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To appear in Journal of Number Theory
CERTAIN ETA-QUOTIENTS AND ARITHMETIC DENSITY OF
ANDREWS’ SINGULAR OVERPARTITIONS
AJIT SINGH AND RUPAM BARMAN
Abstract. In order to give overpartition analogues of Rogers-Ramanujan
type theorems for the ordinary partition function, Andrews defined the so-
called singular overpartitions. Singular overpartition function Ck,i(n) counts
the number of overpartitions of nin which no part is divisible by kand only
parts ≡ ±i(mod k) may be overlined. Andrews also proved two beautiful Ra-
manujan type congruences modulo 3 satisfied by C3,1(n). Later on, Aricheta
proved that for an infinite family of `,C3`,`(n) is almost always divisible by 2.
In this article, for an infinite subfamily of `considered by Aricheta, we prove
that C3`,`(n) is almost always divisible by arbitrary powers of 2. We also
prove that C3`,`(n) is almost always divisible by arbitrary powers of 3 when
`= 3,6,12,24. Proofs of our density results rely on the modularity of certain
eta-quotients which arise naturally as generating functions for the Andrews’
singular overpartion functions.
1. Introduction and statement of results
Beginning with the paper [7], Corteel and Lovejoy introduced and developed
the theory of overpartitions. An overpartition of nis a non-increasing sequence of
natural numbers whose sum is nin which the first occurrence of a number may be
overlined. Thus the eight overpartitions of 3 are 3,3,2 + 1,2 + 1,2 + 1,2 + 1,1 +
1+1,1 + 1 + 1. In order to give overpartition analogues of Rogers-Ramanujan
type theorems for the ordinary partition function with restricted successive ranks,
Andrews [2] defined the so-called singular overpartitions. Andrews’ singular over-
partition function Ck,i(n) counts the number of overpartitions of nin which no
part is divisible by kand only parts ≡ ±i(mod k) may be overlined. For example,
C3,1(4) = 10 with the relevant partitions being 4,4,2 + 2,2 + 2,2+1+1,2 + 1 +
1,2+1+1,2+1+1,1+1+1+1,1 + 1 + 1 + 1. For k≥3 and 1 ≤i≤k
2, the
generating function for Ck,i(n) is given by
∞
X
n=0
Ck,i(n)qn=(qk;qk)∞(−qi;qk)∞(−qk−i;qk)∞
(q;q)∞
,(1.1)
where (a;q)∞:=
∞
Y
j=0
(1 −aqj). Andrews proved the following Ramanujan-type
congruences satisfied by C3,1(n): For n≥0,
C3,1(9n+ 3) ≡C3,1(9n+ 6) ≡0 (mod 3).
Date: March 31, 2020, Revised: November 11, 2020.
1991 Mathematics Subject Classification. Primary 05A17, 11P83.
Key words and phrases. Singular overpartitions; Eta-quotients; modular forms; arithmetic
density.
1
2 AJIT SINGH AND RUPAM BARMAN
Numerous other congruences for Andrews’ singular overpartitions are obtained by
many authors, see for example [1, 5, 6, 11].
In [6], Chen, Hirschhorn and Sellers studied the parity of Ck,i(n). They showed
that C3,1(n) is always even and that C6,2(n) is even (or odd) if and only if n
is not (or is) a pentagonal number. Recently, Aricheta [3] studied the parity of
C3`,`(n). He proved that C3`,`(n)≡b`(n) (mod 2), where b`(n) denotes the number
of partitions of ninto parts none of which are multiples of `. He then used the
density result of Gordon and Ono [8] regarding b`(n) to prove a density result
about C3`,`(n) modulo 2 for an infinite family of `. More precisely, represent any
positive integer `as `= 2αmwhere the integer α≥0 and mis positive odd. Assume
further that 2α≥m. Then Aricheta proved that the set {n∈Z≥0:C3`,`(n)≡0
(mod 2)}has arithmetic density 1.
