Ramanujan series for arithmetical functions

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We give a short survey of old and new results in the theory of Ramanujan expansions for arithmetical functions.

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Ramanujan series for arithmetical functions
M. Ram Murty
To cite this version:
M. Ram Murty. Ramanujan series for arithmetical functions. Hardy-Ramanujan Journal,
Hardy-Ramanujan Society, 2013, 36, pp.21 - 33. <hal-01112690>
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HARDY-RAMANUJAN JOURNAL 36 (2013), 21-33
RAMANUJAN SERIES FOR ARITHMETICAL FUNCTIONS
M. RAM MURTY
Abstract. We give a short survey of old and new results in the theory of Ra-
manujan expansions for arithmetical functions.
1. Introduction
In 1918, Ramanujan [17] published a seminal paper entitled “On certain trigono-
metric sums and their applications in the theory of numbers” in which he introduced
sums (now called Ramanujan sums) defined as
(1) cq(n) =
q
X
a=1
(a,q)=1
cos 2πan
q
for any two natural numbers qand n. It is easy to see that this can be re-written as
(2) cq(n) =
q
X
a=1
(a,q)=1
e2πian/q
since (a, q) = 1 if and only if (q−a, q) = 1 so that we can pair up elements in (2) to
derive (1).
These sums have remarkable properties. First, for fixed n,cq(n) is a multiplicative
function. In other words, if q1, q2are relatively prime, then
cq1(n)cq2(n) = cq1q2(n).
Second, from (2), it is easily seen that cq(n) is a periodic function of nwith period
q. That is, the value of cq(n) depends only on the arithmetic progression of n(mod
q). Third, using the familiar M¨obius function, one can derive an explicit formula for
cq(n):
(3) cq(n) = X
d|(q,n)
µ(q/d)d,
2010 Mathematics Subject Classification. Primary: 11-02, Secondary: 11A25.
Key words and phrases. Ramanujan sums, Ramanujan expansions, mean values, trigonometric
series.
Research partially supported by a Simons Fellowship and an NSERC Discovery grant.

22 M. RAM MURTY
where µdenotes the M¨obius function. This is easily seen by using the characteristic
property of the M¨obius function:
(4) X
d|n
µ(d)=0, n > 1
and is equal to 1 if n= 1. Hence, from (2),
cq(n) =
q
X
a=1
e2πian/q 
X
d|(a,q)
µ(d)
=X
d|q
µ(d)

q
X
a=1,d|a
e2πian/q 
.
The inner sum can be re-written as
q/d
X
b=1
e2πibn/(q/d)
which equals q/d if nis divisible by q/d and zero otherwise. Thus,
cq(n) = X
d|q,(q/d)|n
µ(d)(q/d).
As druns over divisors of qso does q/d and we may re-write the above sum as
cq(n) = X
d|q,d|n
µ(q/d)d,
which is (3). In his work on the cyclotomic polynomial, H¨older [12] derived in 1936
the explicit formula:
cq(n) = φ(q)
φ(q/(q, n))µ(q/(q, n)),
where φdenotes Euler’s function. More generally, one can study generalized Ra-
manujan sums of the form X
d|q,d|n
f(d)g(q/d),
with fand garbitrary arithmetical functions, and derive some interesting results as in
Apostol [1]. In a forthcoming paper, Fowler, Garcia and Karaali [4] show that many
properties of Ramanujan sums can be deduced using the theory of supercharacters
which seems to be an emerging new topic of group theory.
A convenient function to introduce is εd(q) which is dif d|qand is zero otherwise.
This allows us to write (3) as
cq(n) = X
d|q
εd(n)µ(q/d)
so that by M¨obius inversion we have
εq(n) = X
d|q
cd(n).

