A NOTE ON SPECIALLY MULTIPLICATIVE ARITHMETIC FUNCTIONS
PENTTS HAUKKANEN
of Tampere,
SF-33101
Tampere,
University
(Submitted
December
Finland
1986)
An arithmetic function / is called multiplicative if
(1)
f{mn) = f(m)f(n),
whenever (m, ri) = 1. A multiplicative function / is called completely multiplicative if (1) holds for all m9 ft. Further, a multiplicative function / is
said to be a quadratic (see [1], [3], [8]) or a specially multiplicative function (see [2], [4], [6], [7]) if
(2)
f = aob,
where a, b are completely multiplicative functions and ° denotes the Dirichlet
product.
It is known that (2) is equivalent to
f(mn) =
Z
f(m/d)f(n/d)g(d)\i(d),
d\(jn,n)
where g is a completely multiplicative function and y denotes the Mobius function.
The completely multiplicative function g is defined for every prime by
= (ab)(p)
g(p)
where f'1
or g(p)
= f(p)2
- f(p2)
or g(p)
f^ip2),
denotes the Dirichlet inverse of /. Since a quadratic / is multipli-
cative, the values /(ft) are known if the values f(pm)
p and all positive integers m.
2
values f(p) , f(p )
known.
=
Furthermore, the values f(pm)
[or the values f(p)s
m
The values f(p )
are known for all primes
1
2
f~ (p )
are known if the
or the values a(p) , b(p)] are
are given recursively by
/ ( D = 1,
f(p2)
f(p)5
m
are
arbitrary,
m 1
~ g(p)f(pm-2),
= f(p)f(p ' )
f(p )
m
Consequently, if we put f(p )
m = 3, 4, ... .
(3)
= Sm, we obtain a generalized Fibonacci sequence
determined by
S Q - 1,
S1,
S2 are arbitrary,
Sm + i
= S
i^m ~ ((^1)
~ ^ 2 ^ m - i ' m = 2S 3, 4, ...
.
If we let Si = 1, #2 = 2 , we obtain the Fibonacci sequence.
1988]
325
A NOTE ON SPECIALLY MULTIPLICATIVE ARITHMETIC FUNCTIONS
If / is specially
multiplicative and / = a°b,
where a, b are completely
multiplicative, then the generating series of / to the base p is given by
f
(P)(x)
=
where a = a(p) , g = b(p) .
^(p) (J:)
(P a p r i m e ) >
(1 - coOd - to)
=
Then
g(p)x2'
1 - /(p)* +
where jT(p) = a + 3 and g'(p) = a@.
Fibonacci sequence {Fn}
Y<nFnXn
=
1 _
n=0
f(P)(x)
wil1
Noting that the generating function of the
is
- ^2'
x
generate {F n } if j(p) = 1 and #(p) = - 1 .
If a is any nonzero complex number, one could consider / for which f(p)
and g(p)
2
It will follow that
= -a .
2-T = £ ^ n ^^ n .
frn,(ai) = i
J
1 - ax - a ^cz
(p)
H e n c e , f(pn)
of t h e
z
= anFn.
Fibonacci
s e q u e n c e {Gn}
-a
n
„~0
Write j ( p n )
= Gn.
U s i n g known p r o p e r t i e s
s e q u e n c e {Fn } , f o r e x a m p l e ,
can be
tan~k^Gk
the following
(see [ 5 ] ,
[9])
p r o p e r t i e s of
the
derived:
=Gn
- a -
+ 2
2
,
k =0
(-l)kan~kGk
E
= (-DnaGn_1
an ,
+
k =0
a
^2k
~ ^in
+ l'-
fc = 0
V-
2(n-fc) + l /
n
L
jL*a
fe
k == li
Y3(n-fc)
+2
^3/c - 1
-k+3^
(n - k)an-nk+3
Gk
E
U
~
3n+1
_
/-,
_ / w i Q N;^ + 3
= G
(n + 3)a
w+ 3 -
£=0
In
^
E ^„ 22 (( 22 nn --ffce))++ll ^k k+l
k =o
1
r
a
2n-l
y-
U U
2 ( 2 n - / c ) - l r -r
_
r2
~
^2
_
r2
_
hn
k =0
£a*-"+1fffc2
-GHGn
+ 1
.
k =0
326
[Nov.
A NOTE ON SPECIALLY MULTIPLICATIVE ARITHMETIC FUNCTIONS
l o f a * ^ ' ^ = G3n+k + (-l)n6a2n+5Gn_1
+ 5a3n+ \
k=0
G
n+m
G
G G
+
n m
l ~ n-^n+k
aG
aG
=
G
sn + z
2n+l
= G
alG
G
n-l m-l>
= H r ^
1
^ ^ ^ ^ ^
n + i + a Gn - a Gn_1,
=
G
n +1 "
a
G
n-l'
The proofs of the above relations are omitted.
ACKNOWLEDGMENT
The author wishes to thank the referee for his valuable comments.
REFERENCES
1. T. B. Carroll & A. A. Gioia. "On a Subgroup of the Group of Multiplicative
Arithmetic Functions." J. Austral.
Math. Soc. , Ser. A, 20, no. 3 (1975): 348358.
2.
D. H. Lehmer. "Some Functions of Ramanujan." Math. Student
116.
3.
P. J. McCarthy.
(1960):966-970.
"Busche-Ramanujan Identities."
Amer.
27 (1959):105-
Math.
Monthly 67
4. A. Mercier. "Remarques sur les Fonctions Specialement Multiplicatives."
Ann. Sc. Math. Quebec 65 no. 1 (1982):99-107.
5. K. Subba Rao. "Some Properties of Fibonacci Numbers." Amer. Math.
60 (1953):680-684.
Monthly
6.
D. Redmond & R. Sivaramakrishnan. "Some Properties of Specially Multiplicative Functions." J. Number Theory 13, no. 2 (1981):210-227.
7.
R. Sivaramakrishnan. "On a Class of Multiplicative Arithmetic Functions."
J. Reine Angew. Math. 280 (1976):157-162.
8.
R. Vaidyanathaswamy. "The Theory of Multiplicative Arithmetic Functions."
Trans. Amer. Math. Soc. 33 (1931):579-662.
9.
N. N. Vorob'ev*
Fibonacci
Numbers.
New Yorki Pergamon Press, 1961.
40^0^
1988]
327