UNIV.
Ser.
BEOGRAD.
Mat.
593.
PUBL
Fiz. X. 577-X.
ELEKTROTEHN.
598 (1977),
FAK.
74-78.
A PROPERTY OF RAMANUJAN'S
TOTALLY MULTIPLICATIVE
SUMS CONCERNING
FUNCTIONS*
Aleksandar Ivie
ABSTRACT.Two necessary and sufficient conditions for a multiplicative
function to be totally multiplicative are given in terms of Ramanujan's
sums,
and an e1l:pansion involving Ramanujan's sums is given for the generalized von
[L (d) logknjd.
Mangoldt's function Ak (n)
din
L
~
Let Cn(m)
IT
~am)
exp
=
C
l::;~n
(~n)~1
denote RAMANUJAN'S sum.
n
Un (m) =
exp
a~l
then
If we write
2 IT iam
( )
---;;-
by the sum formula for geometric progressions we have
if n{m
if n 1m'
(1)
and the MOBIUS inversion formula gives the folloving well-known
the evaluation of RAMANUJAN'Ssum (see [1]):
(2)
cn (m) =
L
formula
for
d fL(n/d),
dl m,dln
where
fL(n) is the
MOBIUS
function.
In his original paper [5] S. RAMANUJANsummed the series
L
Cn
(m)
n-S
n~l
00
and obtained expansions
of the type f(m)=
L cn(m)
a (n) for many number-
n~1
-theoretic functions which gave rise to much subsequent research (see [3], Ch. 7
for a more detailed account). This note deals with a property of RAMANUJAN'S
sums Cn (m) concerning totally multiplicative functions (an arithmetical function
is multiplicative if f (mn)= f(m)f (n) for m and n relatively prime, and f (n) is
totally multiplicative if f (mn)= f(m)f (n) for all natural numbers m and n). The
identities contained in the following theorem may be also used to obtain series
expansions
for certain arithmetical
functions, and this is done in theorem 2 for
(d)
logk
n/d
which represents a generalization of the
the function Ak (n) =
fL
din
VON MANGOLDT function
A (n) from the theory of prime numbers (see [2]).
L
* Presented
May.16,
1977 by D. S. MITRINOVIC.
74
A property
of Ramanujan's
sums concerning
totalIy multiplicative
functions
75
00
Theorem 1. Let F (s) =
Lf
n~l
(n) n-S be the Dirichlet series of a multiplicative func-
tion f (n) with a finite abscissa of convergence. Then we have
(3)
and
(4)
if and only if f (n) is a totally multiplicative arithmetical function.
Proof. The identity (3) may be written as
00
(5)
F(s)
00
L
f (n) cn (m) n-S =
n~l
L gf(n)
n~l
n-S
where for a given m we define
if n [m
if n{m'
or by the properties
of DIRICHLETseries (see [4]) equivalently as
Lf
(6)
(d) f (n/d) Cn/d(m) = gf(n).
din
If f(n)
is totally
Lf(d)f(n/d)
multiplicative
cn/d(m) = f(n)
din
then
L cn/d(m)
= f(n)
un(m) =gf(n).
din
Suppose now that (3) or equivalently (6) holds. Since f (n) is by hypothesis
multiplicative, in order to show that it is totally multiplicative it suffices to
show that f (pa) = (f (p»a for every prime p and every natural number a (this
being trivially true for a = 1). We now keep p fixed and use induction on a supposing that f(pb)=(f(p»b
for b<a-l.
In (6) we may take a>2, n=m=pa,
and then from (2) we see that cpb(m)=pb_pb-l=rp(pb)
(rp(n) is the EULER
totient function) and so (6) becomes
Lf
dl pa
(d) f (pa/d) rp(d) = pa f (pa),
or after expansion and rearanging
(7)
-2
L
[f(pi)f(pa-i)-
f(pi+l)f(pa-i-l)]pi+(f(pa-l)f(p)-
f(pa»(pa-l_1)=O.
i~l
By the induction hypothesis each term in the square brackets of (7) is
zero, and we are left with f (pa) = f (pa-i) f (p) = (f (p»a, which was to be proved.
76
A. Ivie
To prove (4) note that if f (n) is totally multiplicative
L'"
1/F(s)=
then
!J-(n)f(n)n-S,
n~!
and if for a fixed m we define hf(n) as
(I) (n)
if :3IE N Iln = m
hf(n) = !J- ~f
{
otherwise
'
then (4) may be written as
'"
(1/F(s»). Lf(n)cm(n)n-s=
'"
L hj(n)n-S,
n~!
n=!
of DIRICHLET series equivalently
as
and by the properties
L !J-(d) f(d)
(8)
Cm(n/d) = hf(n).
f(n/d)
din
If f (n) is totally multiplicative then
L !J-(d)f(d)f(n/d)cm(n/d)
din
L
=f(n)
!J-(d)
L
= fen)
Cm (n/d)
din
!J- (d)
din
L
r!J-(m/r)
rim, r InJd
L r!J-(m/r) d LInJr !J-(d)=hj(n),
=f(n)
rim
rln
L!J- (d) =
by the property of the MOBIUS function that
1
k= 1
{
.
