Certain weighted averages of generalized Ramanujan sums

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by  K Vishnu Namboothiri

CERTAIN WEIGHTED AVERAGES OF GENERALIZED
RAMANUJAN SUMS
K VISHNUNAMBOOTHIRI
6
1
0 Abstract. Wederivecertain identitiesinvolving variousknown arith-
2 meticalfunctionsandageneralizedversionofRamanujansum. L.T´oth
constructedcertain weighted averages ofRamanujan sumswithvarious
n
arithmeticfunctionsasweights. WechooseageneralizationofRamanu-
a
J jansumgivenbyE.Cohenandderivetheweightedaveragescorrespond-
ing to theversions of theweighted averages established by T´oth.
6
Keywords: Generalized Ramanujan sum, weighted averages, Jordan
2
totient function, Gamma function, Bernoulli numbers, Holder evalua-

] tion
T Mathematics Subject Classification 2010: 11A25 11B68 33B15
N
.
h 1. Introduction
t
a
Ramanujan sum is a certain sum of powers of roots of unity defined and
m
used seriously for the first time in the literature by Srinivasa Ramanujan.
[
His paper On certain trigonometric sums published in 1918 ([15]) discusses

1
variousinfiniteseriesexpressionsformanywellknownarithmeticalfunctions
v
intheformofapowerseries. Hisinterestinthesumsoriginated inhisdesire
9
7 to“obtainexpressionsforavariety ofwell-knownarithmetical functionsofn
8 intheformof aseries a c (n)”. Therehavebeenalotofanalysis dealing
6 s s s
with this particular aspect of Ramanujan sums (See, for example, [13]). On
0
P
. the other hand, it found immense use in in classical character theory. It
1
was proved that the sums can be used to establish the integrality of the
0
6 character values for the symmetric group ([12]). However, perhaps the most
1 famous appearance of Ramanujan sums is their crucial role in Vinogradov’s
:
v proof (See [14]) that every sufficiently large odd number is the sum of three
i primes.
X
In more recent years, Ramanujan sums have appeared in various other
r
a seemingly unrelated problems. Being interested in the applications of the
Ramanujan sums, many mathematicians later tried to generalize it to find
more and more applications. One of the most popular generalization was
given by E. Cohen. After that, various other generalizations were discussed
in many papers including that of C. S. Venkataraman and R. Sivaramakris-
nan [19] and J. Chidambaraswamy ([5]).
Baby John Memorial Government College, Chavara, Kollam, Kerala, India
Affiliated to theUniversity of Kerala, Thiruvananthapuram
Department of Collegiate Education, Government of Kerala
Email : [email protected]
Theauthoracknowledgessupportfrom theUniversityGrantsCommission ofIndiaunder
its Minor Research Project Grant scheme (XII Plan) .
1
2 KVISHNUNAMBOOTHIRI
In this paper we are interestd in establishing certain weighted averages of
generalized Ramanujan sums. Thegeneralization we choose was given by E.
Cohen. We remark that our identities are based on similar kind of averages
derived by L. To´th in [18] using the classical Ramanujan sums. In between,
we derive another identity using the Jordan totient function which could be
of independent interest.
2. Notations
Most of the notations, functions, and identities we introduce below are
standard and can be found in [4] or [9]. We use the following standard
notations: N the set of all natural numbers, Z the set of all integers, R the
set of all real numbers and C the set of all complex numbers. µ will denote
the Mobius function on N. Note that for any given k N, we have
∈
µ(d) = 0
d|k
X
Let x denote the integer part of x, and τ(n) and σ(n) denote the number
⌊ ⌋
and sum of divisors of n respectively. For two arithmetical functions f and
g, f g denote their Dirichlet convolution (Dirichlet product). Note that
∗
this product is commutative. The arithmetic function N is defined as, for
n N,
∈
Nα(n):= nα
for any α R. For α = 1, we will simply write N(n) instead of writing
∈
N1(n), and N0(n) := 1 for all n.
The von Mangoldt function over N is denoted by Λ(n). The Euler totient
function is denoted by ϕ and it satisfies the identity
1
ϕ(n) = n 1
− p
p|n (cid:18) (cid:19)
Y
pprime
Awell knowngeneralization of this identity gives the Jordantotient func-
tion of order s N:
∈
1
J (n) := ns 1
s − ps
p|n (cid:18) (cid:19)
Y
pprime
where n N.