In this article, we study the arithmetic densities of C3`,`(n) modulo arbitrary
powers of 2 and 3 when `=p·2α, where pis a prime. Let kbe a fixed positive
integer. In a recent paper, Barman and Ray [4] prove that C3·2α,2α(n) is almost
always divisible by 2kand 3kfor α= 0, that is, the sets {n∈Z≥0:C3,1(n)≡0
(mod 2k)}and {n∈Z≥0:C3,1(n)≡0 (mod 3k)}have arithmetic density 1. Very
recently, we [14] prove the same for C3·2α,2α(n) when α= 1,2. In this article we
prove that, for all α≥1 satisfying 2α≥p, the set {n∈Z≥0:C3p·2α,p·2α(n)≡0
(mod 2k)}has arithmetic density 1. To be specific, we prove the following result.
Theorem 1.1. Let kbe a fixed positive integer. Then for all α≥1and all prime
psatisfying 2α≥p,C3p·2α,p·2α(n)is almost always divisible by 2k, namely,
lim
X→∞
#0< n ≤X:C3p·2α,p·2α(n)≡0 (mod 2k)
X= 1.
Remark 1.2.Let kbe a fixed positive integer. If we take p= 2 in Theorem 1.1,
then for all α≥2, we have that C3·2α,2α(n) is almost always divisible by 2k. In
[4, 14], the same has been proved for C3·2α,2α(n) when α= 0,1. Hence, C3·2α,2α(n)
is almost always divisible by 2kfor all α≥0.
Remark 1.3.Let `=p·2α. To find the density of {n∈Z≥0:C3`,`(n)≡0 (mod 2)},
Aricheta used the condition 2α≥pto apply [8, Theorem 1] of Gordon and Ono.
Interestingly, we also need the condition 2α≥pin Theorem 1.1. However, we
need this condition to prove the modularity of cetain eta-quotients appearing in
the proof of Theorem 1.1.
We next study the divisibility of Andrews’ singular overpartitions by powers of
3. In the following theorem we find the arithmetic density of the set {n∈Z≥0:
C3·3·2α,3·2α(n)≡0 (mod 3k)}when α= 0,1,2,3.
Theorem 1.4. Let kbe a fixed positive integer. Then for each α,0≤α≤3,
C3·3·2α,3·2α(n)is almost always divisible by 3k, namely,
lim
X→∞
#0< n ≤X:C3·3·2α,3·2α(n)≡0 (mod 3k)
X= 1.
It should be noted that only a few results analogous to Theorem 1.1 and Theorem
1.4 are known for the overpartition function. In fact Mahlburg [10] conjectured
that the overpartition function is almost always divisible by arbitrary powers of
2, which is still open. The generating function of the overpartition function is a
modular form of half-integral weight. The coefficients for such functions are poorly
ARITHMETIC DENSITY OF ANDREWS’ SINGULAR OVERPARTITIONS 3
understood, and conjectures such as this are considered to be difficult with present
techniques. However, our proofs rely on the fact that, for certain values of αand p,
the generating functions for C3p·2α,p·2α(n) are modular forms of integral weights,
and this allows us to use the Serre’s Theorem regarding divisibility of the coefficients
of such functions.
We also prove that the eta-quotient which arises naturally as generating function
for C3·3·2α,3·2α(n) is not a modular form if α≥4. Therefore, we couldn’t find the
density of the set {n∈Z≥0:C3·3·2α,3·2α(n)≡0 (mod 3k)}if α≥4.
2. Preliminaries
In this section, we recall some definitions and basic facts on modular forms and
eta-quotients. For more details, see for example [13, 9].
2.1. Spaces of modular forms. We first define the matrix groups
SL2(Z) := a b
c d:a, b, c, d ∈Z, ad −bc = 1,
Γ0(N) := a b
c d∈SL2(Z) : c≡0 (mod N),
Γ1(N) := a b
c d∈Γ0(N) : a≡d≡1 (mod N),
and
Γ(N) := a b
c d∈SL2(Z) : a≡d≡1 (mod N),and b≡c≡0 (mod N),
where Nis a positive integer. A subgroup Γ of SL2(Z) is called a congruence
subgroup if Γ(N)⊆Γ for some N. The smallest Nsuch that Γ(N)⊆Γ is called
the level of Γ. For example, Γ0(N) and Γ1(N) are congruence subgroups of level
N.
