Directory UMM :Journals:Journal_of_mathematics:OTHER:

13

ON SOME EXTENSIONS OF JORDAN’S

ARITHMETIC FUNCTIONS

by

Dorin Andrica and Mihai Piticari

Abstract. In this paper we introduce and study the arithmetic functions ,J_k(1) and J_k(2)of Jordan's type. The basic theory of Jordan's totient function J_k is reobtained by using some properties of our second function.

Keywords: Jordan totient, greatest common divisor

Mathematics Subject Classification (2000): 11A25,11A05

1. Introduction

An arithmetic function generalizing the well-known Euler totient function ϕis the Jordan's function of order k, where k is a positive integer. This function is denoted by J_k and it is defined by

J

_k

(

n

)

= the number of all vectors (a₁,...,

k

a

)

∈

Z

₊k with the properties a_i ≤n, i= 1,2, . . . , k and gcd(a1,... ,a1,n) = 1.

It is clear that Ji=ϕ. The early history of the function

J

k is presented in [4].

The function

J

_k has some interesting properties and numerous applications. In what follows we recall few of them.

1.The function

J

_k is multiplicative, i.e. for any positive integers m, n with gcd(m,n) = 1 the relation

J

_k

(

mn

)

=

J

_k

(

m

)

J

_k

(

n

)

holds ([7], [8]).

2. If p is a primp and a is a positive integer, then ) ( )

( α = α − kα−

k p p p

14

3. If the unique prime decomposition of n is m

m

p

p

n

α1

K

α

1

=

then

  

 −   

 −

= _k

m k

k k

p p

n n

J ( ) 1 1 1 1

1

K

An easy argument for this formula is the inclusion-exclusion principle (see [7], [8]).

4. (Gauss' type formula) The following formula holds

∑

=

n d

k k d n

J ( ) (see [7] and [8]).

5. The following formula holds

∑

=

n d

k k

k _n

n J

d

d) ( ) (

µ

where n is the Möbius inversion function. That is for all positive integers n

(

)( )

n d

n d d

d n n

J _k

n d

k n

d k

k µ µ =ζ ×µ

     =

     

=

∑

( )

∑

) (

where ζ_k

( )

n =nk and "*" is the Dirichlet convolution defined by

(

)( )

∑



     =

n

d d

n g d f n

g f

/ ) ( *

for any functions f,g : Z₊ →C ([8, pp. 12-13]).

6. Recall that the Riemann

ζ

function is defined by

( )

1 , Re 1

1

> =

∑

∞

=

z n

z

n z ζ

The following formula holds

(

)

( )

( )

, Re 1.

1

> =

−

∑

∞

=

z n

n J z

k z

n z k ζ

15

7. The following asymptotic formula holds ([8, pp. 265-272])

( )

∑

₌

+ ∞

→ = + +

n s

k k

n _n J s _{k k}

1

1 _{( 1) 1}

1 )

( 1

lim

ζ

In the case k=2m-1 we get

∑

₌ − −

∞ →

− =

n s

m m m m

m

n _B

m s

J

n2 ₁ 2 1 2 1 ₂ 2 )! 1 2 ( ) ( 1

lim

π

8. The von Sterneck function Hkis defined by [

∑

]

≤

≤ =

=

n k s s

k n

s s

k

k n s s

H

, , 1 1

,

1 ,

1) ( ) (

) (

K

K

K

where [s1,...., sk] denotes the latest common multiple of integers s1,..., sk. For all

positive integers k the following formula is true ([8. Proposition 1.7, pp. 15]):

k k H

J =

9. The interpretation of the integer Jk(t) in the theory of finite groups is the

following. Consider the Abelian group defined as the cross product

Z

_nk

=

Z

_n

×

K

Z

_n, where (Zn, +) is the well-known group of residues modulo n. Then for t\n we have

(see [11])

} ) ( : {

# )

(t g Z ord g t

J_k = ∈ _nk =

10. Some interesting applications in determining the order of some matrices finite groups are given by

∏

₌

−

= m

k k m

m

n n J n

Z m GL

1 2

) 1 (

) ( )

