: ll*llll"ll 1 : fl

Annales Academia Scientiarum Fennicre Series A. I. Mathematica

Volumen 15, 1990 , ^37-52

SOME FUFUIHER ARITHMETICAL IDENTITIES

INVOLVING A GENERALIZATION

OF RAMANUJAN'S ^SUM

Pentti

Haukkanen

1. Introduction

Let G

be a commutative semigroup

with identity 1, with

respect

to

a mul- tiplication denoted by

juxtaposition.

Suppose there exists a

finite or

countable infinite set

P(e G)

of primes ^such that each

n € G

^{can be} represented uniquely

in

the form

,r:

_p€P

fl

^o,,(n),

where the exponents

n(p)

^are non-negative integers of which all but a finite num- ber are zero. (Define

1(p):0 for all p € P.)

F\rrther, suppose there exists a real-valued

norm

_ll ^.

ll

^{defined on G such that}

(i) _lllll : _1,

_llpll

> 1 ^(p

^e

^P),

(ii)

_llmnll

: ll*llll"ll ^{(m, n}

^{e G),}

(iii)

the set

{n

^e

G:

_llnll

< r}

^{is finite} for all real numbers

s.

Then

G

is called _[18,

p.

L1] an arithmetical semigroup. Throughout this paper elements

in

an arbitrary arithmetical semigroup a.re typed in boldface.

Let

G be an arbitrary but fixed arithmetical semigroup. By an arithmetical function we mean a complex-valued function defined on the arithmetical semigroup

G. Let L

be a mapping from the set G into the set of subsets of G such that for each

n

€

G,

^.4(n) is a subset of the set of divisors

of n.

Then the A-convolution of two arithmetical functions

/

^and

^g

is defined by

(f eil: » /(d)e(n/a).

d€,{(n)

In this paper ^we confine ourselves to regular convolutions and we shall assume the reader to be familiar with this notion (see e.g. _[13, Section 7.31,122, Chapter 4],

[27]).

^{For example,}

^the

^Dirichlet convolution

D,

where

D(n) is the

set

of

all divisors

of n,

and the unitary convolutiou

[/,

where

U(n): {a: aln,(d,n/d)

: 1),

are regular.

doi:10.5186/aasfm.1990.1513

38 Pentti Haukl<anen For a positive integer k , ^{we define}

Ax(n): _{d

e

G:

d& e

.A(n};}.

It

has been shown _{(see [13,}

p.

10], _[32,

p.

267]) that the:4,7,-convolution is regular whenever the l.-convolution is regular. The symbol

(., b)a,r

denotes the greatest

kth

power divisor

of a

which belongs

to

.a(b).

The classical Ramanujan's sum

C(n;r)

is defined by

C(n;r): »

exp(2rimnf r),

?,*-1'=?

where

n is a

non-negative integer and

r is a

positive

integer. Its

well-known arithmetical representation is given by

C(n;r): \

^apg1a1.

dl(n,r)

In

_[14] the author together with P.J. McCarthy defined a generalized Ramanujan's sum by

C.e,,*(nr,...,n6r) - »

exp(zri(m1nt

*...lmunu)f

^rk),

ii;,;;!ll.Yi)

where r7Ls...,nu a,re non-negative integers,

r is

a positive integer and

(m;) : (*rr.

^{. .}

_,mr),

the greatest common divisor of m1r. ^.

.tffiu.

We noted _[14] that

Ce,x(ur...,tflr) r)- »

dkeÄ(((ni),r&)a,r)

where

lter ^{is the}

^{inverse of.}

E, the function = ^{L, with}

^respect

^{to the}

^ä7.-

convolution. For an arithmetical semigroup this suggests we define

dk" FA.Q ld)

- »

^d,k"

^ttArU

^ld),

de Ar(r)

d* l(";)

ca,r(nr,...,Dr;r)- » lldllr"tro*(r/d).

de €A(((ni),re)a,r)

In

_{[13] we} defined a generalized Ramanujan's sum by

» f@)g("/d).

sfÅ:r('r,"',ru;r) -

dk €A(((";),r&)a,*)

Arithmetical identities involving a generalization

of

Ramarrujan's sum 39

In

other ^words,

SfÅi(t, _).

.., rr;r) -

(1) /(t,,frz,...,tr)A(

-»

/(d)g(, ld) - ^(x((r,);

^(.)n)

^f

^oks)

^(r), i'-tl:;'

where

X(t;d) :

1

if dln,

and

:0

otherwise.

The purpose

of

_{[13] was}

to

derive arithmetical identities

of

classical type involving that sum. The ^purpose of the present paper is to give more arithmetical identities for

that

sum. We shall ^also

list

a large number of known special ^cases of the given identities.

At

the end of this paper ^we shall note

that

two identities here can be extended to

totally

A-b-even functions

(modr).

2. Preliminaries

We define an arithmetical function

/

to be quasi-A-multiplicative _[13,

p.

1a]

if _/(1) l0

^and

/(r)/(mn): ^/(m)/(n)

^whenever

^{m,n € A(mn).}

^Quasi-.A-

multiplicative functions

/ ^with /(1) : L

are called A-muliiplicative [45].