Let H:= {z∈C: Im(z)>0}be the upper half of the complex plane. The
group
GL+
2(R) = a b
c d:a, b, c, d ∈Rand ad −bc > 0
acts on Hby a b
c dz=az +b
cz +d. We identify ∞with 1
0and define a b
c dr
s=
ar +bs
cr +ds, where r
s∈Q∪ {∞}. This gives an action of GL+
2(R) on the extended
upper half-plane H∗=H∪Q∪ {∞}. Suppose that Γ is a congruence subgroup of
SL2(Z). A cusp of Γ is an equivalence class in P1=Q∪ {∞} under the action of Γ.
The group GL+
2(R) also acts on functions f:H→C. In particular, suppose
that γ=a b
c d∈GL+
2(R). If f(z) is a meromorphic function on Hand `is an
integer, then define the slash operator |`by
(f|`γ)(z) := (det γ)`/2(cz +d)−`f(γz).
Definition 2.1. Let Γ be a congruence subgroup of level N. A holomorphic func-
tion f:H→Cis called a modular form with integer weight `on Γ if the following
hold:
4 AJIT SINGH AND RUPAM BARMAN
(1) We have
faz +b
cz +d= (cz +d)`f(z)
for all z∈Hand all a b
c d∈Γ.
(2) If γ∈SL2(Z), then (f|`γ)(z) has a Fourier expansion of the form
(f|`γ)(z) = X
n≥0
aγ(n)qn
N,
where qN:= e2πiz/N .
For a positive integer `, the complex vector space of modular forms of weight `
with respect to a congruence subgroup Γ is denoted by M`(Γ).
Definition 2.2. [13, Definition 1.15] If χis a Dirichlet character modulo N, then
we say that a modular form f∈M`(Γ1(N)) has Nebentypus character χif
faz +b
cz +d=χ(d)(cz +d)`f(z)
for all z∈Hand all a b
c d∈Γ0(N). The space of such modular forms is denoted
by M`(Γ0(N), χ).
2.2. Modularity of eta-quotients. In this paper, the relevant modular forms are
those that arise from eta-quotients. Recall that the Dedekind’s eta-function η(z) is
defined by
η(z) := q1/24(q;q)∞=q1/24
∞
Y
n=1
(1 −qn),
where q:= e2πiz and z∈H. A function f(z) is called an eta-quotient if it is of the
form
f(z) = Y
δ|N
η(δz)rδ,
where Nis a positive integer and rδis an integer.
We now recall two theorems from [13, p. 18] on modularity of eta-quotients. We
will use these two results to verify modularity of certain eta-quotients appearing in
the proof of our main result.
Theorem 2.3. [13, Theorem 1.64] If f(z) = Qδ|Nη(δz)rδis an eta-quotient such
that `=1
2Pδ|Nrδ∈Z,
X
δ|N
δrδ≡0 (mod 24)
and
X
δ|N
N
δrδ≡0 (mod 24),
then f(z)satisfies
faz +b
cz +d=χ(d)(cz +d)`f(z)
ARITHMETIC DENSITY OF ANDREWS’ SINGULAR OVERPARTITIONS 5
for every a b
c d∈Γ0(N). Here the character χis defined by χ(d) := (−1)`s
d,
where s:= Qδ|Nδrδ.
Suppose that fis an eta-quotient satisfying the conditions of Theorem 2.3 and
that the associated weight `is a positive integer. If f(z) is holomorphic at all of
the cusps of Γ0(N), then f(z)∈M`(Γ0(N), χ). The following theorem gives the
necessary criterion for determining orders of an eta-quotient at cusps.