, (

∏

₌

−

= m

k k m

m

n n J n

Z m SL

2 2

) 1 (

) ( )

, (

∏

₌

= m

k k m

n n J n

Z m Sp

1 2 ( ) )

, 2

( 2

where GL(m,Zn), SL(m,Zn), Sp(2m, Zn) are the general linear group, the special linear

16

in the ring Zn. The first two formulas are obtained by C. Jordan [7] and they are also contained in [1]. The third formula is given in [11]. The multiplicative group G(n) is defined by

} 1 ,

, , : {

)

( ∈ − =±

   

= α_{γ δ}β α β γ δ Z andαδ βγ

n

G _n

For any positive integer n≥3the order of G(n) is given by |G(n)| = 2nJ2(n).

11. Other applications of the Jordan's function J2 are given in Diophantine Anal-ysis (see [3]). Some special properties of Jkare obtained in the paper [5], [6] and [10].

There are few generalizations of Jordan's totient function. We mention here the recent one given in [12] and defined by

1 ) , , , gcd(

, , 1

1 1

1 )

(

= ≤ ≤

∑

=

k a a

n a a k

m

m m

n S

K K

where m and k are fixed positive integers. It is dear that. S_kk(n)=J_k(n).

In this paper we introduce two functions of Jordan' type and we make the con-nection with the function Jk. The basic theory for Jordan's function Jk is reobtainedby

using our second function.

2. The arithmetic function J_k(1)

For a fixed positive integer k define ₁₄₂K₄₃

times k

k _{Z Z}

Z

1 1

+ + +

+

+ = × × and consider the sets

(

)

(

, ,

)

1}

gcd

1 : ,

{ ) (

1 1

1 1

1 1

1 1

=

≤ ≤ ≤ ≤ ∈

=

+

+ +

+ + +

n a a

and n a a

Z a a n M

k

k k

k k

K

K K

(

)

(

, ,

)

1} gcd

1 : ,

, { ) (

1

1 1

1 1

=

≤ ≤ ≤ ≤ ∈

= ++ +

n a a

and n a a

Z n a a n N

k

k k

k k

K

K K

17

( )

# ₁( ) (1)

( )

# ( )

1 n M n and J n N n

F_k+ = _{k+ k} = _k

It is clear that for k = 1 we obtainJ₁(1) = J₁ =φ, the well-known Euler totient function.

The Gauss' type formula for the function

J

_k(1)isgiven in Theorem 2.1. The following formula holds

∑

=_ + − _ n d k k k n d

J(1)( ) 1 (2.1) Proof. First of all let us note the following relation

( )

n F n J

( )

n

F_k+1 = _k+1( −1) + _k(1) (2.2) Consider the set

(

)

(

,

)

} gcd 1 : , { ) ( 1 1 1 1 1 1 1 d a a and n a a Z a a n S k k k k d = ≤ ≤ ≤ ≤ ∈ = + + + + + K K K

We have the relations

∑

∑

₌ + =         = =     + + n s k n d d d n F n S k k n 1 1 1 ) ( #

1 (2.3)

Replacing n by n-1 in the above relation we get

∑

−₌ +  −  =     + − + 1 1 1 1 1 1 n s k s n F k k n (2.4) From (2.3) and (2.4) and then by using (2.2) it follows

∑

_ __ __− _ −_{ }_{} =     + − + −     + + =     + − + + + + n d k k k d n F d n F F k k n k k n k k n / 1 1 1 1 1 1 1 1 1

( )

( )

(

)

∑

( )

∑

− − = = + + n d k n d k

k d F d J d

F / ) 1 ( / 1 1 1

For k = 1. from (2.1) we obtain the classical Gauss' formula.

Theorem 2.2. Fur any positive, integer k

≥

2 the following relation is satisfied

( )

∑

₌ − = 2 ) 1 ( 1 ) ( m k

k n J m

F (2.5)

18

( )

∑



     =

n d

k

k f d

d n n

J

/ )

1

( _{( )} µ _(2.6)

where









+

−

=

k

k

m

)

m

(

f

_k

1

i.e. F_k(1)(n)= f_k *µ. Using the relations (2.2) and (2.6) the formula (2.5) quickly follows.