It

^is

easy to see that an arithmetical function

/ ^with

f (L)

*

^{0 is} quasi-,4,-multiplicative

if,

and only

if, f lfO) ^is

A-multiplicative. Quasi-[/-multiplicative ^{functions are} called quasi-multiplicative [19]. For ^those functions

/(L) I ⁰

^and

/(l)/(mn):

/(*)/(r)

^whenever

(*,r): 1. All

quasi-,A-multiplicative functions are quasi- multiplicative.

Let

A(1)

,

^A@),

...,

A@) be regular convolutions. Then we define 65u

a(t) aQ)

....4(')-convolution _of arithmetical functions.f(nr,

rl2r...,

nr,) and g(nr, nz, .. .,

nr)

^bY

1) eQ) . ., _6(") g(r11 _r flz t. ^. ., nrr)

» »

^{/(.1, ,dz). . .,}

^dr).

drۀ(1)(nr) d2eAQ)(n) d,6A(")1n.;

'

g(\

_I d1, nz I ^{d2, . .} ., nu I ^du).

It

is easy to see

that an

a(1) aQ) ^{. . .} a@) -convolution of arithmetical functions ^is associative. Also,

if ä

is an ,4(i)-multiplicative function, then

(2) (/("r, rt2t...,nu)eo) aQ)'''

r(")91nr,

rr2;...,n,))å(n1)

: _/(nr,

rt2t . . ., n,)ä(n6),a(r) ^AQ) .. .

,(")g(*r,

tt2t . . .,

nr)h(n;).

Let /

be an arithmetical function of one rariable and

e, ffi,

u

e N,

u

)

^2,

0(-m < u.

Then we define

P"(/Xnr,...,r-irlrn*tr...,nr)

to be the arithmeti- cal function

of u

variables such that

(

f(n"), ^if

ⁿ¹

: "' ;

^Irrn

: (n-+r)"

(3) P"("fXnr,...r rmirm*rr...,nu): { :

' ' '

: (tr)",

(

o

^otherwise.

40 Pentti Hauk]ranen

Iu particular, we denote

Pr(/Xnr,...,

rrni

tlrn*1r..., nr) : P("fXnr,...,nr).

If m:0,

then we have

(4) P"(/Xnr,...,rrninrn*r:...,u,) : P("f)(nr,...,n,).

We note that some special cases of the function .P"(/)

"*

^{be found}

ⁱⁿ

^{[39, p. 627!}

and _[42, p. 86].

It

is easy

to

see

that if _{f ,} g

are arithmetical functions of one variable and

u)2,1<i<u,then

(5) 9(q)P(/)(nLswzs..

^. ,

n*) :

P(/s)(n1 ,n2,. ^.. , n,,).

Also,if f ,g

^arc arithmeticalfunctionsof onevariable,

T< j< u

and

Au)(n) 9,4(,)(n)

whenever

n

€

G,

L

<

s

< u, i+j,then

(6) P(.fXrr, rt2t...,

n*)a(l)a(z)

-..

a@)p1nXrr,

rr,...,n,) : _{P(f ,(i)}

gXor, ns, _{. . . ,}

tr,).

3. Ideutities

Theorem L.

Suppose

I

^eurd

^g

^are a.rithmetical functions and 0

I m I

u.

Thenfar nlr...,[*, r€G

S{,1(rrr,

...

_)ttn)

(r*+r)0,. .., (nu)e;r)

: E(nr)...E(n")g(r)o.

^-.

».tpPp(/Xrr,. ..,

^D"ni

rrn*r,. .., ru,r),

where.E(n)

: 1

for aJl

n €

G.

Proof. Bv (1) and (3),

t(*r)...

E(n")g(") D -.

.DAxPx(f)(*,

_).

..

tfrmifrrn*17.. .,

rr, r)

_ »... » » E(*, ldr).."8(nuld*)geld)

dr Jrr ^d,. Iro ^{d €.4r} (r)

'

Pp(/)(dr, . . .,

d*i

d*+1, . . -,

d*,d)

- f

_L

geld)Px(/)(d0,...,d*id,...d,d)

d€-4n ("),d It r,+1,...,nu a* _[r,t r...rDnz

: o"*;Li'::ljj »

_'*"

^g$ld)/(d)

- S*!*(rrr?,..rn*,

(Drr+r)k,

...) (ro)u;")

_, which was to be proved.

Arithmetical identities involving

a

generalization of Ramanuja.n's

sum

⁴¹

Theorem 2.

suppose ht, . . _{. ,}

hu

^{are quasi- D} -multiplicative functions, h is ^a quasi- A1, -multiplicative function

ffid f

^{, g} ,

H

are arbitrary arithmetical functions.

Then for rltr

tl2r...,

^nu,

r

€ ^G

(t , ^{. .}

.ä,hX1)

oä", s*'gk(O\ld)0,.

^. . ,

(n"/d)e;r/d)

dl('r)

.

ä1(n1ld)'''

h*(n* _I

d)ä(r/d)Ir(d)

: »

^h1(n1ld)

^... h,(n*ld)(lzs)(r/a)($n' "'

ä"1?).4åIr)(d).

d€-4*(r)

dl('r)

Proof.

It

sufrces

to

consider the case

är(1): "': I,"(1): h(1):1'

^Let

.L denote the left-hand ^side of the identity

in

Theorem

2.

