On the Kesava Menon norm of semimultiplicative functions

Aequa . Ma h.
c
The Au ho (s) 2019
h ps://doi.o g/10.1007/s00010-019-00660-x Aequa iones Ma hema icae
On he Kesa a Menon no m o semimul iplica i e unc ions
Pen i Haukkanen
Abs ac . The Kesa a Menon no m o an a i hme ical unc ion is de ined by N( )(n)=
( ∗λ )(n2), whe e ∗deno es he Di ichle con olu ion and λdeno es Liou ille’s unc ion.
The m hpowe Kesa aMenonno mo is de ined induc i ely by N0( )= ,Nm( )=
NNm−1( ),m=1,2,... In his pape we p o e ha he m hpowe Kesa aMenonno m
o a semimul iplica i e unc ion is semimul iplica i e and ha he m h powe Kesa a Menon
no m dis ibu es o e he Di ichle con olu ion o semimul iplica i e unc ions. In addi ion
we show ha he m h powe Kesa a Menon no m o a a ional a i hme ical unc ion o
deg ee ( , s) is a a ional a i hme ical unc ion o he same deg ee.
Ma hema ics Subjec Classi ica ion. 11A25.
Keywo ds. Semimul iplica i e unc ion, Kesa a Menon no m, Di ichle con olu ion, Ra ional
a i hme ical unc ion.
1. In oduc ion
Le be an a i hme ical unc ion ( ha is, a eal- o complex- alued unc ion
on he se o posi i e in ege s). In 1963, Kesa a Menon [4, Sec ion 3] defined
he no m o as he a i hme ical unc ion N( ) gi en by
N( )(n)=( ∗ )(n2),
whe e ∗is he Di ichle con olu ion (see (1)) and is he conjuga e o .The
conjuga e is defined as
=λ ,
whe e λis Liou ille’s unc ion (see (3)). The no m N( ) is e e ed o as he
Kesa a Menon no m in he li e a u e [6, Sec ion 5]. Redmond and Si a a-
mak ishnan [9, Sec ion 4] defined he m h powe Kesa a Menon no m o
induc i ely by
P. Haukkanen AEM
N0( )=
Nm( )=NNm−1( ) o m=1,2,...
In his pape we in es iga e he conjuga e, he Kesa a Menon no m and
he m h powe Kesa a Menon no m o semimul iplica i e unc ions. Semimul-
iplica i e unc ions o m a supe class o he class o he usual mul iplica i e
unc ions. Quasimul iplica i e unc ions lie be ween mul iplica i e and semi-
mul iplica i e unc ions. Ra ional a i hme ical unc ions o m he subg oup o
he g oup o mul iplica i e unc ions unde he Di ichle con olu ion gene a ed
by comple ely mul iplica i e unc ions. An a i hme ical unc ion is said o
be a a ional a i hme ical unc ion o deg ee ( , s) i i is he Di ichle con o-
lu ion o comple ely mul iplica i e unc ions and he in e se o scomple ely
mul iplica i e unc ions. Fo de ails o hese a ious ypes o mul iplica i i y,
see Sec . 2.
This pape is o ganized as ollows. In Sec . 2we e iew he known p ope -
ies o a i hme ical unc ions needed in his pape . In Sec . 3we p esen new
esul s. In Sec . 3we fi s no e ha he conjuga e o a semimul iplica i e unc-
ion is semimul iplica i e and ha he conjuga e dis ibu es o e he Di ichle
con olu ion o any wo a i hme ical unc ions. We con inue by applying hese
esul s o show ha he m h powe Kesa a Menon no m o a semimul iplica-
i e unc ion is semimul iplica i e, ha is, he m h powe Kesa a Menon no m
p ese es semimul iplica i i y. As special cases we ob ain he same p ope ies
o quasimul iplica i e and mul iplica i e unc ions. The e o e ou esul gen-
e alizes he esul o Si a amak ishnan [11, Sec ion 2], namely ha he usual
Kesa a Menon no m o a mul iplica i e unc ion is mul iplica i e. In Sec . 3
we also p o e ha he m h powe Kesa a Menon no m dis ibu es o e he
Di ichle con olu ion o semimul iplica i e unc ions, ex ending he esul o
Laohakosol and Pabhapo e [6, Sec ion 5], who p o ed ha he m h powe Ke-
sa a Menon no m dis ibu es o e he Di ichle con olu ion o a ional a i h-
me ical unc ions. We apply ou dis ibu i i y p ope y o show ha he m h
powe Kesa a Menon no m p ese es he Di ichle in e se o a quasimul iplica-
i e unc ion.
In Sec . 4o his pape we u ilize he p ope ies p esen ed in Sec . 3 o p o e
ha he m h powe Kesa a Menon no m o a a ional a i hme ical unc ion o
deg ee ( , s) is a a ional a i hme ical unc ion o he same deg ee. Laohakosol
and Pabhapo e [6, Sec ion 5] p o ed he same esul in a diffe en way. We also
p esen he analogous esul s o he conjuga e o semimul iplica i e unc ions
and a ional a i hme ical unc ions o deg ee ( , s).
On he Kesa a Menon no m o semimul iplica i e unc ions
2. P elimina ies on a i hme ical unc ions
The Di ichle con olu ion o a i hme ical unc ions and gis defined as
( ∗g)(n)=
d|n
(d)g(n/d).(1)
We may also in e p e ha an a i hme ical unc ion is defined on he se
o posi i e eal numbe s so ha (x)=0i xis no a posi i e in ege . This
makes i possible o p esen he Di ichle con olu ion in he o m
( ∗g)(n)=
∞