RAMANUJAN SERIES FOR ARITHMETICAL FUNCTIONS 23
This permits us to deduce several elegant Ramanujan expansions. For instance, let
σs(n) = X
d|n
ds.
Then,
σs(n)
ns=X
d|n
1
ds=
∞
X
d=1
1
ds+1 εd(n) =
∞
X
d=1
1
ds+1 X
q|d
cq(n).
Interchanging summation, we find the sum is
σs(n)
ns=ζ(s+ 1)
∞
X
q=1
cq(n)
qs+1
and all the sums are absolutely convergent for <(s)>0 since |cq(n)| ≤ σ1(n). This
formula appears in [17].
One can derive several variations of this result. For instance, if we let
σs(n, χ) = X
d|n
χ(d)ds,
then essentially the same argument as above shows that for any Dirichlet character
χ(mod q), and (n, q) = 1, we have
χ(n)σs(n, χ)
ns=L(s+ 1, χ)
∞
X
q=1
χ(q)cq(n)
qs+1 .
More elaborate examples of this nature can be found in [2]. Ivic [13] has shown that
generally, for any completely multiplicative function f, we have by essentially the
same argument that
X
d|n
f(d)d−s= ∞
X
q=1
f(q)
qs+1 ! ∞
X
q=1
f(q)cq(n)
ns+1 !.
Formula (3) also allows us to deduce that cq(n) is an ordinary integer though this
is not obvious from the definition of (1) or (2). (One can also deduce this by viewing
(2) as the trace of the algebraic integer ζn
qin the cyclotomic field Q(ζq).) Thus, if
(q, n) = 1, then cq(n) = µ(q). Many of the basic properties of the Ramanujan sums
are collected in [9].
In his paper, Ramanujan [17] derives a variety of expressions of the form
(5)
∞
X
q=1
aqcq(n)

24 M. RAM MURTY
for some arithmetical functions. For instance, for the divisor function, Ramanujan
showed that
d(n) = −
∞
X
q=1
log q
qcq(n).
If r(n) denotes the number of ways of writing nas a sum of two squares, Ramanujan
proves that
r(n) = π
∞
X
q=1
(−1)q
2q+ 1c2q+1(n).
We call series of the form (5) Ramanujan series (or Ramanujan expansion or some-
times Ramanujan-Fourier series) since such series mimic the notion of a Fourier ex-
pansion of an L1-function. More precisely, given an arithmetical function f, we say
that fadmits a Ramanujan expansion if
f(n) =
∞
X
q=1 b
f(q)cq(n)
for appropriate complex numbers b
f(q) and the series on the right hand side converges.
We say that b
f(q) is the q-th Ramanujan coefficient of f.
Several natural questions now arise. First, for which arithmetical functions do
we have such a series. If such a series exists for a given function f, then how can
we determine the Ramanujan coefficients b
f(q). What can we say about the rate of
convergence of such a series? These are the questions we explore in this (largely
survey) paper.
Some of these questions can be quite subtle. As Ramanujan observes in his paper
[17], the assertion
∞
X
q=1
cq(n)
q= 0
is equivalent to the prime number theorem. This equation also shows that the Ra-
manujan coefficients of a given function need not be unique (since the above is an
expansion of the zero function).
A more complicated example was given by Hardy [7] for the von Mangold function
Λ(n) defined as log pwhen nis a power of the prime pand zero otherwise:
φ(n)
nΛ(n) =
∞
X
q=1
µ(q)
φ(q)cq(n).
This expression, in conjunction with a Wiener-Khinchine type conjecture for Ra-
manujan expansions, was applied by Gadiyar and Padma [5] to give a heuristic
derivation of the celebrated conjectural formula of Hardy and Littlewood on the

RAMANUJAN SERIES FOR ARITHMETICAL FUNCTIONS 25
number of twin primes up to x. We will describe their work below and take this op-
portunity to formulate a general question of Wiener-Khinchine type for a wide class
of arithmetic functions. In a forthcoming paper, we plan to explore this in greater
detail.
Regarding the first question of when such series expansions exist, there are a num-
ber of results for which we refer the reader to [18]. For instance, using functional
analysis, Spilker [21] has proved that every bounded arithmetical function admits
a Ramanujan expansion. Hildebrand [10] gave a surprisingly simpler proof of this
result without the boundedness assumption. However, the Ramanujan coefficients in
many of these cases are not “natural.”
2. An orthogonality principle
It was Carmichael [2] who noticed an orthogonality principle for the Ramanujan
sums. This observation led him to predict what the Ramanujan coefficients of a
given arithmetical function should be, if such an expansion exists. Indeed, given an
arithmetical function f, we write M(f) for the limit
lim
x→∞
1
xX
n≤x
f(n)
when it exists and call it the mean value of f. We also write Thfor the shift operator:
Th(f)(n) = f(n+h). Then,
Theorem 1. (Orthogonality relations)
lim
x→∞
1
xX
n≤x
cr(n)cs(n) = φ(r)
if r=sand zero otherwise. More generally,
lim
x→∞
1
xX
n≤x
cr(n)cs(n+h) = cr(h)
if r=sand zero otherwise. In other words, M(crTh(cs)) = cr(h)if r=sand zero
otherwise.
Proof. We evaluateX
n≤x
cr(n)cs(n) = X
(a,r)=1 X
(b,s)=1 X
n≤x
e2πi(n(a/r+b/s)) .
Now, if r6=sthen a/r +b/s is not an integer. Indeed, if it were equal to m(say),
then
as +br =mrs =br +as,
from which we see that r|sand s|rso that r=s, a contradiction. Thus, if r6=s,
the innermost sum is a geometric sum which is bounded. Hence the limit in question