Then,

by

(1)-(6) and Theorem 1,

We thus arrive at our result.

Theorem 3.

Suppose

H is a

quasi-Ap-multiplicative function,

Ht,

Hz,

..., ^Eo

^a,re quasi-D-multiplicative functions ^and

f , 9, h, hr, h2, ..., hu

are arbiftary arithmetical functions. ^Then for ¹¹¹ , r12, . . .,

[r, t e G, att.

_{.. ,au}

: 0rl

z,

: _(a(nr

₎ ^{. . .}

E(n*)s(r)

o ^{. .}

. oepP(/Xrr,

^r12 : . ^. ., rr,,

"))

.

är(nr)

^{. -.}

h*(n,)h(")r. "

oepP(H)(rrrr n2,..

.,n*,r)

: (nrlrrr) ...h*(n*)(h.qxr)p

^.

''

oe1,h1(n1)'

''h,(n,)h(r)

.

P(.fXrr, D2s...,r,,r))

D

"' DAlP(EXrr,rtz,..., tr,,r)

: _(nrlrrr;

^{. ..}

h,(n,)(h.cxr)D "'

^oe1,P(f

^h1'''

^{h,å,)(n 1sn2s.. ., rr,,}

^*))

' D

"'DApP(äXnr, rl2:..., ^n,, r)

: är(nr)''' h"(n")(hsx")P "'

^DA*

.

(r$U,.

^..

h*h)(n1,r^zt..., n,,r)D'''

o,+1,P(HXrr,

rr,. .', t,,

"))

: är(nr) "'h*(n,)(h.gxr)a "'oe*P((f

fu

"'huh)e*Il)(n1, ^rt2t...,

^tr,,

r)'

42 Pentti Haukl<anen

(Hr' " ^Huäx1) » "' » » s*':r(af"',...,d1,""

_;6)

ar

lnr

^d., ln" 6gar (")

är(dr)..

^.

Hu(d")ä(6)hr(n1/d, )..

^.hu(*,,

Id,)n$I6)

(f

H)(d)ä,

(a*'-41 ) ^{. . . Hu(a*L-au} ) @or(g

u)) (*/d)

dۀr (r) a* l(nf "')

.

(HrDht)(*,

_{ldrl-ot )} ^{. .}

.(HuDhu)(rr,

ldn'-"" _).

Proof. It

^suffi.ces

to

consider the case

I/r(1) _ ... _ H"(L)

Further,without1ossofgenera1itywemayaSSumea1

Theorem 3 can be written as

1.

in the

ä(1) -

0,

^am*l

identity

h(rr)

^{. . .}

hr(*r)ä(")

D . . .

DAkS*!r(rr,

. H1

(rr).

^{. .}

H"(n")ff(r).

,,frm,

(t*+r)k,

.. . _,

(rr)*;")

Thus, applying Theorem 1 and formulas (2), (B), we have the theorem.

Remark.

The functions ä1 , ^{. .}

.,

hu,

h

^axe of great importance in Theorem B.

In

fact,

if h : Es,

defined bV

Eo(f) : 1

and Eo(n)

: 0 for n * ^L,

^{then the}

summation over d1

,

^{, .}

., dr,

6reduces

to

the summation over d1

,

^{, .}

., dr.

A similar reduction is valid with respect to any subset of the set of functions ä1

,

^{. . . ,}

hu,

h.

Notation.

For an arithmetical function

/,

^denote

f ^{n (A,,e;} n)

- /(")

@ € R),

ll'r ll <,

n €A(ne)

f"(*) - f"(D,€in) - ^(,

^{e R).}

1m u)

are arithmetical ,U €R), llrn*l r,..)frut

Theorem 4.

Suppose

f

,

g, h, ht,

r€ G, &!t..., a,m:0,

¹

» ^/(,,)

ll'll <,

hz, ..

., h*

(0

0 (*t) ...,

rrn

(7)

s*io(rf"'

_,

lli, ll

1tr

_lli* ll 1r,n _lUllSy

,iX* rfl**lr...rr.r;j)

(B)

Arithmetical identities involving ^a generaJization of Ramanuja,n's

sum

⁴³

'

hi

_@,

I

lli,

ll)''' ^tri(**l lli*ll)ä"(a*,i;v I

^llill)

_ » /(d) (hrDE)"

^(*

,l lldlt*'-'l

)

lld lle (min {yr ,*1"' ,...,x!-l* } do lt rn*Lr...rDu

. . . ₍

h*oE)"

(*

*/

lld ll*'- "*

)U,

^{Aks)" (Ä* ,, d; y llld ll) ,}

» "' » s*:r(tf"',''',iI:*''n**1:''',"';")

rri'lr

s'r

''ä'ä"Jt

tli, ll)

" ^'tri(**/ lli*ll)

- » ^{/(d )g(, ldXä,}

^{DE)" (,}

^,l

lldllkL-" ) ^{. . .}

":llf#l,f,i" ;oåiä'

Proof.

Let L

^denote the left-hand ^side of (7). Then, using the notation of

x

given

in

the introduction, ^{we can} write

L-»...»»