∈
We would be using the following identity for the Jordan totient function:
Proposition 1. ([4], Chapter 2, Exer. 17) For n 1, we have
≥
J (n)= dsµ(n/d) = (n/d)sµ(d)
s
d|n d|n
X X
For x R,x > 0, the Gamma function Γ is defined by
∈
∞
Γ(x)= e−ttx−1dt
Z0
CERTAIN WEIGHTED AVERAGES OF GENERALIZED RAMANUJAN SUMS 3
Now we recall the definition of the Bernoulli Polynomials and Bernoulli
numbers. For any x C define the functions B (x) by the equation
n
∈
zexz ∞ B (x)
= n zn where z <2π
ez 1 n! | |
− n=0
X
If we put x = 0, we get (with the convention that B := B (0)), then
n n
∞
z B
= nzn where z < 2π
ez 1 n! | |
− n=0
X
B (x) are called Bernoulli polynomials and B = B (0) are called Bernoulli
n n n
numbers.
Following are some important properties of Bernoulli polynomials (See
section 12.2 of [4]) which we use later:
Proposition 2. For n N,n 2, the Bernoulli numbers satisfy
∈ ≥
(1) B (0) = B (1), and
n n
(2) B = n n B
n k=0 k k
Next we dePfine th(cid:0)e(cid:1)generalized gcd function:
Definition 3. The generalized GCD function on N N denoted by (j,ks)
s
is defined to give the largest d N such that dk×and ds j. Therefore,
∈ | |
(j,ks) = 1 means that no natural number d 1 exists such that dk and
s
≥ |
ds j.
|
When s = 1, the generalized GCD function becomes the usual GCD
function.
For k N,j Z, the Ramanujan sum defined as the sum
∈ ∈
k
c (j) = e2πijm/k
k
m=1
X
gcd(m,k)=1
In [6], E. Cohen generalized this to
Definition4. Fors,k N,j Z, thegeneralizedRamanujan sumisdefined
∈ ∈
as
k
c(s)(j) = e2πijm/ks
k
m=1
(j,kXs)s=1
Recall that, for Ramanujan sum, we have
k k
c (j) = dµ( )= µ(d)
k
d d
d|(k,j) d|(k,j)
X X
The following similar identity holds for the generalized Ramanujan sum ([6]
Section 2):
Proposition 5.
k k
(1) c(s)(j) = dsµ( ) = dsµ( )
k d d
d|(Xj,ks)s Xd|js
d|k
4 KVISHNUNAMBOOTHIRI
Holder’s (See [11]) evaluation of the Ramanujan sums is given by
ϕ(k)µ(k/gcd(k,j))
c (j) = where k N,j Z
k
ϕ(k/gcd(k,j)) ∈ ∈
For the generalized sum, Cohen showed that([8], section 2)
Proposition 6. (Generalized Holder’s identity)
J (k)µ(k/(j,ks) )
(2) c (s)(j) = s s where j,k,s N
k J (k/(j,ks) ) ∈
s s
Definition 7. For k,n N, the function θ (n) is defined as
k
∈
1,if (k,n) = 1
θ (n):=
k
(0,if (k,n) > 1
Let denotetheC linearspaceofallarithmeticfunctionswiththeusual
F −
operations.
Definition 8. An arithmetic function f is said to be k periodic for a
fixed k N if f(n+k) = f(k) for all n ∈NF. −
∈ ∈
The following proposition which we use later, can be found, for example,
in [16].
Proposition 9. Fork N, everyk periodic f hasaFourierexpansion
∈ − ∈ F
of the form
k
f(k) = g(j)exp(2πijn/k)
j=1
X
where the Fourier coefficients g(j) are unique, and are infact given by the
equation
k
1
g(m) = f(k)exp( 2πijn/k)
k −
j=1
X
Anyother notations anddefinitionsweusewillbeintroducedlater asand
when required.
3. Problems
For r N, E. Alkan established an identity for the weighted average
∈
k
1
jrc (j)
kr+1 k
j=1
X
based on which he could establish many results in [1] and [2]. If we denote
the above weighted average by S (k), the identity given by E. Alkan is
r
precisely the following:
Proposition 10.