k=1
(k)g(n/k).
This exp ession is use ul in some calcula ions p esen ed in his pape . The
unc ion δ, defined as δ(1) = 1 and δ(n) = 0 o he wise, se es as he iden i y
unde he Di ichle con olu ion. The Di ichle in e se o exis s i and only
i (1) = 0, and i is deno ed by −1.
An a i hme ical unc ion is said o be mul iplica i e i (1) = 1 and
(mn)= (m) (n) o all cop ime posi i e in ege s m, n. An a i hme ical
unc ion is said o be semimul iplica i e [8] i he e exis s a nonze o cons an
c , a posi i e in ege a and a mul iplica i e unc ion Msuch ha
(n)=c M(n/a ).
Semimul iplica i e unc ions can also be cha ac e ized as he a i hme ical unc-
ions (no iden ically ze o) sa is ying he unc ional equa ion
(m) (n)= ((m, n)) ([m, n])
o all posi i e in ege s mand n, whe e (m, n) and [m, n] a e he gcd and lcm o
mand n. Semimul iplica i e unc ions a e he same as Selbe g mul iplica i e
unc ions (see [2, Sec ion 2.1] and [10]). We do no p esen he de ails he e.
Quasimul iplica i e unc ions (see [2, Sec ion 2.1] and [12, Sec ion XI.2]) a e
he a i hme ical unc ions such ha (1) =0and (1) (mn)= (m) (n)
o all cop ime posi i e in ege s m, n. Lahi i [5] e e s o hese unc ions as
hypo-mul iplica i e unc ions. Quasimul iplica i e unc ions a e, in ac , he
semimul iplica i e unc ions wi h a = 1 (i.e. wi h (1) = 0), c = (1)
and M(n)= (n)/ (1). No e ha is mul iplica i e i and only i is
semimul iplica i e wi h a =c =1and M= .
The Di ichle con olu ion o mul iplica i e unc ions is mul iplica i e. The
same applies o quasimul iplica i e and semimul iplica i e unc ions. To be
mo e p ecise [8, Sec ion 5], i and ga e semimul iplica i e, hen ∗gis
semimul iplica i e wi h
c ∗g=c cg,a
∗g=a ag,( ∗g)M= M∗gM.(2)
P. Haukkanen AEM
A mul iplica i e unc ion is said o be comple ely mul iplica i e i
(mn)= (m) (n) o all posi i e in ege s m, n. Liou ille’s unc ion λis
an example o a comple ely mul iplica i e unc ion. I is defined as
λ(n)=(−1)Ω(n),(3)
whe e Ω(n) ep esen s he o al numbe o p ime ac o s o n, each coun ed
acco ding o mul iplici y [1, Sec ion 2.12].
A mul iplica i e unc ion is said o be a a ional a i hme ical unc ion o
deg ee ( , s)i
=g1∗···∗g ∗(h1∗···∗hs)−1
o some comple ely mul iplica i e unc ions g1,...,g
,h
1,...,h
s(see [6]and
[13, Sec ion III]). A a ional a i hme ical unc ion o deg ee (2,0) is e e ed
o as a specially mul iplica i e unc ion [9]. I =g1∗g2is a specially mul-
iplica i e unc ion, we deno e A=g1g2. The unc ion Ais e med as he
associa ed comple ely mul iplica i e unc ion. Fo example, he di iso unc-
ions σaand Ramanujan’s τ- unc ion a e specially mul iplica i e unc ions.
Eule ’s o ien unc ion φis a a ional a i hme ical unc ion o deg ee (1,1).
Fo gene al accoun s on a i hme ical unc ions, we e e o [1,7,12].
3. The m h powe Kesa a Menon no m o semimul iplica i e
unc ions
In his sec ion we fi s no e in Theo ems 3.1 and 3.2 ha he conjuga e o a
semimul iplica i e unc ion is semimul iplica i e and ha he conjuga e dis-
ibu es o e he Di ichle con olu ion o any wo a i hme ical unc ions. We
hen apply hese heo ems o p o e Theo ems 3.3,3.4 and 3.5, which s a e
ha he m h powe Kesa a Menon no m o a semimul iplica i e unc ion is
semimul iplica i e and ha he m h powe Kesa a Menon no m dis ibu es
o e he Di ichle con olu ion o semimul iplica i e unc ions.
Theo em 3.1. I is semimul iplica i e, hen is semimul iplica i e wi h c =
λ(a )c ,a =a and  M= ( M)=λ M.
Theo em 3.2. Fo all a i hme ical unc ions and g,
∗g= ∗g.
Theo ems 3.1 and 3.2 ollow di ec ly om he defini ions o conjuga e and
semimul iplica i e unc ion and om comple e mul iplica i i y o λ.
In o de o p o e ha he m h powe Kesa a Menon no m p ese es semi-
mul iplica i i y, we fi s p esen his esul in he case m= 1, since his case
is needed in a ious s ages o he p oo o he gene al case.
On he Kesa a Menon no m o semimul iplica i e unc ions
Theo em 3.3. I is semimul iplica i e, hen N( )is semimul iplica i e wi h
cN( )=c c =λ(a )c 2
aN( )=a
N( )M=N( M).
P oo . By he defini ions o he Kesa a Menon no m and he Di ichle con o-
lu ion,
N( )(n)=( ∗ )(n2)=
∞