Ili,ll

sr,

^{lli,-ll 10,n} llill<u

orr_,1,il:j: t,...,n7

./(d)g6ld) » h,(b,)... » h*(b*) » ^ä(").

llbl ll<c1lllil

ll

llb-

ll<z-llli-ll ll.lfl{lill

Now we shall change the order of summation.

It

can be proved that the rule

("r,.

^. . _{tcmty td,} br, . . .,

b-,

a) ^--+ (c1/b rr. ^. .,

c*lb*rv

_{/ ard,}

bl,

^{. .} .,

b-,

a) defines abijection from the set of (2rn+3)-tuples

("r,... ,cm,Y,d,br,...,b-,a)

satisfying

ll"rll S tt;.

^. ., 11"-ll

1 t*, _ll"ll

S ^y,

d

e

.46(v), dr'-o' _l"r,

^{. . .}

,6&'-"*

1c-, b1lcr61-&1-",..

-,b*|"*d-nt-"-,

^{a e A*(v}

/d)

onto the set

of

(2m

*

3)-tuples (ir, . . .,

i*, j,

d, br, ^{. .} .,

b-, a)

satisfying

x((,,**1:..., ^t r);d*)

lli,

ll<117...,11i*ll 1n*, _lljll ^1y,

44 Pentti Haukkanen

d

€

,at6),

det-"t lil ,...,d*t-cln.,

li*,

llb,

ll 1xrlllirll,...,llb*ll sr*llli*ll, ll"ll <yllljll, a€A*0*).

Thus we obtain

L- » » » » x((t,,*r,...,n,.)iao)/(a)

ll"rll(r' llc-llS,-,,r,,=r"rr_"tjlj:l,,*

h{b1)... »

t- o1

br, l"rrrd- ^e L- arn

h*(b*) » h@)s((.,/a)1")

»

d-&

br l"r ^a€Ar Q ld)

X((nnz*r, ^{. .} ., r,.);

d*)f(a)

_»...»»»

ll"rll<r,

^llc*ll(c_

"r,,=,

o.r_"1ij:j:I,...,_

. (fu o E) (c14-&r- ^"r ) ^{. . .} @* ^o a) (",,,

6-r'

^{- "}

^*

₎ (h e,'/.)O _/ d).

Further,

it

can be proved that the rule

("r,

^{. . .} _,e

^rtrd)

^-+ (e1dfr'-o' _, . . . _r

€-d&l-o-

, td, d)

defines.a bijection from the set of (rn

*

^2)-tuples

("r,... ,e*,t d)

^satisfying

lldllo

< *ir,{uft,

^*1"' ,. ^{. .}

,*li^ _},

ll"rll <rrllldlle'-"',...,11"-ll _< *^/lldll*'-"* , ^lltll Sylllall, t€L&(td),

onto the set

of

(rn

+

2)-tuples

("r,... _tcrn,ysd)

satisfying

ll"rll < &.,t...,11"-ll lomt _ll"ll Sy, ^de,41(v), 6&1-o11"r,...,6&'-"-;c_.

Thus

L_ _{» x((nm*l}

_1. .

.1n,-);d*)f(a)

lldll& (min _{yk ,rl"t ,...tt#," }

» (fu»EX"') "' » (h*DE)("-)

It.' ll 1r _t /lldllo' ^- "'

. »

^Qree.ext)

lltll< ^y /lldll

t 6 Ao 1ta;

11.,, ll <, ,- llld ll ^{e L} - ^{a rn}

/(dXä,

^DE)n(*, llldll*1-o1 ) ^{. . .} lldll& (min _{yu ,*r"' s...st** }

do lrr-*r r...,ru

.

(h*,DE)^(, *llldll

^kL-o* X he1,s)"(A*, d; v llldll).

This proves

(7).The

proof of (8) goes through on similar lines.

Arithmetical identities involving

a

generalization of Ramanujan's

sum

⁴⁵

Theorem 5.

^Suppose

hr, hz, ..., h* ^{(0 <} * 1u)

^are quasi'D-multipli- cative functions,

h

is

a

quasi-Ap-multiplicative functioa

ffid _f , ^g

^{ate arbitraty}

a,rithmeticalfunctions. Thenfor

tLt ...t

^om,,y

)0 (q,...;onty € R)t an*rt ,.,, \u, t

e

G, dtt... ra* : 0rl

llir ^ll

Sr,

lli".ll ^1x,n llj llSY

.hr(ir)

-..

,h-(i-)

^ä6)

I ^/(d)ä(d) fu(dk'-"') "'h*(dtr'-"^

)

llall& §rnin{ sk,*f "t , ...,rfi* I d* lt **1r.,.rru

.

hl (*, llldll*'-"'

) ^{. . .}

hh@*

llldllo'-orn

)kD" (l*,

^{d; y}

I

lldll),

(10)

(ä,

...ä-X1) » »

stÅg:u$|"'

,... ,ilå^, rrn*r,..., n,,ir)

lli,

ll<rr

lli_ll<o_

.är(ir)...h*(i^)

: » f(d)s(rld)hr(6t'-").'.ä-1dt'-'-,

llalll <-i.{,f ^"1 ,...,,#^ t

dL ln-11,...,n';d€A* (r)

.

äl(lrr/lldllo'-"') .'.hi(r-llldlln'-'-), m27.

Theorem 5 can be proved

in

a similar way to Theorem 4.

Theorem 6.

^Suppose

ztt zzt ...,

^zs

€ C (1 ( _s ( _u)

and denote

zt*

zz*...*2": z. Let f beanarithmeticalfunctionsuchthat f(m)*0

^foraJl

m€G.Thenforn,r€G

nf' ,F.

Si,i*o*

(*,

₎ . . ., hu;

r) -

s'o',trtltA1'(', _: lls*1 7''', ^{[?ri dr} )''' dr,...'d. ۀ* (r)

[d1,...,d!l=t

.

Sli,'r''oo(ne, nsarr ^{. .} .,

nri dr),

where the symbol

[...] i"

used for the ^least common multiple.

Proof. Let .B(r)

denote the right-hand side of the identity

of

Theorem 6.

46

Then

Pentti Haukkanen

» rr(d)- »

S'o,'rtqA7'(rr, _r

[s*1,

^. . .,

[ri dr)

^{. . .} d€A&(r) d€Al(r) dr,...,d,€är(d)

[d1, " ',d6 ]=d

' s'i.,"0''o*(trs, ne+t,

'''

^,

^n,ridr)

il »

s'i.,to'ro'(ny,

n,.u1,..., n,idj)

j=1 _d; §ä1(r)

: fI

I

x(("i, ^rs*r,

^{. . .} ,

n,)i ,k)f

ⁱ

(r) :

_x((n;)

_{;ru)f '(r),}

i=t that

is,

(nerE)(r) : x((r,); r&).f'("),

where X is the function defined in the iutroduction. Thus

.R(r)

: (x((",); (.)u)l'ootro*) (") : Sr;,:{"*(rr,..., n,ir).

This completes the proof.

Remark. It

is easy to see that the value of the sum

sj,l(rrr,...,n,ir)

is independent of the order of the variables n1 _{, .} . . , Du

.

Thus Theorems 1, 4, 5 and 6 can be further generalized by rearranging'the

first u

lariables into an arbitrary order.

Theorem 7.

Suppose g

, ^h

^a,nd

H

^a,re arithmetical functions such that h,

H

^a,re quasi-Ap-multiplicative and he1,gH

: (hgH)(L)Es,

where Eo(1)

:

^'J- and

Eo(n) : 0 for n + l. Let f

^be

^an

^arbitrary a,rithmetical

function.

Then for

r€Gro:0,

¹

(11) » St;lo(@n";r/d)å.(d)Il(r/d):e(r)(/ä)(m)ä(mk'-")

dۀe(r)

if r-

rnk,--"+l

,rn

^€

^.46(r),

^and,

- 0

^otherwise.

Proof. Denote

by L

the left-hand side of (11). Then

L-» » f?)g(rl(dd)) h(d)H("/d).

d€.a*(r) 6ee*G/d)

6'* _;6u "

Arithmetical identities involving ^a generalization of Ramanujan's

sum

⁴⁷

It

can be proved that

de.4*(r), ⁶ eA1,(rld),

^6k14r'

if,

and only if,

6

e.41(r),

6&1-"+r

€.41(r), d:6å'-oe, eeA1,(r/6*'-"*1).

Therefore

T

»»

6e ao

(")

_{ee Ax$ /6o1-} ^{" +1)}

6*t-o*1€Ar(r)

f(6)g(("/C*'-o+t)/") h(60'

e)H (,

l(6*'-" "))

: \- ^gäX6) h(60'-"),,

-L Wfffn^r tr)(rl6k'-"+r1

6eA*(") 6*r_ ^" *,

a o* 1,y

: » UH)(6)h(60'-")n(t)Er(r/ak'-"+t;.

6ee.(') 6*t-" *'€,4* (")

We thus arrive at our result.

Theorem 8.

Suppose

f ^{is a}

quasi-Ap-multiplicative function and

a, b,

r€G ^witha, be -41(r).

?åen

sr;:{- (("/a)*;*)

sr;:r"- ((

rlb)*;d) :

{ {,r)/(") ',lri}l'

d€,,a.3 (r)

Theorem 9.

Suppose

f ^{is a}

quasi-Ap-multiplicative function such that

f(r) * ⁰

^{for aJl}

^r

^e

^G.

Then foraJ.l

n, r

e

G

and integers

a,

^b

»

Sro,{"r

6k ;r1sl,{o

^(n;

^r/d) : /'(r)/(1)u(1"-u o,tro-X6)/å-"(6),

d€Ar(r)

where 6ft

: (r,

rk),+,x.

Theorem 10.

Suppose

/ ^{is an}

Ax-multiplicative function

with

(f elp'a*)

(") I

^{O for aJI}

reG.

^Thenfor

^{all nt, n, r}

^{€ G}

.L _(f nuFAxXd)

(f er

FA.X"/6)

dۀr (r)

if (r,

^rk

_)o,r -

^{(m, rfr )o,*}

:

6r

,

^and

- 0

^otherwise.

48 Pentti Haukkanen

Theorem 11.

Suppose

f

^is

^a

quasi-Ap-multiplicative function ar'd

n, r

e

G.

Denote

a:rf1ar(r), be: (r,Z(r)o)o,1,,where 1,Ab(L):L

^ar.dfor

r{1,

7e*(r)

^{is the product}

^of

distinct prime divisors of

r.