⌊r/2⌋
ϕ(k) 1 r+1 1
(3) S (k) = + B (1 )
r 2k r+1 2m 2m − p2m
m=0(cid:18) (cid:19) p|k
X Y
CERTAIN WEIGHTED AVERAGES OF GENERALIZED RAMANUJAN SUMS 5
This identity appeared as Eq. 2.19 in [2] . He used this identity to
prove exact formulas for certain mean square averages of special values of
L functions. Alkan described these mean square averages in [1] and proved
−
an asymptotic formula for S k in [2] based on the identity given above.
r
k≤x
How did he proceed to estabPlish the identity is, that, he used Holder’s eval-
uation of the Ramanujan sums [11] and applied the formula
(4)
n nr+1 ⌊r/2⌋ r+1 B
jr = 2m (1 p2m−1) (n,r N,n > 1)
r+1 2m n2m − ∈
j=1 m=0(cid:18) (cid:19) p|n
X X Y
gcd(j,n)=1
(See [17] for details) and then considered the cases r even and r odd seper-
ately. The identity (3) and its proof appears in [3] also.
The proof of the identity (3) given in [1] was a little bit complicated. A
simpler proof this identity (3) appeared in a paper [18] of L.T´oth. In ad-
dition, in the same paper, To´th derived identities for some other weighted
averages of theRamanujan sumswith weights concerninglogarithms, values
of arithmetic functions for gcd’s, the Gamma function, the Bernoulli num-
bers, binomial coefficients, and the exponential function. Our focus will be
onestablishingtheweightedaverages ofTo´thforthegeneralizedRamanujan
sums (of E. Cohen) replacing the classical Ramanujan sums. Considering
that the genralized Ramanujan sum with s = 1 becomes the classical Ra-
manujan sum, it may not be very surprising to note that, the identities we
establish here will resemble To´th’s identities when we put s = 1. In fact,
the way in which we proceed to establish the identities closely resembles the
steps given in [18].
4. Alkan’s Identity
As we mentioned earlier, the most important computations in [1] and
[2] of E. Alkan depends heavily on the identity given in Proposition (10).
The identity in its final form consists of the Euler totient function and the
Jordan totient function. We propose and prove a similar identity for the
generalized Ramanujan sum consisting of similar type of functions. Our
identity is precisely the following:
Proposition 11. For s,k,r N, define the weighted average S(s)(k) by
r
∈
ks
1
(5) S(s)(k) = jrc(s)(j)
r ks(r+1) k
j=1
X
Then,
⌊r/2⌋
J (k) 1 r+1 J (k)
(6) S(s)(k) = s + B s
r 2k r+1 2m 2m k2ms
m=0(cid:18) (cid:19)
X
6 KVISHNUNAMBOOTHIRI
Proof. Using identity (1), we get
ks
1 k
S(s)(k) = jr dsµ( )
r ks(r+1) d
j=1 d|js
X X
d|k
ks/ds
1 k
= dsµ( ) (nds)r
ks(r+1) d
d|k n=1
X X
ks/ds
ds(r+1) k
= µ( ) nr
ks(r+1) d
d|k n=1
X X
Now from [9], Prop. 9.2.12, for every N,r N, we have
∈
N r
1 r+1 r+1
nr = Nr+1+ Nr + B Nr+1−m
m
r+1 2 m
!