k=1
(k) (n2/k).
Applying he defini ion o a semimul iplica i e unc ion and Theo em 3.1,we
ob ain
N( )(n)=
∞

k=1
c M(k/a )λ(a )c (λ M)n2/(ka )
=λ(a )(c )2
∞

k=1
M(k)(λ M)n2/(a )2/k
=λ(a )(c )2( M∗(λ M))((n/a )2).
By he defini ions o he conjuga e and he Kesa a Menon no m, we see ha
N( )(n)=λ(a )(c )2N( M)(n/a ).
By Theo em 3.1,λ(a )(c )2=c c (= 0). Since he Kesa a Menon no m o
a mul iplica i e unc ion is mul iplica i e, N( M) is mul iplica i e. We hus
ob ain he esul . 
Theo em 3.4. I is semimul iplica i e, hen Nm( )(whe e m≥0)issemi-
mul iplica i e wi h
cNm( )=c (c )2m−1=λ(a )2m−1c 2m
aNm( )=a
Nm( )M=Nm( M).
P oo . We p oceed by induc ion on m.Fo m= 0 he heo em holds, since
N0( )= . The case m= 1 is p esen ed in Theo em 3.3.
Suppose ha he heo em is ue o m=k.ThusNk( ) is semimul iplica-
i e, and hen applying Theo em 3.3 we see ha N(Nk( )) is semimul iplica-
i e, ha is, he unc ion Nk+1( ) is semimul iplica i e.
Fu he , om Theo em 3.3,weha e
cNk+1( )=cN(Nk( )) =λ(aNk( ))cNk( )2.
By he induc ion hypo hesis,
cNk+1( )=λ(a )λ(a )2k−1c 2k2=λ(a )2k+1−1c 2k+1
.

