Extension of a proof of the Ramanujan congruences for multipartitions

0.2 MB·

by  Oleg Lazarev

EXTENSION OF A PROOF OF THE RAMANUJAN
CONGRUENCES FOR MULTIPARTITIONS
6
1
OLEGLAZAREV,MATTHEWS.MIZUHARA,BENJAMINREID,ANDHOLLYSWISHER
0
2

n
a
J
Abstract. RecentlyLachterman,Schayer,andYoungerpublishedanelegant
0 proof of the Ramanujan congruences for the partition function p(n). Their
2 proof uses only the classical theory of modular forms as well as a beautiful
resultof Choie, Kohnen, andOno, without need forHecke operators. Inthis
] paper wegiveamethod forgeneralizingLachterman, Schayer, andYounger’s
T
proof to include Ramanujan congruences for multipartition functions pk(n),
N andRamanujancongruences forp(n)modulocertainprimepowers.
.
h
t
a
m 1. Introduction and Statement of Results
[ The subject of partitions has a long fascinating history, including connections

1 to many areas of mathematics, and mathematical physics. For example, [And98]
v and [AO01a] provide a glimpse into the history of partitions. In particular, the
7 generalizationofpartitionstok-componentmultipartitions,alsoknownask-colored
8
partitions,isarichsubjectinitsownright(see[And08]foranice survey). We will
2
begin by reviewing partitions and multipartitions.
5
0
.
1 1.1. Partitions. We recall that a partition of a positive integer n is defined as a
0 nonincreasingsequenceofpositiveintegerscalledparts,thatsumton(oftenwritten
6
as a sum). For n = 0 we define the empty set as the unique “empty partition” of
1
0. We write |λ|=n todenote thatλisa partitionofn. For example,the following
:
v gives all the partitions λ such that |λ|=5:
i
X
5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1.
r
a
The partition function p(n) counts the total number of partitions of n. In order to
definep(n)onallintegerswefurtherdefinethatp(n)=0whenn<0. Weseefrom
our example above that p(5)=7.
The generating function for p(n) has the following infinite product form, due to
Euler:
∞ ∞
1
p(n)qn = .
(1−qn)
n=0 n=1
X Y
2010 Mathematics Subject Classification. Primary11P83.
Key words and phrases. partitions,multipartitions,Ramanujan,congruences.
Theauthors weresupportedinpartbytheNSFgrantDMS-0852030.
1
2 OLEGLAZAREV,MATTHEWS.MIZUHARA,BENJAMINREID,ANDHOLLYSWISHER
1.2. Multipartitions. Onenaturalgeneralizationofpartitionsisbythefollowing.
Define a k-component multipartition of a nonnegative integer n to be a k-tuple of
partitions λ=(λ ,...,λ ) such that
1 k
k
|λ |=n.
i
i=1
X
We will write |λ| = n if λ is a k-component multipartition of n. The following
k
gives all the 2-component multipartitions of 3, i.e., all λ such that |λ| =3:
2
(3,∅),(2+1,∅),(1+1+1,∅),(2,1),(1+1,1),(1,2),(1,1+1),(∅,3),(∅,2+1),(∅,1+1+1).
We define p (n)asthe number ofk-componentmultipartitionsofn, againdefining
k
p (n)=0 for n<0. From our example above we see that p (3)=10.
k 2
Remark 1. Wenotethatorderingmattersinthisdefinition,inthatarearrangement