n=1 m=2(cid:18) (cid:19)
X X
Note that B = 1, and B = 1 (See [4], section 12.12). Also, for m 1,
0 1 −2 ≥
B = 0 (See [4] Theorem 12.16). Therefore, B ,B ,... are all equal to
2m+1 3 5
0. So we can rewrite the above expression as
N
1 r+1 r+1
nr = ( 1)0B Nr+1−0( 1)1B Nr+1−1
0 1
r+1 − 0 − 1
n=1 (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)
X
r
r+1
+ B Nr+1−m
m
m
m=2(cid:18) (cid:19) (cid:19)
X
In this expression, after B , only even order Bernoulli numbers survive and
1
so we need to consider only such terms in the summation. Rewriting using
this information
N 1 r+1 1 ⌊r2⌋ r+1
nr = ( 1)1B Nr+1−1+ B Nr+1−2m
1 2j
r+1 − 1 r+1 2m
n=1 (cid:18) (cid:19) m=0(cid:18) (cid:19)
X X
⌊r⌋
Nr 1 2 r+1
= + B Nr+1−2m
2m
2 r+1 2m
m=0(cid:18) (cid:19)
X
Using this new expression, we get
⌊r⌋
ds(r+1) k ks r 1 1 2 r+1 ks r+1−2m
S(s)(k) = µ( ) + B
r ks(r+1) d  ds 2 r+1 2m 2m ds 
d|k (cid:18) (cid:19) m=0(cid:18) (cid:19) (cid:18) (cid:19)
X X
 ⌊r⌋ 
1 ds k 1 2 r+1 d2ms k
= µ( )+ B µ( )
2 ks d r+1 2m 2m k2ms d
d|k m=0(cid:18) (cid:19) d|k
X X X
⌊r⌋
1J (k) 1 2 r+1 J (k)
s 2ms
= + B
2 ks r+1 2m 2m k2ms
m=0(cid:18) (cid:19)
X
and this is what we claimed. (cid:3)
CERTAIN WEIGHTED AVERAGES OF GENERALIZED RAMANUJAN SUMS 7
When s = 1, the above identity becomes the one given by Alkan (Prop.
10).
5. log as weight
We proceed now to generalize the other weighted averages computed by
To´th([18]). The next one uses the logarithmic function as weight.
Proposition 12. For every s,k N, we have
∈
1 k ds k k
(s)
logjc (j) = sΛ(k)+ µ( )log !
k k k d ds
j=1 d|k (cid:18) (cid:19)
X X
Proof. We use identity (1) to proceed:
k k
k
logjc(s)(j) = logj dsµ( )
k d
j=1 j=1 d|js
X X X
d|k
k
k ds
= dsµ( ) log(mds)
d
d|k m=1
X X
k
k ds k
= dsµ( ) [log(1.ds)+log(2.ds)+...+log( ds)]
d ds
d|k m=1
X X
k k
= dsµ( )log[( )!(ds)dks]
d ds
d|k
X
k k k k
= dss logdµ( )+ dsµ( )log( )!
ds d d ds
d|k d|k
X X
k k k
= sk µ( )logd+ dsµ( )log( )!
d d ds
d|k d|k
X X
Note that µ(k)logd = (µ log)(k) = Λ(k). So we get
d ∗
d|k
P
1 k ds k k
(s)
logjc (j) = sΛ(k)+ µ( )log( )!
k k k d ds
j=1 d|k
X X
(cid:3)
6. Function of generalized GCD as weight
Now we proceed to establish an identity with weight as the generalized
GCD function. But instead, we prove it generally for arithmetic functions
composed with the GCD function. This, on taking the first function to be
the function N as a special case, we get the weight as the generalized GCD
function.
8 KVISHNUNAMBOOTHIRI
Proposition 13. For s,k,j N
∈
ks
f((j,ks) )c(s)(j) = J (k)[(f Ns) (µ Ns)](k)
s k s ◦ ∗ ◦
j=1
X
Proof. We use the generalized Holder’s evaluation (2) to start off.
ks ks J (k)µ(k/(j,ks) )
f((j,ks) )c(s)(j) = f((j,ks) ) s s
s k s J (k/(j,ks) )
s s
j=1 j=1
X X
Now we group the terms in the sum according to ds = (j,ks) ,
s
ks µ(ks) kdss
f((j,ks) )c(s)(j) = J (k) f(ds) ds 1
s k s J (ks)
j=1 ds|ks s ds m=1
X X (m,Xkdss)s=1
µ(ks) ks
= J (k) f(ds) ds J ( )
s J (ks) s ds
ds|ks s ds
X
ks
= J (k) f(ds)µ( )
s ds
ds|ks
X
which is equivalent to the identity we have to prove. (cid:3)
7. The Gamma function as weight
To establish the next average, we need a few preliminary lemmas. We
have the following identity which appears in [9] as exercise 1.8.45:
Lemma 14. For s,k N
∈
µ(d) ϕ(k) logp
logd=
d − k p 1
d|k p|k −
X X
pprime
We will give a generalization of this lemma in terms of Jordan totient
function. This lemma could be of some other independent interest.