Then

ItlSt;i{r

(a&n;r)

:

_f

(a)per(tor!)) po*(b)(/a"ra*

_Xb).

By

quasi-multiplicativity _Theorems 8-1L can be proved

by

considering the case in which

r

is a prime power. We omit the details.

Theorem

12. Suppose

f

,

g,

h and

H

are adthmeticalfunctions and

n

e G.

Let w

denote the arithmetical function such

that

u.,(f)

: ₀

andfor

r t' ^1,

^ur(r)

is tåe number

of

distinct pfime divisors of

r.

Then

»

^{srÅi(d, ,. ..r du;e)}

h(e)H(nl") - /(1)(("- H)u(gh))(*),

d1,...,dr,e

where

the

summation

is

over

d1,...,d'e € G

^sueh

^that

^d1

^...dr,e : n

^and

dr,..

., dr,

€

are pairwise relatively prime,

Proof. The left-hand side of the identity

in

Theorem 12 is

» /(1)g(") h(")ä( nle) - /(1) »

u-('/e)g(e)

h(e)H("/")

d1,.,,,dr,e _e€U(n)

: /(1) ((eh)u(u-ä))

^(n);

hence the theorem is valid.

Theorem L3. Let f , 9, ^h

^and

H

^be

afithmeticil

functions and

n €

G.

Then

» ^S*i(d,

^{, . . . ,}

^d,;

^{e) h(u)H (:nle)}

- ^/(1)

^((8"

H)u(gh))("),

d1"'d11e:n

(dr "'d.r,e):1

where

Eu: EDEry... DE (u

^factors).

Proof. The left-hand side of the identity is

/(1)g(")h(u)H(nle)-/(1) I ( » ^,), G)h(")a( nle)

e€U(n) d1"'du:n/e

- /(1) » E,(nle)g(e)ä(e)ä( nle) -

^f

(1)((E uH)u(gä))(*).

e€U(n)

d1...dse=n

(dr...d.,,e):1

We thus arrive at our result.

Arithmetical identities involving

a

generalization

of

Ramanuian's sum 49

Remark. A

large number

of

special ^cases

of

our results can be found in the literature.

In

fact, special ^cases of Theorem L can be found

in

_[33, Corollary (2.1.5),

p.

170, Theorem

(2.2.6),p.

^774), _[38,

pp.

15 and 721 ar,d {40, Chapter

i.t].

special ^cases of Theorem 2 can be found

in

_[2, equation (2.7)], _[20, equation (6)], [33, Theorem (2.2.72),

p.

176], [34, Theorem 3.1], [35, equation (4.3)], ^[41, Theorem

3]

alrd [44, Theorem

5.3].

special ^cases

of

Theorem

3

can be found

in

[1, Theorems 1-4], [2, equations (2.8), (2.10)], [4, Theorem 8], [5, Corollaries 2,

i, ^4, 5,

^{LO.z and} 10.4], [6, equation (5'4)], [8, equations (3.1), (5.1)], _[10, ^p.

2031,l12,equation (3.16)], [18, equation (2.3),

p.

194], _[20, equations (4), (5)], _[23, Lemma U, [28, Theorem 1l,l2g, equation ^{(1a) and} Theorem 8], _[30, equation (2.6)], [31, Theorern 2.4], [33, Theorem (2.1.8),

p.

^772, Theorem (2.2-77),

p.

176], _[34, Theorems 3.2, 3.3], _[36, equation (3.3)], _[41, Theorems ^1, 2] and _[44, Theorem 5.4].

Special cases of Theorem 4 can be found

in

[1, Theorems 5, 6], [28, Theorem ^2]

and _[2g, Theorem (1b)]. special cases of Theorem 5 ^{can be} found

in

_[1, ^Theorems

7,

81, 12, Theorem 3.21, [3, Corollary 2.7], ^177, Lemma ^2.61, _129, Theorem 3], _[40, Theorem 4.1.2] and _[44, Theorem 5.5]. Special cases of Theorem 6 can be found in [7, Lemma 4], U4, Lemma ^3] and _[25, Theorem 8]. Special cases of Theorem 7 can be found

in

[2, Theorem 2.6], [L6, Theorem 1], ^[20, equation (3)], _[37' equations (2.10), (2.11)] and _[40, equation (4.14)]. special ^cases of Theorem 8 can ^be found

in

[3, Theorem 2], [8, equation (4.2)], [15,

Theore*

1], [18, Lemma 2.2,

p.

1941,

[21, Theorem

5],

[24, Theorem

3],

[26, equation (4.1.6)] and [32, Theorem ^7.2].

A

special ^case of Theorem 9 can be found

in

_[37, equation (2.1,2)1. special ^cases of Theorem 10 can be found

in

[4, Theorem 6], [8, equation (4.5)], [9, Theorem 3.3], [21, Theorem 41, 132, Theorem 7.4] and [34, Theorem

3.4]'

Special cases

of

Theorem 11 can be found

in

[11, Theorem 3] and [25, Theorem

3].

FinallS Theorem 6 of _[25] is a special ^case of Theorems 12 and 13.

4. Totally l.-k-even functions (modr)

Let r € G

^be

fixed.