ofcomponentsλ mayyieldadistinctmultipartition. Inaddition,wedrawattention
i
to the fact that since the empty set is a partition of 0 some λ may equal ∅.
i
The generating function for p (n) follows from the generating function for p(n)
k
by taking the kth power of the product form. Namely,
∞ ∞
1
(1) p (n)qn = .
k (1−qn)k
n=0 n=1
X Y
1.3. Ramanujan Congruences. Among the most celebrated results in partition
theoryarethefollowingcongruencesofRamanujanforp(n). Forallintegersn≥0,
p(5n+4) ≡ 0 (mod 5)
p(7n+5) ≡ 0 (mod 7)
p(11n+6) ≡ 0 (mod 11).
Work of Ono and Ahlgren [Ono00], [AO01b], [Ahl00] has shown that for any
m coprime to 6 there exist infinitely many nonnested arithmetic progressions for
whichp(an+b)≡0 (mod m). However,ithasbeenshownby AhlgrenandBoylan
[AB03] that the three Ramanujan congruences above are the only congruences for
p(n) of the form
p(ℓn+b)≡0 (mod ℓ),
for ℓ prime.
Many congruences for multipartitions have been proven as well, see Andrews
[And08], Gandhi [Gan63], Atkin [Atk68], and Kiming and Olsson [KO92]. Recent
work by Folsom, Kent, and Ono [FKO] discusses some of the underlying reasons
for such congruences.
Using notation of Kiming and Olsson [KO92], for a prime ℓ ≥ 5, an integer a,
andpositiveintegersk,m,wesaythere is a congruence at(ℓm,k,a)ifforalln≥0,
p (ℓmn+a)≡0 (mod ℓm).
k
There have been a number of proofs of the Ramanujan congruences. The proof
by Lachterman, Schayer, and Younger in [LSY08] is notable because it uses only
the classical theory of modular forms, without relying on Hecke operators. In this
paper we extend their proof to include Ramanujan congruencesfor multipartitions
modulo prime powers. In particular, we prove the following theorem.
A PROOF OF THE RAMANUJAN CONGRUENCES FOR MULTIPARTITIONS 3
Definition 1.1. Let ℓ ≥ 5 be prime, and let k,m be positive integers. Then we
define
k(ℓ2m−1)
δ := .
k,ℓ,m
24
We note that the δ are all positive integers, since ℓ2 ≡ 1 (mod 24) for all
k,ℓ,m
primes ℓ≥5.
Theorem 1.2. Let ℓ ≥ 5 be prime, and let k,m be positive integers such that
k ≡−4 (mod ℓm−1). Ifforeach1≤r <mthereisacongruenceat(ℓr,k,−δ ),
k,ℓ,m
then there is a congruence at (ℓm,k,−δ ) if and only if
k,ℓ,m
p (ℓmn−δ )≡0 (mod ℓm)
k k,ℓ,m
for all 0≤n< kℓm+2ℓ+2.
24
Remark 2. Note that whenever 1 ≤ r ≤ m we have δ ≡ δ (mod ℓr).
k,ℓ,m k,ℓ,r
Thus there is a congruence at (ℓr,k,−δ ) if and only if there is a congruence
k,ℓ,m
at (ℓr,k,−δ ). Here we are using the fact that δ is a positive integer for
k,ℓ,r k,ℓ,r
each r, and that we defined p to take the value 0 when evaluated at any negative
k
integer. More specifically, the condition that p (ℓrn − δ ) ≡ 0 (mod ℓr) for
k k,ℓ,r
0 ≤ n ≤ N is equivalent to the condition that p (ℓrn − δ ) ≡ 0 (mod ℓr)
k k,ℓ,m
for 0 ≤ n ≤ N + δk,ℓ,m−δk,ℓ,r. In particular, in Theorem 1.2 one can replace the
ℓr
condition that there is a congruence at (ℓr,k,−δ ) for all 1 ≤ r < m with the
k,ℓ,m