Theorem 2.4. [13, Theorem 1.65] Let c, d and Nbe positive integers with d|N
and gcd(c, d) = 1. If fis an eta-quotient satisfying the conditions of Theorem 2.3
for N, then the order of vanishing of f(z)at the cusp c
dis
N
24 X
δ|N
gcd(d, δ)2rδ
gcd(d, N
d)dδ .
3. Proof of Theorem 1.1
Here αis a positive integer and pis a prime. The generating function for
C3p·2α,p·2α(n) is given by
∞
X
n=0
C3p·2α,p·2α(n)qn=(q2p·2α;q2p·2α)∞(q3p·2α;q3p·2α)2
∞
(q;q)∞(qp·2α;qp·2α)∞(q6p·2α;q6p·2α)∞
.(3.1)
We note that η(3p·2α+3z) = qp·2αQ∞
n=1(1 −q(3p·2α+3 )n) is a power series of q. Let
Aα,p(z) =
∞
Y
n=1
(1 −q(3p·2α+3)n)2
(1 −q(3p·2α+4)n)=η2(3p·2α+3 z)
η(3p·2α+4z).
Then using binomial theorem we have
A2k
α,p(z) = η2k+1 (3p·2α+3 z)
η2k(3p·2α+4z)≡1 (mod 2k+1 ).
Define Bα,p,k(z) by
Bα,p,k(z) = η(3p·2α+4 z)η(9p·2α+3 z)2
η(24z)η(3p·2α+3z)η(9p·2α+4 z)A2k
α,p(z).(3.2)
Modulo 2k+1, we have
Bα,p,k(z)≡η(3p·2α+4 z)η(9p·2α+3 z)2
η(24z)η(3p·2α+3z)η(9p·2α+4 z)
=qp·2α−1 (q3p·2α+4 ;q3p·2α+4 )∞(q9p·2α+3 ;q9p·2α+3 )2
∞
(q24;q24 )∞(q3p·2α+3 ;q3p·2α+3 )∞(q9p·2α+4 ;q9p·2α+4 )∞!.(3.3)
Combining (3.1) and (3.3), we obtain
Bα,p,k(z)≡
∞
X
n=0
C3p·2α,p·2α(n)q24n+p·2α−1(mod 2k+1).(3.4)
In the following lemma, we prove that Bα,p,k(z) is a modular form for certain values
of α, p and k.
6 AJIT SINGH AND RUPAM BARMAN
Lemma 3.1. Let pbe a prime and αbe a positive integer satisfying 2α≥p. Then
Bα,p,k(z)∈M2k−1Γ0(9p·2α+5 ), χfor all k≥2α, where the character χis given
by χ= (2α+2k
·(α+2)32k+1 p2k+1
•).
Proof. From (3.2) we have
Bα,p,k(z) = η(3p·2α+4 z)η(9p·2α+3 z)2
η(24z)η(3p·2α+3z)η(9p·2α+4 z)A2k
α,p(z)
=η(3p·2α+3z)2k+1 −1η(9p·2α+3z)2
η(3p·2α+4z)2k−1η(24z)η(9p·2α+4 z).
We now calculate the level of the eta-quotient Bα,p,k(z) by using Theorem 2.3.
Thus, the level of Bα,p,k(z) is equal to 9p·2α+4 ·m, where mis the smallest positive
integer such that
9p·2α+4m2k+1 −1
3p·2α+3 +2
9p·2α+3 −2k−1
3p·2α+4 −1
24 −1
9p·2α+4 ≡0 (mod 24).
Equivalently,
m9·2k−3p·2α+1≡0 (mod 24).
Therefore, if k≥2, then m= 2 and the level of the eta-quotient is 9p·2α+5.
The cusps of Γ0(9p·2α+5) are represented by fractions c
dwhere d|9p·2α+5 and
gcd(c, d) = 1. For example, see [12, p. 5]. By Theorem 2.4, we find that Bα,p,k(z)
is holomorphic at a cusp c
dif and only if
gcd(d, 3p·2α+3)2
3p·2α+3 (2k+1 −1) + gcd(d, 3p·2α+4 )2
3p·2α+4 (1 −2k)
+ 2gcd(d, 9p·2α+3 )2
9p·2α+3 −gcd(d, 24)2
24 −gcd(d, 9p·2α+4)2
9p·2α+4 ≥0.