Lemma 15. For s,k N
∈
µ(d) J (k) logp
s
logd=
ds − ks ps 1
d|k p|k −
X X
pprime
Proof. When we take all the divisors of k, only those µ(d) survive in the
computation where d is square free. Now recall that
J (k) 1
s
= 1 .
ks − ps
p|k (cid:18) (cid:19)
Y
pprime
Sothequotient Js(k) considerseachprimefactorinadivisordofkonlyonce.
ks
Therefore, without any loss of generality we may assume that k is square
free. Letk = p wherep aredistinctprimesandD = p ,...,p denote
i i 1 r
{ }
pi|k
Q
CERTAIN WEIGHTED AVERAGES OF GENERALIZED RAMANUJAN SUMS 9
the set of all prime divisors of k. With this assumption and notation, the
the RHS of the expression we want to verify becomes
J (k) logp 1 logp
s i i
= (1 )
ks ps 1 − ps ps 1
pXi∈D i − pYi∈D i pXi∈D i −
(ps 1)logp
 j − i
= psi −1pPi∈DppjjQ6=∈Dpi 
ps  (ps 1) 
pYi∈D i pi∈D i −
Q
(ps 1)logp
 j − i
pi∈D pj∈D
P pjQ6=pi 
=
 ps 
i
pi∈D
Q
We introduce a notation for convenience. S will stand for the set of n
n,pi
primes (where n < r) from D and contains p and S will stand for the set
i n
of all S where p D. Therefore, S = D. With this notation, we get
n,pi i ∈ r,pi
J (k) logp 1
sks ps 1i =  ps logpi (psj −1)
pXi∈D i − piQ∈D ipXi∈D ppYjj6=∈Dpi 
  
1
=  ps logpi psj − psj +...+
piQ∈D ipXi∈D ppYjj6=∈Dpi Sr−1X,pi∈Sr−1pjp∈Yj6=Srp−i1 
  
1 1
= logp +...+
ips − ps 
pXi∈D  i S2,Xpi∈S2 pj∈S2 j 
 Q 
Computing the LHS now:
µ(d) logp
1
logd =
ds − ps
1
d|k
X
logp p logp p
1 2 r−1 r
+ +...+
− psps ps ps
(cid:18) 1 2 r−1 r (cid:19)
logp p p logp p p
1 2 3 r−2 r−1 r
+ +...+
pspsps ps ps ps
1 2 3 r−2 r−1 r
logp ...p
+ ...+( 1)r 1 r
− ps...ps
1 r
10 KVISHNUNAMBOOTHIRI
grouping the terms on p ,
i
µ(d) 1 1 1
logd = logp +
ds i − ps ps − ps
Xd|k pXi∈D (cid:18) i S2,Xpi∈S2 pj∈S2,pi j S3,Xpi∈S3 pj∈S3,pi j
Q Q
+ ...
1
+ ( 1)k
− ps
Sk,Xpi∈Sk pj∈Sk,pi j
Q
1
+ ...+( 1)r−1
− ps
Sr−1,Xpi∈Sr−1 pj∈Sr−1,pi j
Q
1
+ ( 1)r
− ps
j(cid:19)
pj∈D
Q
which is the same as the RHS computed above. (cid:3)
The above lemma will be used in the proof of the next proposition.
Proposition 16. For s,k N
∈
ks
1 s logp log2π
logΓ(j/ks)c(s)(j) =
J (k) k 2 ps 1 − 2
s
j=1 p|k −
X X
Proof.
ks ks
k
logΓ(j/ks)c(s)(j) = logΓ(j/ks) dsµ( )
k d
j=1 j=1 ds|j,ds|ks
X X X
ks
k ds
= dsµ( ) logΓ(mds/ks)
d
ds|ks m=1
X X
ks
k ds m
= dsµ( )log Γ( )
d ks
ds|ks m=1 ds
X Y
Now [9], Proposition 9.6.33 tells that
(2π)(n−1)/2
Γ(j/N) =
√n
1≤j≤N
Y
Using this, we get