0
k~2
To prove converesely that f (n) is totally multiplicative if (8) holds we
proceed as before by induction on a supposing that f (pb) (f (p»b for b ~ a - 1
and take in (8) n = m = pa, a ~ 2, so that (8) becomes
dlk
~
paf(pa)=
L
!J-(d) f (d) f(pafd)
Cpa(pafd)
d I pa
Cp a (pa)
= f (1)f(pa)
Now
from (2) it follows
that
-
f (p) f (pa-I) Cp a (pa-I).
Cp a (pa) = pa- pa-I
and
Cp a (pa-I)
=
- pa-I so
that we obtain
pa f (pa)
= f (pa) (pa- pa-I) + pa-I f (p) f (pa-I)
which by the induction hypothesis gives f (pa) = f (p) f (pa-I) = (f (p»a, and
this finishes the proof of the theorem, since p is an arbitrary prime number.
In his paper [5] S. RAMANUJANobtained the expansion (m ~ 2)
A (m) =
where
A (m) =
L !J-(d)
log mid
-
L'"
Cm(n) n-I,
n~ !
is the
dim
well-known
VON MANGOLDT
function.
As stated at the beginning, theorem 1 may be used to obtain an expansion for the
generalized VON MANGOLDT Ak (m) = L !J-(d) log k mid (see [2] for more properties of this function).
dim
A property
of Ramanujan's
Theorem 2. If Ak (m)
totalIy multiplicative
77
functions
[J.(d) log k mid, then for integer m> 2 we have
L
=
sums concerning
dim
00
L cm (n) gk (n),
n~1
Ak (m) =
(9)
where
00
gkCn)
L
n-1
=
[J.(r),-l
(log nr)k.
r ~I
00
Write (4) as
Proof.
00
L
r~1
[J.(r)f(r),-S
( - I)k
L
[J. (d)
L
n~1
set f (n) = I (which is certainly)
Then for Res> I we have
(10)
Lf(n)cm(n)n-s=
totally
[J.(mld)d1-Sf(d),
dim
multiplicative
and differentiate
k times.
(mld)I-S logk mid
dim
k
= .L
_
1-0
( )~
~
(-
l)k-i
Cm (n)logk-in.
n-S
~
n -I
-
00
If
s-+
1
+0
L
then
r~
have
~
l)i [J.(r) r~S logi r.
00
[J.(r) logi r. r-S
converges
L
to
1
[J.(r) r-1 logi r and we
r~1
(
(see [4], Ch. III).
!:- il/~(S)
)
[J.(r)r-IIOgir=(-l)i
Moreover
for a
ds
r~ 1
fixed
(-
r~1
m> 1 we have
LCm(n)=L
ll~X
n~x
=
L
d[J.(mld)=Ld[J.(mld)L
dim, d : n
dim
L d[J.(mld)(xld + 0
(I»=
x
dim
uniformly
in x since
1
d l-:£x
L
[J.(mld)
+ 0(1)=
0(1),
dim
L
[J.(mid)
=
dim
L
[J.(d) = 0 for m> 1.
By partial
summa-
dim
00
tion
it follows
that
.
L
Cm (n) logk-1n.
n-I
converges
and so by the properties
of
n~1
DIRICHLET series (see [4]) we may take
in (10) the limit
as s-+ 1 + 0 to obtain
00
=
This clearly
reduces
L cm (n) gk (n).
n~1
to RAMANUJAN'S result
about
A (m) when k = 1 since
78
A. Ivi6
because (see [4] Ch. III)
L fL(r)r-l=O
and
r~l
L fL(r)logr.,-l=-1
and
so
r~l
for m ~ 2
A(m)=A1(m)=
L'"
cm(n)gl(n)=
n~l
which
was obtained
'" cm(n)n-l,
- L
n~l
by RAMANUJAN.
REFERENCES
1. G. H. HARDY, E. M. WRIGHT: An introduction
2. A. IVIe: An application of Dirichlet
Balkanica 3 (1973), 158-165.
3. J. KNOPFMACHER:Ahstract
analytic
4. K. PRACHAR: Primzahlverteilung.
series
to the theory of numbers. Oxford, 1960.
to certain
arithmetical
functions.
number theory. Amsterdam-Oxford,
Berlin-Heidelberg-New
York,
1975.
1957.
5. S. RAMANUJAN: On certain trigonometrical sums and their applications
of numbers. Trans. Cambridge Phil. Soc., N2 22. 13 (1918), 259-276,
S. RAMANUJAN: Collected Papers. New York, 1962.
Rudarsko-geoloski
fakultet
Dusina 7, 11000 Beograd
Math.
in the theory
reprinted in: