ∈N,
1 r
φ(d )···φ(d )
1 r
A(m ,...,m )= , (21)
1 r
lcm[d ,...,d ]
1 r
d1|m1X,...,dr|mr
Proof. SimilartotheproofofProposition3,thisisobtainedbyinsertinggcd(k,m )= φ(d )
i di|gcd(k,mi) i
(1≤i≤r).
P
Corollary 13. The function A is multiplicative (as a function of several variables).
Proof. According to (21), A is the convolution of the functions G and the constant 1 function, where G
is given by
φ(m )···φ(m )
1 r
G(m ,...,m )= ,
1 r
lcm[m ,...,m ]
1 r
both being multiplicative. Therefore A is also multiplicative.
Proposition 14. For any m ,...,m ∈N,
1 r
1
A(m ,...,m )= gcd(d,m )···gcd(d,m )φ(m/d). (22)
1 r 1 r
m
Xd|m
Proof. Similar to the proof of Proposition 9. Using that gcd(k,m )=gcd(gcd(k,m),m ) (1≤i≤r) we
i i
have
m
1
A(m ,...,m )= gcd(gcd(k,m),m )···gcd(gcd(k,m),m ),
1 r 1 r
m
k=1
X
and by grouping the terms according to the values gcd(k,m)=d we obtain the formula.
Corollary 15. Let p be a prime and let a ,...,a ∈ N. Assume that a := 0 < a ≤ a ≤ ... ≤ a .
1 r 0 1 2 r
Then
1 r aℓ−1
A(pa1,...,par)=pa0+a1+...+ar−1 + 1− pa0+a1+...+aℓ−1 p(r−ℓ)j. (23)
p
(cid:18) (cid:19)Xℓ=1 j=Xaℓ−1
Proof. Let a:=ar. Then lcm[pa1,...,par]=pa. From (22) we obtain
a
1
A(pa1,...,par)= gcd(pj,pa1)···gcd(pj,par)φ(pa−j)
pa
j=0
X
a−1
1
=pa0+a1+...+ar−1 + 1− pmin(j,a1)+...+min(j,ar)−j,
p
(cid:18) (cid:19)j=0
X
where the last sum is
a1−1 a2−1 ar−1
prj−j + pa1+(r−1)j−j +...+ pa1+a2+...+ar−1+1j−j
Xj=0 jX=a1 j=Xar−1
r aℓ−1
= pa0+a1+...+aℓ−1+(r−ℓ)j,
Xℓ=1j=Xaℓ−1
finishing the proof.
6
Application 16. From (21) we have for any m ,...,m ∈N,
1 r
φ(d )···φ(d ) φ(d ) φ(d )
1 r 1 r
A(m ,...,m )≥ = ··· =A(m )···A(m ), (24)
1 r 1 r
d ···d d d
1 r 1 r
d1|m1X,...,dr|mr dX1|m1 dXr|mr
cf. (6), with equality if m ,...,m are pairwise relatively prime.
1 r
Note that if m =...=m =m, then from its definition,
1 r
m
1 1
A (m):=A(m,...,m)= (gcd(k,m))r = drφ(m/d), (25)
r
m m
Xk=1 Xd|m
which is a multiplicative function. An asymptotic formula for A (m) was given by Alladi [3]. See
m≤x r
also [22, Section 2].
P
A simple inequality for the functions E and A is given by
Proposition 17. For any m ,...,m ∈N,
1 r
E(m ,...,m )≤A(m ,...,m ). (26)
1 r 1 r
Proof. Using the inequality |c (k)|≤gcd(k,n), mentioned in the Introduction, we obtain
n
m n
1 1
E(m ,...,m )≤ |c (k)|···|c (k)|≤ gcd(k,m )···gcd(k,m )=A(m ,...,m ).
1 r m m1 mr m 1 r 1 r
k=1 k=1
X X
5 Generalizations
Let f ∈F be a function of two variables and consider the function
2
m
1
F (m ,...,m ):= f(k,m )···f(k,m ). (27)
f 1 r 1 r
m
k=1
X
Proposition 18. ([10, Lemma 5]) If n 7→ f(k,n) is multiplicative for any k ∈ N and k 7→ f(k,n) is
periodic (mod n) for any n∈N, then F is multiplicative.
f
Now suppose that f has the following representation:
f(k,n)= g(d)h(n/d) (k,n∈N) (28)
d|gcXd(k,n)
where g,h ∈ F are arbitrary functions. Functions f defined in this way, as generalizations of the
1
Ramanujan sums, were investigated in [4, 5]. See also Apostol [6, Section 8.3].
Proposition 19. Assume that f is given by (28). Then
i) F has the representations
f
g(d )···g(d )
1 r
F (m ,...,m )= h(m /d )···h(m /d ), (29)
f 1 r 1 1 r r
lcm[d ,...,d ]
1 r
d1|m1X,...,dr|mr
1
F (m ,...,m )= f(gcd(d,m ))···f(gcd(d,m ))φ(m/d). (30)
f 1 r 1 r
m
Xd|m
ii) Ifg andharemultiplicative functions,thenF is multiplicative (asafunctionof severalvariables).
f
7
Proof. i) Follows by the same arguments as in the proofs of Propositions 3, 9, 12 and 14.
ii) Direct consequence of (29).