Equivalently, if and only if
L:= 6G2
1·(2k+1 −1) + 3G2
2·(1 −2k)+4G2
3−3p·2α+1G2
4−1≥0,
where G1=gcd(d, 3p·2α+3)
gcd(d, 9p·2α+4), G2=gcd(d, 3p·2α+4 )
gcd(d, 9p·2α+4), G3=gcd(d, 9p·2α+3 )
gcd(d, 9p·2α+4), and
G4=gcd(d, 24)
gcd(d, 9p·2α+4), respectively.
In the following table, we find all the possible values of L.
ARITHMETIC DENSITY OF ANDREWS’ SINGULAR OVERPARTITIONS 7
d|9p·2α+5 G1G2G3G4L
1,2,3,4,6,8,12,24 1 1 1 1 9 ·2k−3p·2α+1
p, 2p, 3p, 4p, 6p, 8p, 12p, 24p1 1 1 1/p 9·2k−3·2α+1/p
9,18,36,72 1/3 1/3 1 1/3 2k+ 8/3−
2α+1p/3
9p, 18p, 36p, 72p1/3 1/3 1 1/3p2k+ 8/3−
2α+1/3p
2α+4,2α+5 ,3·2α+4,3·2α+5 1/2 1 1/2 2−1−α1.5−3p·2−1−α
p·2α+4, p ·2α+5,3p·2α+4,3p·2α+5 1/2 1 1/2 2−1−α/p 1.5−3·2−1−α/p
2r,3·2r: 4 ≤r≤α+ 3 1 1 1 23−r9·2k−3p·
27+α−2r
p·2r,3p·2r: 4 ≤r≤α+ 3 1 1 1 23−r/p 9·2k−3·
27+α−2r/p
9·2r: 4 ≤r≤α+ 3 1/3 1/3 1 23−r/3 2k+ 8/3−p·
27+α−2r/3
9p·2r: 4 ≤r≤α+ 3 1/3 1/3 1 23−r/3p2k+ 8/3−
27+α−2r/3p
9·2α+4,9·2α+5 1/6 1/3 1/2 2−1−α/3 1/6−p·2−1−α/3
9p·2α+4,9p·2α+5 1/6 1/3 1/2 2−1−α/3p1/6−2−1−α/3p
Using the given condition 2α≥p, we now find that L≥0 for all d|9p·2α+5 and for
all k≥2α. Hence, Bα,p,k(z) is holomorphic at every cusp c
d. Now, from Theorem
2.3 we find that the weight of Bα,p,k(z) is `= 2k−1. Also, the associated character
for Bα,p,k(z) is given by χ= ( 2α+2k
·(α+2)32k+1 p2k+1
•). Finally, Theorem 2.3 yields
that Bα,p,k(z)∈M2k−1Γ0(9p·2α+5 ), χfor all k≥2α.
Proof of Theorem 1.1. For a fixed α≥1, it is enough to prove Theorem 1.1 for all
k≥2α. For given any positive integer m, applying a deep theorem of Serre [13, p.
43], if f(z)∈M`(Γ0(N), χ) has Fourier expansion
f(z) =
∞
X
n=0
c(n)qn∈Z[[q]],
then there is a constant a > 0 such that
#{n≤X:c(n)6≡ 0 (mod m)}=OX
(log X)a.
This yields
lim
X→∞
#{0< n ≤X:c(n)≡0 (mod m)}
X= 1.(3.5)
Now, if 2α≥p, then from Lemma 3.1 we have Bα,p,k(z)∈M2k−1Γ0(9p·2α+5 ), χ
for all k≥2α. Hence, the Fourier coefficients of Bα,p,k(z) are almost always
divisible by m= 2k. We now complete the proof of the theorem by using (3.4).