26 M. RAM MURTY
is zero. If r=s, then the innermost sum is again bounded if (a+b)/r is not an
integer. Thus, the limit is non-zero only in the case a≡ −b(mod r) and the result
is now immediate. For the second assertion, we proceed as before. Namely, we write
out the Ramanujan sums. Interchanging summations, we see the limit in question is
r
X
b=1
(b,r)=1
e2πihb/r
q
X
a=1
(a,q)=1
lim
x→∞
1
xX
n≤x
e2πin(a/q+b/r).
As before, if r6=q, then a/q +b/r is not an integer and the innermost sum is bounded
so that the limit is zero. If r=q, then the innermost sum is bounded unless a+b≡0
(mod q), in which case the limit is 1. This completes the proof. 
This orthogonality principle allows one to (heuristically) write down possible can-
didates for the Ramanujan coefficients of any given arithmetical function. Indeed,
if
f(n) =
∞
X
q=1 b
f(q)cq(n),
then, multiplying both sides of the equation by cr(n) and taking the mean value of
both sides, we find on interchanging the sum that
b
f(r)φ(r) = M(fcr).
The second property allows us to deduce that if f(n) admits a Ramanujan expansion
with Ramanujan coefficients b
f(q), then, so does the function f(n+h) which has
Ramanujan coefficients b
f(q)cq(h)/φ(q).
There has been extensive study of when Ramanujan expansions exist and we refer
the reader to some excellent surveys like [15] and [18] for details and additional
references. However, in the next section, we highlight two important theorems due
to Wintner and Delange (see for example, Cor. 2.3 of [18] or [23]) in this context.
Generalizations of this theory to functions of several variables have been investigated
by several authors, most notably, Ushiroya [22], but the theory is still in its infancy.
For instance, one can investigate Ramanujan expansions of arithmetical functions of
several variables and see under what conditions these expansions are valid.
3. The theorems of Wintner and Delange
The theorem of Wintner discussed below allows us to determine a large number of
Ramanujan expansions.
Theorem 2. (Wintner, 1943) Suppose that
f(n) = X
d|n
g(d),

RAMANUJAN SERIES FOR ARITHMETICAL FUNCTIONS 27
and that ∞
X
n=1
|g(n)|
n<∞.
Then, M(f) = P∞
n=1 g(n)/n.
Proof. We have
1
xX
n≤x
f(n) = 1
xX
d≤x
g(d)[x/d] =
∞
X
d=1
g(d)
d+O X
d>x
|g(d)|
d!+O 1
xX
d≤x
|g(d)|!.
By our hypothesis, the first big O-term goes to zero as xtends to infinity. To study
the last term, we need only apply partial summation. Let S(x) = Pn≤x|g(n)|/n.
Then, X
n≤x
|g(n)|=S(x)x−Zx
1
S(t)dt =Zx
1
(S(x)−S(t))dt.
Now fix  > 0. Then, there is an x0such that |S(x)−S(t)| ≤ for x>t>x0. Thus,
splitting the integral into two parts, we see that our sum in question is
≤2αx0+x,
which completes the proof. 
More generally, Wintner showed that if
∞
X
n=1
|g(n)|
n<∞,
then the Ramanujan coefficients M(fcr) exist for every r. If in addition,
∞
X
n=1
|g(n)|d(n)
n<∞,
then the Ramanujan expansion converges pointwise to f(n). This was later improved
by Delange [3] who proved the following theorem regarding Ramanujan expansions.
Theorem 3. (Delange, 1976) Suppose that
f(n) = X
d|n
g(d),
and that ∞
X
n=1
2ω(n)|g(n)|
n<∞,
where ω(n)is the number of distinct prime divisors of n. Then, fadmits a Ramanu-
jan expansion with
b
f(q) =
∞
X
m=1
g(qm)
qm .