The functions E and A are recovered choosing g = id, h = µ and g = φ, h = 1, respectively, where
note that gcd(k,n)= φ(d).
d|gcd(k,n)
Now let f(k,n) = f(gcd(k,n)) (k,n ∈ N), where f ∈ F is an arbitrary function. Then f is of type
P 1
(28) with g =f ∗µ, h=1, since f(gcd(k,n))= (f ∗µ)(d).
d|gcd(k,n)
Special choices of f can be considered. For f =id we reobtain the function A. As another example,
P
let f =τ, with g =h=1. Then we obtain
Corollary 20. The function F ∈F is multiplicative and
τ r
1
F (m ,...,m )= , (31)
τ 1 r
lcm[d ,...,d ]
1 r
d1|m1X,...,dr|mr
1
F (m ,...,m )= τ(gcd(d,m ))···τ(gcd(d,m ))φ(m/d). (32)
f 1 r 1 r
m
Xd|m
Further common generalizations of the functions E and A can be given using r-even functions. See
[12, 19, 23] for their definitions and properties.
6 Linear congruences with constraints
A direct generalization of the interpretation of E given in the Introduction is the following. Let M ∈N
and let D (M) (1≤k ≤r) be arbitrary nonempty subsets of the set of (positive) divisors of M. For an
k
integer n let N (M,D ,...,D ) stand for the number of solutions (x ,...,x )∈Zr of the congruence
n 1 r 1 r M
x +...+x ≡n (mod M) (33)
1 r
satisfying gcd(x ,M)∈D ,...,gcd(x ,M)∈D .
1 1 r r
If D ={M/m } (1≤k ≤r) and n=0, then we reobtain the function E.
k k
Special cases of the function N (M,D ,...,D ) were investigated earlier by several authors.
n 1 r
The case D ={1},i.e., gcd(x ,M)=1 (1≤k≤r) was consideredfor the first time by Rademacher
k k
[17] in 1925 and Brauer [7] in 1926. It was recovered by Nicol and Vandiver [16] in 1954, Cohen [8] in
1955, Rearick [18] in 1963, and others. The case r =2 was treated by Alder [1] in 1958.
The general case of arbitrary subsets D was investigated, among others, by McCarthy [11] in 1975
k
and by Spilker [21] in 1996. One has
r
1
N (M,D ,...,D )= c (n) c (d). (34)
n 1 r M M/d M/e
dX|M iY=1e∈DXi(M)
For M =m, D ={m/m } (1≤k ≤r) and n=0 (34) reduces to our Proposition 9.
k k
The proof of formula (34) given in [21, Section 4], see also [12, Ch. 3], uses properties of r-even
functions, Cauchy products and Ramanujan–Fourier expansions of functions.
See Chapter 3 of the book of McCarthy [12] for a survey of this topic.
It is well known that the number of solutions of polynomial congruences can be expressed using
exponentialsums, see for ex. [15, Th. 1.31]. Althoughthe obtainedexpressioncan be easilytransformed
by means of Ramanujan sums in case of linear congruences with side conditions, this is not used in the
literature cited in this Section.
In whatfollows we givea simple directproofof (34) in the caseD (M)={d } (1≤k ≤r), applying
k k
the method of above.
8
Proposition 21. Let M ∈N, n∈Z and let d ,...,d |M. Then
1 r
M
1
N (M,{d },...,{d })= c (k)···c (k)exp(−2πikn/M) (35)
n 1 r M M/d1 M/dr
k=1
X
1
= c (δ)···c (δ)c (n). (36)
M M/d1 M/dr M/δ
δX|M
Proof. Only the simple fact
M
M, M |n,
exp(2πikn/M)= (37)
(0, M ∤n.
k=1
X
and the definition of Ramanujan sums are required. By the definition of N (M,{d },...,{d }),
n 1 r
n
1
N :=N (M,{d },...,{d })= ··· exp(2πik(x +...+x −n)/M)
n 1 r 1 r
M
1≤Xx1≤M 1≤Xxr≤M kX=1
gcd(x1,M)=d1 gcd(x1,M)=dr
M
1
= exp(−2πikn/M) exp(2πikx /M)··· exp(2πikx /M),
1 r
M
Xk=1 1≤Xx1≤M 1≤Xxr≤M
gcd(x1,M)=d1 gcd(xr,M)=dr
and denoting x =d y , gcd(y ,M/d )=1 (1≤k ≤M),
k k k k k
M
1
N = exp(−2πikn/M) exp(2πiky /(M/d ))··· exp(2πiky /(M/d ))
1 1 r r
M
Xk=1 1≤y1X≤M/d1 1≤yrX≤M/dr
gcd(y1,M/d1)=1 gcd(yr,M/dr)=1
M
1
= exp(−2πikn/M)c (k)···c (k)
M M/d1 M/dr
k=1
X
To obtain the second formula use that c (k) = c (gcd(k,M)) and group the terms according
M/di M/di
to the values of gcd(k,M)=δ, cf. proof of Proposition 9.
For n=0 and d =M/m (1≤k ≤r) we reobtain the interpretation of the function E and formula
k k
(17). We remark that the proof of Proposition 21 is a slight simplification of the proof of [13, Lemma
4.1].
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