Then an arithmetical

function /(";r) ^of

^{one vari-}

able is said

to

be A-k-even

(modr) if /(n;r): /((n,rk)e,x;r) fot all n €

^G.

An

arithmetical function

"f(nr,...rruir) ^{of u}

^{variables is said}

^to

^be

^totally

^.4-

fr-even

(modr) if

there exists

an

A-la-even function

g(n;r) (modr)

such that

.f(rr,...,r,ir) :

^g((n1

,...,n,)i r) ^{for all}

ⁿ¹

,...,nu e G.

^{The concept}

^of

^a

totally

A-k-even function

(modr)

originates from _[14]

in

the case of the arith- metical semigroup of positive integers.

It

can be proved

(cf.

_[14, Theorem

f])

that an arithmetical function

/(tr,.. .,ll,ir)

^is

^totally

.4.-&-even

(modr)

if, and only

if, it

has a unique representation of the form

/(rr,

^{. . . : ilu;}

^r) - »

^o(d ;r)Ce,*(nr? ^{. . .} r ru; d),

where

d€An (r)

50 Pentti Haukkanen

o(d;

r) : *-ku » ^gek l6r;r)Ce,r(rk ldr,... ,tk

ldr; ^6).

6eao1";

The

coefi.cients

o(d;r)

are called

the

Fourier coefEcients

of /(n1,...,n,ir).

It can

also

be

proved

(cf.

_[1.4,

Theore* 2]) that an

arithmetical function

.f(tr,...,ruir)

is totally,4.-/c-even

(modr) if,

and only

if, it

has the form (12)

In

this case

a(d;

r) : 1-&u » f'Gl";")ek".

e€A*(r/d)

By (12) we find that the generalized Ramanujan's sum considered in this paper is a

totally

,4.-/c-even function

(modr).

The purpose of this section is to note

that

the equations (8) and (10) can be extended to

totally

A-lc-even functions

(modr).

In fact, if we replace the general- ized Ramanujan's sum by an arbitrary totally ^{,4.- & -even} function

.f(r,

^{, . . . , n, i}

r) (modr)

in the left-hand sides of equations (8) and (10), we must replace the factor

of

_f

(d)g(t/d) bv "f'(d;") ⁱⁿ

the right-hand sides of the equations.

Referencee

t1] ^Arosrol,

^T.M,: Arithmetical properties of generalized Ramanujan sums.

-

Pacific J.

Math. 41, 1972, 28L-293.

12)

CtttoltvtslnA.swAMy, J.: Generalized Ramauujan's sum.

-

^Period. Math. Ilungar. ^1.0, L979,7t-87.

t3]

Couutt, E.: An extension of Ramanujan's sum. II. ^Additive properties. - Duke Math. J.

22, L955,543-550.

t4]

CottoN, E.: An extension of Ramanujan's sum. III. Connections with totient functions. - Duke Math. J.23, 1956, 623-630.

15]

Cottott, E.: Representations of even functions (modr), I. Arithmetical identities. - Duke Math. J. 25, 1958, 401-42L.

t6]

Cottotr, E.: A class of residue systems (modr) and related arithmetical functions, I. A generalization of Möbius inversion. - Pacific J. Math. 9, 1959, 13-23.

17)

^{CottEN, E.: Ä class of} arithmetical functions in several variables with applications to congruences. - Tlans. Amer. Math. Soc. 96, 1960, 355-381.

t8]

^{CoEEN, E.:} On the inversion of even functions of finite abelian groups (modIy'). - J. Reine Angew. Math. 207, 1.961, 192-202.

t9]

^CoHoN, E.: Unitary functions (modr). - Duke Math. J. 28, 1961, 475-485.

[10]

DELS.o,RTE, J.: Essai sur l'application de la theorie des fonctions presque p6riodiques a I'arithm6tique. - Ann. Sci. Iicole Norm. Sup. 62, Lg45,185-204.

tl1]

^{DoNovAN, G.S., and D. RpaRrcx:} On Ramanujan's sum. - Norske Vid. Selsk. Forh. 3g, 1966, 1-2.

/(*r,...,ru;r)- » ^/'(d;

^r).

dft €A (««"i),rk )o,*)

Arithmetical identities involving ^a generalization of Ramanujan's

surn

51

ll2l

HANUMANTHAoTTART, J., and V.V. SUsnA,HMANYA Snsru: Certain totient functions-

allied Ramanujan sums and applications to certain linear congruences. - Math. Stu- dent 42, 1974,214-222.

[13]

H.l,uxxervEN, P.: Classical arithmetical identities involving a generalization of Ramanu- jan's _sum.- Ann. Acad. Sci. Fenn. Ser. A. I. Math. Dissertationes 68, 1988, 1-69.

[14]

HluKxAr.lolv, P. and P.J. McC.q,ntsv: Sums of values of even functions . - Portugal.

Math. (To appear.)

[15]

^{HoReoeM, E.M.:} Ramanujan's sum for generalised integers. - Duke Math. J. 31, 1964, 697-702.

[16j

^{HoReolu, E.M.:} Exponential functions for arithmetical semi-groups. - J. Reine Angew.

Math. 222, 1966, 14-19.

[17]

^{JonNSoN, K.R.:} Reciprocity and multiplicativity in a class of arithmetic functions. - Ph.