P. Haukkanen AEM
By Theo em 3.3 and he induc ion hypo hesis,
aNk+1( )=aNNk( )=aNk( )=a
and
Nk+1( )M=NNk( )M=NNk( )M
=NNk( M)=Nk+1( M).
This comple es he p oo . 
Co olla y 3.1. I is quasimul iplica i e, hen Nm( )(whe e m≥0)isquasi-
mul iplica i e wi h Nm( )(1) = (1)2m.
P oo . I is quasimul iplica i e, hen a = 1, and hus
aNm( )=a =1.
This shows ha Nm( ) is quasimul iplica i e. Since λ(a )=λ(1) = 1,
Nm( )(1) = cNm( )=λ(a )2m−1c 2m
=c 2m
= (1)2m.
This comple es he p oo . 
Co olla y 3.2. I is mul iplica i e, hen Nm( )(whe e m≥0) is mul iplica-
i e.
Co olla y 3.2 ollows di ec ly om Co olla y 3.1, since each mul iplica i e
unc ion is quasimul iplica i e wi h (1) = 1.
Theo em 3.5. I and ga e semimul iplica i e, hen
Nm( ∗g)=Nm( )∗Nm(g),m≥0.
P oo . Suppose fi s ha and ga e mul iplica i e. Then, by he defini ion
o he Kesa a Menon no m, o all p ime powe s pe,
N( ∗g)(pe)=( ∗g)∗( ∗g)(p2e).
By Theo em 3.2, we ob ain
N( ∗g)(pe)=( ∗ )∗(g∗g)(p2e)=
2e

i=0
( ∗ )(pi)(g∗g)(p2e−i).
Bu ( ∗ )(pi)=0i iis odd; hence we ha e
N( ∗g)(pe)=
e

i=0
( ∗ )(p2i)(g∗g)(p2(e−i))
=
e

i=0
N( )(pi)N(g)(pe−i)=N( )∗N(g)(pe).
On he Kesa a Menon no m o semimul iplica i e unc ions
Since and ga e mul iplica i e, N( ∗g)andN( )∗N(g) a e also mul iplica-
i e. A mul iplica i e unc ion is o ally de e mined by i s alues a p ime
powe s. The e o e
N( ∗g)=N( )∗N(g).
Now, applying induc ion on mgi es
Nm( ∗g)=Nm( )∗Nm(g).
Conside now he gene al case ha and ga e semimul iplica i e. Then, by
Theo em 3.4 and Eq. (2), Nm( ∗g)andNm( )∗Nm(g) a e semimul iplica i e.
In addi ion, using Theo em 3.4 we ha e
cNm( ∗g)=λ(a ∗g)2m−1(c ∗g)2m.
On he basis o (2),
cNm( ∗g)=λ(a ag)2m−1(c cg)2m.
Since λis comple ely mul iplica i e, λ(a ag)=λ(a )λ(ag). The e o e
cNm( ∗g)=λ(a )2m−1(c )2mλ(ag)2m−1(cg)2m.
Using Theo em 3.4 and Eq. (2)wege
cNm( ∗g)=cNm( )cNm(g)=cNm( )∗Nm(g).(4)
Fu he , applying Theo em 3.4 and Eq. (2)wege
aNm( ∗g)=a ∗g=a ag
=aNm( )aNm(g)
=aNm( )∗Nm(g).(5)
Nex , applying Theo em 3.4 and Eq. (2), we ge
Nm( ∗g)M=Nm( ∗g)M=Nm( M∗gM).
On he basis o he fi s pa o his p oo on mul iplica i e unc ions,
Nm( ∗g)M=Nm( M)∗Nm(gM).
By Theo em 3.4 and Eq. (2),
Nm( ∗g)M=Nm( )M∗Nm(g)M=Nm( )∗Nm(g)M.(6)
Finally, combining (4), (5) and (6) gi es
Nm( ∗g)=Nm( )∗Nm(g).
This comple es he p oo . 
Co olla y 3.3. I and ga e quasimul iplica i e, hen
Nm( ∗g)=Nm( )∗Nm(g),m≥0.
P. Haukkanen AEM
Co olla y 3.4. I and ga e mul iplica i e, hen
Nm( ∗g)=Nm( )∗Nm(g),m≥0.
Co olla ies 3.3 and 3.4 ollow di ec ly om Theo em 3.5, since each quasi-
mul iplica i e unc ion is semimul iplica i e and each mul iplica i e unc ion
is quasimul iplica i e.
Theo em 3.6. I is quasimul iplica i e, hen
Nm( −1)=Nm( )−1,m≥0.
P oo . By Theo em 3.5,
Nm( )∗Nm( −1)=Nm( ∗ −1)=Nm(δ).
Now,
N(δ)=δ∗δ=δ∗δ=δ.
Applying induc ion, we ob ain
Nm(δ)=δ.
This comple es he p oo . 
Co olla y 3.5. I is mul iplica i e, hen
Nm( −1)=Nm( )−1.
Co olla y 3.5 ollows di ec ly om Theo em 3.6, since each mul iplica i e
unc ion is quasimul iplica i e.
Theo em 3.7. Fo all a i hme ical unc ions wi h (1) =0,
( −1)=( )−1.
P oo . We ha e
∗( −1)=λ ∗λ −1=λ( ∗ −1)=λδ =δ.