4. Proof of Theorem 1.4
The generating function for C3·3·2α,3·2α(n) is given by
∞
X
n=0
C3·3·2α,3·2α(n)qn=(q3·2α+1 ;q3·2α+1 )∞(q32·2α;q32·2α)2
∞
(q;q)∞(q3·2α;q3·2α)∞(q32·2α+1 ;q32·2α+1 )∞
.(4.1)
8 AJIT SINGH AND RUPAM BARMAN
For α≥0, let
Aα(z) =
∞
Y
n=1
(1 −q(32·2α+3)n)3
(1 −q(33·2α+3)n)=η3(32·2α+3 z)
η(33·2α+3z).
Then using binomial theorem we have
A3k
α(z) = η3k+1 (32·2α+3z)
η3k(33·2α+3z)≡1 (mod 3k+1 ).
Define Dα,k(z) by
Dα,k(z) = η(32·2α+4 z)η(33·2α+3 z)2
η(24z)η(32·2α+3z)η(33·2α+4 z)A3k
α(z).(4.2)
Modulo 3k+1, we have
Dα,k(z)≡η(32·2α+4 z)η(33·2α+3 z)2
η(24z)η(32·2α+3z)η(33·2α+4 z)
=q3·2α−1 (q32·2α+4 ;q32·2α+4 )∞(q33·2α+3 ;q33·2α+3 )2
∞
(q24;q24 )∞(q32·2α+3 ;q32·2α+3 )∞(q33·2α+4 ;q33·2α+4 )∞!.(4.3)
Combining (4.1) and (4.3), we obtain
Dα,k(z)≡
∞
X
n=0
C3·3·2α,3·2α(n)q24n+3·2α−1(mod 3k+1).(4.4)
We now study the modularity of the eta-quotient Dα,k(z) in the following lemma.
Lemma 4.1. Let α≥0be an integer. For a fixed k≥1, let χαdenote the character
(−2α+3k
·(2α+6)33k+1 +2
•). We have
(1) D0,k ∈M3k(Γ0(1728), χ0)for all k≥1.
(2) D1,k ∈M3k(Γ0(1728), χ1)for all k≥1.
(3) D2,k ∈M3k(Γ0(1728), χ2)for all k≥2.
(4) D3,k ∈M3k(Γ0(3456), χ3)for all k≥2.
Furthermore, Dα,k(z)is not a modular form if α≥4.
Proof. As before, using Theorem 2.3 we find that the levels of the eta-quotients
D0,k(z) and D1,k (z) are equal to 1728 for all k≥1. Also, if α≥2 then the
level of Dα,k(z) is equal to 33·2α+4 for all k≥1. Again, Theorem 2.3 yields
that the weight of the eta-quotient Dα,k(z) is 3kand the associated character is
χα= (−2α+3k
·(2α+6)33k+1 +2
•).
We now first prove that Dα,k(z) is not a modular form if α≥4. If α≥2 then
Theorem 2.4 yields that Dα,k(z) is holomorphic at a cusp c
dif and only if
Q:= 6gcd(d, 32·2α+3 )2
gcd(d, 33·2α+4)2(3k+1 −1) + 2 gcd(d, 33·2α+3)2
gcd(d, 33·2α+4)2(2 −3k)
+ 3gcd(d, 32·2α+4 )2
gcd(d, 33·2α+4)2−32·2α+1 gcd(d, 24)2
gcd(d, 33·2α+4)2−1≥0.
If we take d= 27 then we find that
Q=6
9(3k+1 −1) + 2(2 −3k) + 3
9−2α+1
9−1 = 8
3−2α+1
9<0
for all α≥4. Hence, Dα,k (z) is not a modular form if α≥4.