D. thesis, University of Colorado, ^1980.

[18]

KNoprMAcHEn, J.: Abstract analytic number theory. - North-Ilolland, Amsterdam-New York, 1975.

[19]

^{LAHTRI, D.B.:} Hypo-multiplicative number-theoretic functions. - Aequationes Math. 9, 1973, 184-192.

[20]

^{McClntnv, P.J.:} Some properties of the extended Ramanujan sums. - Arch. Math. L1, 1960, 253-258.

121]

^{McClRtuv, P.J.: Regular} arithmetical convolutions. - Portugal. Math.27, 1968, 1-13.

122)

McC.l,Rruv, P.J.: Introduction to arithmetical functions. - Springer-Verlag, New York- Berlin-Ileidelberg-Tokyo, L 986.

[23]

N.l,cosw,l.Rl Rlo, K.: Unitary class division of integers moda and related arithmetical identities. - J. Indian Math. Soc. 30, 1966, 195-205.

l24l

NecpsweRl Rno, I(.: Some applications of Carlitz's rr-sum.

-

Acta Arith. 12, 1967,

2t3-22t.

125)

N,q,cpswARA. Rao, K.: Some identities involving an extension of Ramanujan's sum, - Norske Vid. Selsk. Forh. 40, 1967, 18-23.

[26]

Nlooswa.Ra Rlo, K., and R. SrveRlvr,lxRrsHNAN: Ramanujan's sum and its applica- tions to some combinatorial problems. - Congr. Numer. 31, 1981, 205-239.

l27l

N.aRrtowIcz, W.: On a class of arithmetical convolutions. - Colloq. Math. 10, 1963, 81-94.

[28]

Nlcor,, C.A.: On restricted partitions and a generalization of the Euler O number and the Moebius functions. - Proc. Nat. Äcad. Sci. U.S.A. 39, 1953, 963-968.

[29]

^{NIcoL, C.A., and H.S.} V.o,NoIvnR: A vou Sterneck arithmetical function and restricted partitions with respect to a modulus. - Proc. Nat. Acad. Sci. U.S.A. 40, L954, 825- 835.

[30]

R.lu.nru.t.lN, S.: On certain trigonometrical sums and their applications in the theory of numbers. - Tlans. Cambridge Philos. Soc. 22, 1918, 259-276.

[31]

^SHIDER, L.E.: Arithmetic functions associated with unitary divisors in GFfq,c] _, I. - Ann.

Mat. Pura Appl. 86, 1970, 79-85.

[32]

SIU Reltet.lu, V.: Arithmetical surns in regular convolutions. - J. Reine Angew. Math.

303/304, 1978, 265-283.

[33]

Srv.ln.o,rrrlxRrsuNAN, R.: Contributions to the study of multiplicative arithmetic func- tions. - Ph. D. thesis, University of l(erala, 1971.

[34j

StvnRltrtnxRtsuNAN, R.: Multiplicative even functions (modr). II. Identities involving Ramanujan sums. - J. Reine Angew. Math. 302, 1978, 44-50.

52 Pentti Haukkanen

[35]

Srv.a,RnvrnrRrsHNAN, R.: On abstract Möbius inversion. - Proceedings of the 2nd Confer- ence on Number Theory (Ootacamund, 1980), Matscience Rep. 1.04, 85-100.

[36]

SusnA.HN{vÄ. SlstRI, V. V.: On a certain totient function for generalised integers. - Math.

Student 37, 1969, 59-66.

[37]

SucuNluul, M.: Eckford Cohen's generalizations of Ramanujan's trigonometrical sum

C(n,r). - Duke Math. J. 27, 1960,323-330.

[38]

Suouu.nuu.l,, M.: Contribution to the study of general arithmetic functions. - Ph. D.

thesis, Sri Venkateswara University, Tirupati, 1965.

[39]

Varovlx.ntnlswAMy, R.: The theory of multiplicative arithmetic functions.

-

tans.

Ämer. Math. Soc. 33, ^1931, 579-662.

[40] VpNxltlRlultt,

C.S.: A new identical equation for multiplicative functions of two argu- ments and its applications to Ramanujan's sum Cu@). - Proc. Indian Acad. Sci.

Sect. A 24, L946,518-529.

t41]

VpxxltnnlIr,r.r,tt, C.S.: F\uther applications of the identical equation of Ramanujan's sum

Cu(N) and Kronecker's fuuction A(M,N). - J. Indian Math. Soc. 10, 1946, 57-61.

l42l

VprremRltr.tnu, C.S.: The ordinal correspondeuce and certain classes of multiplicative functions oftwo variables. - J. Indian Math. Soc. 10, 1946,81-101.

[43]

VENKATARAM.a,N, C.S.: On von Sterneck-Ramanujan function. - J. Indian Math. Soc. ^1.3, t949,65-72.

[44]

VsNrltnRlIllAN, C.S., and R. StvlRnunrRtsHNAN: An extension of Ramanujan's sum.

- Math. Student 40 A,1972,211-216.

[45]

YocoM, K.L.: Totally multiplicative functions in regular convolution rings.

-

^Canad.

Math. Bull. 16, 1973, 119-128.

University of Tampere

Department of Mathematical Sciences P. O. Box 607

SF-33101 Tampere Finland

Received 9 December 1988