equivalent condition that there is a congruence at (ℓr,k,−δ ) for all 1≤r <m.
k,ℓ,r
We observe that the m = 1 case of Theorem 1.2 has trivial conditions. In
particular it yields that when ℓ ≥ 5 is prime and k is a positive integer, there
is a congruence at (ℓ,k,−δ ) if p (ℓn − δ ) ≡ 0 (mod ℓ) for all 0 ≤ n <
k,ℓ,1 k k,ℓ,1
kℓ+2ℓ+2. Wealsoobservethatforageneralpositiveintegerm,theconditionk ≡−4
24
(mod ℓm−1) in Theorem 1.2 implies that k ≡ −4 (mod ℓr) for any 1 ≤ r < m−1
as well. Thus, fixing a prime ℓ ≥ 5, and positive integers k,m such that k ≡ −4
(mod ℓm−1),supposeweknowforeach1≤r ≤mthatp (ℓrn−δ )≡0 (mod ℓr)
k k,ℓ,r
for all 0≤n < kℓr+2ℓ+2. Then, the m=1 case of Theorem 1.2 implies that there
24
is a congruence at (ℓ,k,−δ ), and so by Remark 2 there is a congruence at
k,ℓ,1
(ℓ,k,−δ ). Thus the conditions of the m= 2 case of Theorem 1.2 are satisfied,
k,ℓ,2
so we can conclude in fact that there is also a congruence at (ℓ2,k,−δ ), and
k,ℓ,2
again by Remark 2 a congruence at (ℓ2,k,−δ ). Continuing inductively, we
k,ℓ,3
obtain that there is in fact a congruence at (ℓr,k,−δ ) for each 1 ≤ r ≤ m.
k,ℓ,r
In particular, we have the following corollary which gives a finite condition for the
existence of a congruence at (ℓm,k,−δ ).
k,ℓ,m
Corollary 1.3. Let ℓ ≥ 5 be prime, and let m,k be positive integers such that
k ≡−4 (mod ℓm−1). If for each 1≤r ≤m,
p (ℓrn−δ )≡0 (mod ℓr),
k k,ℓ,r
for all 0 ≤ n < kℓr+2ℓ+2, then there is a congruence at (ℓr,k,−δ ) for all
24 k,ℓ,r
1≤r ≤m.
Remark 3. In light of Remark 2 one can rewrite the conditions in Corollary 1.3 as
p (ℓrn−δ )≡0 (mod ℓr)forall0≤n< kℓ2m−r+2ℓ+2. Likewise,theconclusion
k k,ℓ,m 24
can be replaced with congruences at (ℓr,k,−δ ) for all 1≤r ≤m.
k,ℓ,m
4 OLEGLAZAREV,MATTHEWS.MIZUHARA,BENJAMINREID,ANDHOLLYSWISHER
In Section 2, we establish some preliminaries which enable us to proveTheorem
1.2. InadditionweproveageneralizationofalemmaofKimingandOlsson[KO92]
inLemma2.3,whichcombinedwithCorollary1.3andRemark3givesthefollowing
more general corollary to Theorem 1.2.
Corollary 1.4. Let ℓ≥5 be prime, and let m,k,k′ be positive integers such k′ ≡k
(mod ℓm) and k′ ≡k ≡−4 (mod ℓm−1). If for each 1≤r ≤m,
p (ℓrn−δ )≡0 (mod ℓr)
k k,ℓ,r
for all 0 ≤ n < kℓr+242ℓ+2, then there is a congruence at (ℓr,k′,−δk′,ℓ,m) for all
1≤r ≤m.
Remark 4. As in Remark 3, we can replace the conditions on p (ℓrn−δ ) in
k k,ℓ,r
Corollary1.4 by the appropriate conditions on p (ℓrn−δ ) and the conclusion
k k,ℓ,m
that there are congruences (ℓr,k′,−δk′,ℓ,m) by congruences at (ℓr,k′,−δk′,ℓ,r).
In Section 3, we proveTheorem1.2, and in Section 4 we provide some examples
of congruences for p (n) proved by Theorem 1.2.
k
2. Preliminaries
Recall the ring of formal Laurent series in q with integer coefficients defined by
Z((q)):= f(q)= a(n)qn |n ∈Z,a(n)∈Z for all n≥n .
 0 0
 nX≥n0 
The set of formal power series in q with integer coefficients, denoted Z[[q]], is the
 