28 M. RAM MURTY
The proof of Delange’s theorem is not difficult and we refer the reader to [3] for
complete details. However, we can here indicate the main ideas of the proof. One
first shows that the double series
X
m,q≥1
g(qm)
qm cq(n)
is absolutely convergent. Once this is done, we evaluate
∞
X
k=1
wk,
where
wk=X
qm=k
g(qm)
qm cq(n) = g(k)
kX
q|k
cq(n).
As noted earlier, the latter sum is εk(n) which is kif k|nand zero otherwise. Thus,
wk=g(k) if k|nand zero otherwise, from which the theorem follows immediately.
It is clear that the arithmetical function for <(s)>0, σs(n)/nssatisfies the con-
ditions of Delange’s theorem and thus admits a Ramanujan expansion and the Ra-
manujan coefficients are also determined from the theorem. Despite its simplicity
and beauty, this theorem does not include the more subtle Ramanujan expansions of
functions such as the divisor function or the von Mangoldt function. Several authors
have developed larger theories that would enable one to derive such expansions. We
refer the reader to the recent paper of Lucht [15] for details and additional references.
For instance, Lucht proves:
Theorem 4. Let b
f:N→C. If the series
g(d) := d
∞
X
m=1
µ(m)b
f(dm)
converges for every natural number d, then for
f(n) = X
d|n
g(d),
we have
f(n) =
∞
X
q=1 b
f(q)cq(n),
This theorem allows one to deduce, for example, Ramanujan’s expansion of the
divisor function. Indeed, if we take
b
f(n) = log n
n,

RAMANUJAN SERIES FOR ARITHMETICAL FUNCTIONS 29
then
g(d) =
∞
X
m=1
µ(m)
dm log dm =−1
since
∞
X
m=1
µ(m)
m= 0,
and
∞
X
m=1
µ(m)
mlog m=−1,
both assertions being quite subtle and requiring the theory of the Riemann zeta func-
tion. Indeed, the former is known to be equivalent to the prime number theorem. One
can also deduce the Wintner-Delange theorems from this result. However, Hardy’s
Ramanujan expansion of φ(n)Λ(n)/n noted in the introduction, cannot be deduced
from this result. Indeed, taking the cue from Hardy’s expansion, we set
g(d) = d
∞
X
m=1
µ(dm)
φ(dm)µ(m).
This series needs to converge for us to be able to deduce the Hardy expansion. Un-
fortunately, the series diverges since we may restrict the sum to those mwhich are
coprime to d(since the M¨obius function vanishes otherwise), in which case the sum-
mand is
µ(d)µ2(m)/φ(d)φ(m)
and the series clearly diverges.
Hardy [7] obtains his expansion by considering the Dirichlet series
f(s) =
∞
X
q=1
µ(q)cq(n)
qs−1φ(q)
which converges absolutely for <(s)>1. By the mutliplicative properties of the
Ramanujan sum and the Euler function, we see that
f(s) = Y
p1−cp(n)
ps−1φ(p),
where the product is over all primes p. Now cp(n) = −1 if pis coprime to nand p−1
otherwise. Thus,
f(s) = Y
p|n1−1
ps−1Y
(p,n)=1 1 + 1
ps−1(p−1).

30 M. RAM MURTY
We can rewrite this as
Y
p|n(p−1)(ps−1−1)
ps−ps−1+ 1 Y
p1 + 1
ps−1(p−1)=g(s)h(s) say.
It is evident that if nis a prime power, g(s) has a simple zero at s= 1, and if nis not
a prime power, it has a zero of order at least 2. On the other hand, it is not difficult
to see that
h(s) = ζ(s)h1(s),
where h1(s) is regular for <(s)>0 and non-vanishing there. Thus, f(s) is zero at
s= 1 when nis not a prime power and is equal to
p−1
plog p
if nis a prime power. To complete the proof, we need to establish that the series
converges to f(1). But this can be deduced in several ways. Hardy deduces it by
appealing to his earlier work with Littlewood [8] (see in particular Theorem D on page
218) where he showed that the prime number theorem is equivalent to the assertion
that
∞
X
n=1
µ(n)
n= 0.
A more direct method is to use Perron’s formula for the partial sums and use contour
integration which is a standard method in analytic number theory. The reader can
find details of this method in [16].
4. The Wiener-Khintchine formula and analogues
In the theory of Fourier series, the Wiener-Khintchine formula states that if
f(t) = X
n
fneiλnt,
then
lim
T→∞
1
2TZT
−T
f(t+u)f(t)dt =X
n
|fn|2eiλnu.
Inspired by this theorem, Gadiyar and Padma [5] have asked if it is reasonable to
expect that for a function f(n) with Ramanujan coefficients b
f(q), a similar result
holds. Namely, is it true that
lim
x→∞
1
xX
n≤x
f(n)f(n+h) =
∞
X
q=1 b
f(q)2cq(h)?