ARITHMETIC DENSITY OF ANDREWS’ SINGULAR OVERPARTITIONS 9
We next consider the remaining four values of α, namely α= 0,1,2,3. We prove
that for each of these values of α,Dα,k(z) is a modular form. Putting α= 1 in
(4.2) we have
D1,k(z) = η(288z)η(432z)2
η(24z)η(144z)η(864z)A3k
1(z) = η(144z)3k+1−1η(288z)
η(432z)3k−2η(24z)η(864z).
Now, D1,k is an eta-quotient with N= 1728. As before, the cusps of Γ0(1728) are
represented by fractions c
dwhere d|1728 and gcd(c, d) = 1. By Theorem 2.4, we
find that D1,k(z) is holomorphic at a cusp c
dif and only if
gcd(d, 144)2
144 (3k+1 −1) + gcd(d, 432)2
432 (2 −3k)
+gcd(d, 288)2
288 −gcd(d, 24)2
24 −gcd(d, 864)2
864 ≥0.
Equivalently, if and only if
P:= 6gcd(d, 144)2
gcd(d, 864)2(3k+1 −1) + 2gcd(d, 432)2
gcd(d, 864)2(2 −3k)
+ 3gcd(d, 288)2
gcd(d, 864)2−36 gcd(d, 24)2
gcd(d, 864)2−1≥0.
In the following table, we find all the possible values of P. We use MATLAB to
prepare the table.
d|1728 gcd(d,144)2
gcd(d,864)2
gcd(d,432)2
gcd(d,864)2
gcd(d,288)2
gcd(d,864)2
gcd(d,24)2
gcd(d,864)2P
1,2,3,4,6,8,
12, 24
1 1 1 1 16 ·3k−36
27,54,108,216 0.1111 1 0.1111 0.0123 2.2222
9, 18, 36, 72 1 1 1 0.1111 16 ·3k−4
32, 64, 96, 192 0.2500 0.2500 1 0.0625 4 ·3k−0.7500
16, 48 1 1 1 0.2500 16 ·3k−9
288, 576 0.2500 0.2500 1 0.0069 4 ·3k+ 1.2500
144 1 1 1 0.0278 16 ·3k−1
432 0.1111 1 0.1111 0.0031 2.5556
864, 1728 0.0278 0.2500 0.1111 0.0007 0.1389
Since P≥0 for all d|1728 and for all k≥1, we have that D1,k(z) is holomorphic
at every cusp c
d. Hence, Theorem 2.3 yields that D1,k(z)∈M3k(Γ0(1728), χ1) for
all k≥1. This completes the proof of the lemma when α= 1.
The proof goes along similar lines when α= 0,2,3, and so we omit the details
for reasons of breviety. This completes the proof of the lemma.
Proof of Theorem 1.4. Throughout the proof we assume that α≤3. Without loss
of generality we assume that k≥2. From Lemma 4.1 we know that Dα,k(z) is a
modular form. Hence, (3.5) yields that the Fourier coefficients of Dα,k(z) are almost
always divisible by 3k. Now using (4.4) we find that C3·3·2α,3·2α(n) is almost always
divisible by 3k. This completes the proof of the theorem.
10 AJIT SINGH AND RUPAM BARMAN
5. Concluding remarks
Let kbe a fixed positive integer. In this article we have found the arithmetic
density of the set {n∈Z≥0:C3`,`(n)≡0 (mod 2k)}for an infinite family of `.
But the arithmetic density of the set {n∈Z≥0:C3`,`(n)≡0 (mod 3k)}is known
only for a few values of `till date. In this paper, we have found the density of
{n∈Z≥0:C3`,`(n)≡0 (mod 3k)}when `= 3,6,12,24, and the same is already
known for `= 1,2,4, see for example [4, 14]. Since Dα,k is not a modular form
if α≥4, therefore, using the method used in this article, it won’t be possible to
find the density of the set {n∈Z≥0:C3`,`(n)≡0 (mod 3k)}when `= 3 ·2αand
α≥4. It would be interesting to study this problem for an infinite family of `.
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Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India,
PIN- 781039
Email address:ajit18@iitg.ac.in
Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India,
PIN- 781039
Email address:rupam@iitg.ac.in