subring of Z((q)) obtained by fixing n =0 in the above definition.
0
2.1. Reduction of power series modulo prime powers.
Proposition 2.1. Let f(q) ∈ Z((q)) (resp. Z[[q]]). For any integer k ≥ 0, and
prime ℓ,
f(qℓk)ℓ =g(qℓk+1)+ℓ·h(qℓk),
where g,h∈Z((q)) (resp. Z[[q]]).
Proof. Fix an integer k ≥ 0, and let f(q) = a(n)qn, with a(n) ∈ Z. Then
n≥n0
f(qℓk)= a(n)qnℓk, so that
n≥n0 P
P f(qℓk)ℓ ≡ a(n)qnℓk+1 (mod ℓ).
nX≥n0
ℓ∤a(n)
Thus the coefficients of any terms qm of f(qℓk)ℓ where m is not a multiple of ℓk+1
must be divisible by ℓ. So we can write
f(qℓk)ℓ =g(qℓk+1)+ℓ·h(qℓk),
for series g,h∈Z((q)). (cid:3)
The following lemma is used many times leading up to Theorem 1.2.
Lemma 2.2. Let ℓ be prime, and f ∈ Z((q)) (resp. Z[[q]]). Then for any positive
integer m, there exist f ,f ,...,f ∈Z((q)) (resp. Z[[q]]), such that
0 1 m
f(q)ℓm =f (qℓm)+ℓ·f (qℓm−1)+ℓ2·f (qℓm−2)+···+ℓm−1·f (qℓ)+ℓm·f (q),
0 1 2 m−1 m
A PROOF OF THE RAMANUJAN CONGRUENCES FOR MULTIPARTITIONS 5
Proof. Fix a prime ℓ, and let f ∈Z((q)) with coefficients given by
f(q)= a(n)qn.
nX≥n0
Weinductonm. FromProposition2.1withk =0,weimmediatelyobtainthecase
m=1. Suppose now for an arbitrary positive integer m that
f(q)ℓm =f (qℓm)+ℓ·f (qℓm−1)+···+ℓm−1·f (qℓ)+ℓm·f (q),
0 1 m−1 m
for some f ,f ,...,f ∈Z((q)). Then,
0 1 m
f(q)ℓm+1 =(f(q)ℓm)ℓ = f (qℓm)+ℓf (qℓm−1)+···+ℓm−1f (qℓ)+ℓmf (q) ℓ.
0 1 m−1 m
(cid:16) (cid:17)
By the multinomial theorem, this yields that
f(q)ℓm+1 = ℓ f (qℓm) k0 ℓf (qℓm−1) k1···[ℓmf (q)]km.
k ,...,k 0 1 m
k0+··X·+km=ℓ(cid:18) 0 m(cid:19)h i h i
ki≥0
We break this sum into cases based on the highest index i for which k 6= 0. In
i
particular, letting
F (q):= ℓ f (qℓm) k0 ℓf (qℓm−1) k1··· ℓif (qℓm−i) ki,
i k ,...,k 0 1 i
k0+·X··+ki=ℓ(cid:18) 0 i(cid:19)h i h i h i
ki6=0
for 0≤i≤m, we obtain that
f(q)ℓm+1 =F (q)+F (q)+···+F (q).
0 1 m
WeobservethatF (q)hasonlyoneterm,k =ℓ. ThusF (q)=f (qℓm)ℓ. Applying
0 0 0 0
Proposition 2.1, we see that
F (q)=g (qℓm+1)+ℓ·g (qℓm),
0 0 1
for some g ,g ∈Z((q)). For 1≤i≤m, we separate the term k =ℓ to obtain
0 1 i
F (q)= ℓif (qℓm−i) ℓ+ ℓ f (qℓm) k0··· ℓif (qℓm−i) ki.
i i k ,...,k 0 i
h i k0+·X··+ki=ℓ(cid:18) 0 i(cid:19)h i h i
1≤ki≤ℓ−1
We can see from above that F (q) is a series in qℓm−i. In addition, each term is a
i
multiple of ℓi+1. This is clear in the first term, since we have a factor of ℓiℓ, and
ℓ≥2. For the remaining terms we have a factor of ℓi appearing in ℓif (qℓm−i) ki
i
since k ≥ 1, but in addition ℓ divides the multinomial coefficient h ℓ , sinice
i k0,...,ki
k ≤ℓ−1. Thus we have for each 1≤i≤m, that
i (cid:0) (cid:1)