RAMANUJAN SERIES FOR ARITHMETICAL FUNCTIONS 31
We propose here a more general question. Given two arithmetical functions fand g,
when is it true that
lim
x→∞
1
xX
n≤x
f(n)g(n+h) =
∞
X
q=1 b
f(q)bg(q)cq(h).
In a forthcoming paper [6], we show this is true for a wide class of functions. However,
one must exercise some caution since there are already some contradictions, even in
the simplest cases.
Still, the question is intriguing since as noted in [5], the case f(n) = g(n) =
φ(n)Λ(n)/n gives the Hardy-Littlewood prime-tuplet conjecture (which incidentally,
was conjectured using the more sophisticated circle method). Indeed, from Hardy’s
expansion, we would have
lim
x→∞
1
xX
n≤x
φ(n)
n
φ(n+h)
n+hΛ(n)Λ(n+h) =
∞
X
q=1
µ2(q)
φ2(q)cq(h).
The last sum can be expressed as an infinite product over prime numbers:
Y
p1 + cp(h)
(p−1)2.
Since cp(h) = −1 if pis coprime to hand p−1 if p|h, we see that this product is
Y
p|h1 + 1
p−1Y
(p,h)=1 1−1
(p−1)2
which agrees with the Hardy-Littlewood constant obtained by them using heuristic
reasoning based on the circle method.
The conjecture can be viewed as a special case of a Parseval type formula in a
suitable Hilbert space. Indeed, the authors in [11] consider the C-linear span of all
Ramanujan sums and consider the closure of this space (denoted B2) with respect to
the (semi)-norm
||f||2
2:= lim sup
x→∞
1
xX
n≤x
|f(n)|2.
For functions in this space, they establish using elementary Hilbert space theory, the
Parseval identity:
||f||2
2=
∞
X
q=1
|b
f(q)|2φ(q).
Indeed, the space B2(modulo null functions) can be equipped with an inner product
(f, g) := M(fg).
This makes B2into a Hilbert space and we have the general Parseval theorem from
which our result above follows. This is not particular to our situation but a general

32 M. RAM MURTY
fact about Hilbert spaces that the more specialized version of Parseval’s formula
implies the general one via a useful identity, namely,
(f, g) = 1
4||f+g||2
2− ||f−g||2
2+i||f+ig||2
2−i||f−ig||2
2.
The difficulty in extending this theorem to a wider class is formidable since as we
said above, such a general theorem would lead to a proof of the twin-prime conjecture.
Nevertheless, the theory of Ramanujan expansions does give a viable framework to
understand a wide spectrum of questions and it would be interesting to investigate
this further.
5. Concluding remarks
At the end of his paper, Ramanujan [17] derives the expression
Λ(n) = −
∞
X
q=1
cn(q)
q,
which is “dual” to the expansions we have considered till now. This suggests that
perhaps there is a “dual theory” of such expansions of the form
f(n) =
∞
X
q=1
f∗(q)cn(q).
Several extensions of Ramanujan’s expansions of this type to generalizations of the
von Mangoldt function appear in [13].
Finally, we mention the monograph of Sivaramakrishnan [20] which contains a
wealth of material on Ramanujan sums including a curious “reciprocity law” which
surely will find a place in a larger theory. We also highlight the relevance of the
theory of almost periodic functions as studied in [14] where Ramanujan expansions
are derived for certain strongly multiplicative function.
The interesting thing about the right hand side of (5) is that if the series converges
absolutely and we replace nby a real number x, we obtain a continuous function
which interpolates the given arithmetical function. In this way, we can view the
Ramanujan expansion as a continuous analogue of the discretely defined arithmetical
function.
Acknowledgements. I would like to thank Kumar Murty, S. Gun, P. Rath, G.
Gadiyar and R. Padma for their comments on an earlier version of this article.
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RAMANUJAN SERIES FOR ARITHMETICAL FUNCTIONS 33
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Department of Mathematics, Queen’s University, Kingston, Ontario,
K7L 3N6, Canada
E-mail address: murty@mast.queensu.ca