Theorem 2.3. The following formula holds

( )

∑

∑



  

= − = <

= s

n m

k n

s n s

k n J m

J

1 ) 1 (

1 1 ) , gcd( )

1 (

)

( (2.7)

Proof. We have

( )

∑

∑

∑

∑



  

= − = < =

< =

< =

  

    =

=

= s

n m

k n

s n s k

n s

n s S

n s

n s k

k J m

s n F n

S n

N n J

1 ) 1 (

1 1 ) , gcd( 1

) , gcd( 1

) , gcd( )

1 (

) ( # )

( # ) (

and the formula is proved.

Corollary 2.4. If n is a prime, then

( )

(

2

)

2 1 2 1

1

1

− + =

∑

∑



  

= −

=

n n m

s n m n s

ϕ (2.8)

Proof. Consider k = 2 in (2.7).

3. The arithmetic function Jk(2)and the connection to Jordan's function

J

_k Consider the set

(

,

)

:gcd

(

, ,

)

1} {

)

(n = a₁ a ∈Z₊ a₁ a n =

P_k K _k k K _k

and define

G

_k

(

n

)

=

#

P

_k

(

n

)

. Let us define the integer

(

,

)

: }

{ # )

( ₁

) 2

( _{n a a P atleast acomponenta isequal n}

J_k = K _k ∈ _{k j}

19

Theorem 3.1. The fallowing formula holds

( )

∑

= − −

n d

k k

k d n n

J(2)( ) 1 (3.1)

Proof. Note that the following relation is valid

( )

n G n J

( )

n

G_k = _k( −1) + _k(2) (3.2)

Consider the set

(

,

)

:1 gcd

(

,

)

}

{ )

(n a₁ a Z a₁ a nand a₁ a d

N_d = K _k ∈ ₊k ≤ ≤K≤ _k ≤ K _k =

and obtain

∑

∑

₌ = _{= }_ __

= n

s k n

s s k

s n G n

N n

1 1

) (

# (3.3)

Replacing n by n - 1 we get

(

)

∑

= 

 

     − =

− n

s k k

s n G n

1

1

1 (3.4)

It follows

( )

(

( )

( )

)

k

( )

k

n d

k k

n d

k d G d G d n n

J 1 1

/ /

) 2

( =

∑

− − = − −

∑

Remarks.

1) If k = 2, then

( )

( )

( )

  

= =

−

> =

1 1

1 2

1 2

) 2 (

n if n

n if n

n J_k

and from relation (3.1) we obtain

( )

2

( )

1 2

( )

12 2 1

/ /

) 2

( =

∑

− = − − = −

∑

J d d n n n

n d n

d k

that is the classical Gauss' formula for Eulers totient function.

20

( )

∑

_n d d f n S / )

( . It is easy to see that if f is multiplicative then S is multiplicative. From formulas (2.1) and (3.1) the summation functions of

J

_k(1)and

J

_k(2) are not multiplicative, hence these functions are not multiplicative.

Theorem 3.2. The fallowing formla holds

( )

∑

=

+

= n

m

k n J m

G k 2 ) 2 ( ) (

1 (3.5)

Proof. Applying the Möbius inversion formula, from (3.1) we obtain

( )

∑

      = n d k

k g d

d n n J / ) 2 ( ) (

µ (3.6)

where gk(m) = mk - (m - 1 )k. That is

J

_k(2)

=

g

_k

*

µ

. From (3.6) and (3.2) it follows

relation (3.5).