4. The m h powe Kesa a Menon no m o a ional a i hme ical
unc ions
Laohakosol and Pabhapo e [6]p o ed ha hem h powe Kesa a Menon no m
o a a ional a i hme ical unc ion o deg ee ( , s) is also a a ional a i hme i-
cal unc ion o deg ee ( , s). In his pape we p esen a sho p oo (applying
Co olla ies 3.4 and 3.5; see he p oo o Theo em 4.1). Redmond and Si a a-
mak ishnan [9] p o ed his esul o a ional a i hme ical unc ions o deg ee
(2,0), ha is, o specially mul iplica i e unc ions. We no e in Co olla y 4.2
a simila esul o he conjuga e o a a ional a i hme ical unc ion o deg ee
( , s).
On he Kesa a Menon no m o semimul iplica i e unc ions
Theo em 4.1. Suppose ha is a a ional a i hme ical unc ion o deg ee ( , s)
gi en as
=g1∗···∗g ∗(h1∗···∗hs)−1,
whe e g1,...,g
,h
1,...,h
sa e comple ely mul iplica i e unc ions. Then
Nm( )is a a ional a i hme ical unc ion o deg ee ( , s)such ha
Nm( )=(g1)2m∗···∗(g )2m∗(h1)2m∗···∗(hs)2m−1,m≥0.
P oo . Fo a comple ely mul iplica i e unc ion gwe ha e
N(g)(pe)=g∗(λg)(p2e)=g(u∗λ)(p2e)
o all p ime powe s pe, whe e u(n) = 1 o all posi i e in ege s n. He e
(u∗λ)(p2e)=
2e

i=0
(−1)2e−i=
2e

i=0
(−1)i=1.
The e o e
N(g)(pe)=g(p2e)=g2(pe).
Since N(g)andg2a e mul iplica i e unc ions, his implies
N(g)=g2.
Applying induc ion on mgi es
Nm(g)=g2m.
Now, by Co olla ies 3.4 and 3.5, we ob ain Theo em 4.1.
Co olla y 4.1. Suppose ha is a specially mul iplica i e unc ion. Then
Nm( )is specially mul iplica i e wi h Nm( )A= A2m
. Fu he ,
Nm( )A=Nm( A).
P oo . Le =g1∗g2, whe e g1and g2a e comple ely mul iplica i e unc ions.
Then A=g1g2. On he o he hand, by Theo em 4.1,Nm( )=g2m
1∗g2m
2,
and hus
Nm( )A=g2m
1g2m
2=(g1g2)2m=( A)2m.
Fu he , since Ais comple ely mul iplica i e, by Theo em 4.1, we ob ain
Nm( A)=( A)2m.
This comple es he p oo . 