The connection between J_k(2) and the Jordan's functions Jiis given by

Theorem 3.3.The following relation holds

( )

( )

∑

+₌ +  +−      + − = 1 1 1 1 ) 2

( _{( ) 1} 1

k s

s k s

k J m

s k n

J (3.7)

Proof. Note that we can write

( )

(

)

( )

( )

( )

 =          + − − =       − − = =       − −       = − −       =

∑∑

∑

∑

∑

∑

− + + = + + + + + + + + n d s k k m k k n d k k n

d d n

k k n d k k k d d n s k n J d n d n J d n d d d n d d d n n J / 1 1 0 1 1 / 1 1 / / 1 1 / 1 1 ) 1 ( 1 1 ) ( 1 ) ( 1 1 ) ( µ µ µ µ µ µ

( )

( )

∑

+₌ − +  +  +− = 1 1 1 1 1 1 k S s k s n J s k Remarks.

21

(

)

1

1 1, , + ∈ ++

k

k Z

a

a K such that 1≤a₁ ≤K≤a_k+1 ≤n,gcd

(

a₁,Ka_k+1

)

=1 and n is a component of the vector

(

a

₁

,

K

a

_k+1

)

at least once. Also, consider the sets M, consisting in all vectors

(

a₁,Ka_k+1

)

∈Z₊k+1,

(

,

)

1

gcd ,

1≤a₁ ≤K≤a_k+1 ≤n a₁ Ka_k+1 = , and n is the sth component of the vector, s = 1,2,..., k+1.

It is clear that

U

1

1 +

=

=

k

s s

M

M

and from inclusion-exclusion principle we have

I

+K

−

=

∑

∑

+ ≤ ≤ ≤ +

= i j k i j

k s

s M M

M

M # #

#

1 1 1

1 That is

( )



( )

+K

     + −       +

= k J n k J n

n

J_{k k k}

1 1 1

1 )

( ) 2 (

i.e. the connection between

J

_k(2)and Jordan's functions given in the formula (3.7).

2) Consider

n

=

p

₁α1

K

p

₁α1 the prime factorization of n. Using formula (3.7) and a simple mathematical induction argument it follows

  

 −   

 − =      

=

∑

k

m k

k n

d

k k

p p

n d d n n

J ( ) 1 1 1 1

1 /

) 2

( µ _{K (3.8)}

From formula (3.8) we deduce immediately that the Jordan function Js is

mul-tiplicative.

Also, by using formula (3.8) and Möbius inversion formula weobtain the Gauss' formula for Jk, i.e.

k n

d

k d n

J =

∑

_/ ( )

References

[1] Cohen, E.. Theorie des nombres, Herman, Paris, 1914.

22

[3]Dajani. K., Kraaikamp, C., A Note on the Approximation by Continued Fractions under an Extra Condition, New York ,1. Math. 3A(1998), 69-80.

[4] Dickson. L. E., History of the Theory of Numbers. Chelsea Publishing Company, New York. 1971, vol. 1.

[5] Haukkanen, P., Some. Limits Involving Jordan's Function and the Divisor Function, Octogon Math. Mag., Vol. 3. No. 2(1995), 8-11.

[6] Haukkanen, P., Wang,J., A Generalization of Menon's Identity woth Respect to a Set of Polynomials, Portulagiae Matematica, Vol. 53, Fase. 3 - 1996, 331-337.

[7] Jordan, C., Traitee. substitutions et des equations algebriques, Gauthier-Yillars. Paris, 1957.

[8] McCarthy, P. J., Introduction to Arithmetical Functions, Springer-Verlag, New York, 1986.

[9] Niveu. I., Zuckerman. H. S., Montgomery, H. L., An Introduction in the Theory of Numbers. John Wiley & Sons, Inc., New York – Chichester - Brisbane Toronto Singapore, 1991.

[10] Sandor, J., Note, on Jordan's arithmetical function. Octogon Math. Mag. 1(1993), 1-4.

[11] Schulte, J., Uber die Jordansche Verallgemeinerunq der Eulerschen Funktion, Resultate der Mathemalik, Vol. 36, No. 3/4, 354-364, 1999.

[12] Shonhiwa, T., A generalization of the Euler and Jordan totient, functions. The Fibonacci Quarterly, Vol. 37(1999), No. 1, 67-76.

Authors:

Dorin Andrica – “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Cluj Napoca, Romania, e-mail: dandrica@math.ubbcluj.ro