F (q)=ℓi+1g (qℓm−i),
i i+1
where g ∈Z((q)). Together, we have shown that
i+1
f(q)ℓm+1 =g (qℓm+1)+ℓ·g (qℓm)+ℓ2g (qℓm−1)+···+ℓm+1g (q).
0 1 2 m+1
(cid:3)
ThefollowingisausefulgeneralizationofalemmaofKimingandOlsson[KO92]
that gives an infinite family of congruences for each congruence proved using The-
orem 1.2. In particular, it yields Corollary 1.4 as a consequence of Corollary 1.3.
6 OLEGLAZAREV,MATTHEWS.MIZUHARA,BENJAMINREID,ANDHOLLYSWISHER
Lemma2.3. Letℓ≥5beprime. Foranintegera,andpositive integersm,k,there
is a congruence at (ℓr,k,a) for all 1 ≤ r ≤ m if and only if there is a congruence
at (ℓr,k+ℓm,a) for all 1≤r ≤m.
Proof. Wewillshowmoregenerallythatwheneverthereisacongruenceat(ℓr,k,a)
for all 1 ≤ r ≤ m, there is also a congruence at (ℓr,k+sℓm,a) for all 1 ≤ r ≤ m,
for any s ∈ Z for which k+sℓm ≥ 1. The lemma then follows by letting s = 1 in
the “only if” direction, and s = −1 in the “if” direction. We prove this statement
by induction on m.
The case when m = 1 follows directly from Lemma 1 of Kiming and Olsson in
[KO92]. Suppose the claim holds then for a positive integer m and assume there
is a congruence at (ℓr,k,a) for all 1 ≤ r ≤ m +1. Fix an integer s such that
k + sℓm+1 ≥ 1. Since sℓm+1 = (sℓ)ℓm, by the induction hypothesis there is a
congruence at (ℓr,k+sℓm+1,a) for all 1 ≤ r ≤ m. It remains to show there is a
congruence at (ℓm+1,k+sℓm+1,a).
Since by assumption we know there is a congruence at (ℓm+1,k,a), our goal is
to relate p (ℓm+1n+a) and p (ℓm+1n+a) modulo ℓm+1. Considering the
k k+sℓm+1
generating functions of p (n) and p (n), we see that
k k+sℓm+1
∞ ∞ ∞ s ℓm+1
1
p (n)qn = p (n)qn .
k+sℓm+1 k 1−qn
! ! !
n=0 n=0 n=1
X X Y
Thus, by Lemma 2.2,
∞
p (n)qn =
k+sℓm+1
n=0
X
∞
p (n)qn f (qℓm+1)+ℓ·f (qℓm)+···+ℓm·f (qℓ)+ℓm+1·f (q) ,
k 0 1 m m+1
!
nX=0 (cid:16) (cid:17)
where f ,...,f ∈Z[[q]]. Note that we can apply Lemma 2.2 even if s<0.
0 m+1
Define coefficients c (n)∈Z of f (qℓm+1−i) by
i i
f (qℓm+1−i)= c (n)qnℓm+1−i.
i i
n≥0
X
Then,
∞
(2) p (n)qn =
k+sℓm+1
n=0
X
∞
p (n)qn c (n)qnℓm+1 +ℓ c (n)qnℓm +···+ℓm+1 c (n)qn .
k 0 1 m+1
! 
n=0 n≥0 n≥0 n≥0
X X X X
 
For fixed integers n ≥ 0, a ≥ 0, and 0 ≤ i ≤ m+1, the coefficient of qnℓm+1+a
in the term
∞
ℓi p (j)qj c (j)qjℓm+1−i
k i
  
j=0 j≥0
X X
  
A PROOF OF THE RAMANUJAN CONGRUENCES FOR MULTIPARTITIONS 7
is
nℓi+bi
ℓi p (ℓin−j)ℓm+1−i+a c (j),
k i
j=0
X (cid:0) (cid:1)
where b = ⌊ a ⌋ and so the b satisfy a = ℓm+1−ib +a′ for some 0 ≤ a′ <
i ℓm+1−i i i
ℓm+1−i. By equating the coefficients of qnℓm+1+a from the left and right sides of
(2), we have that for all n≥0,
m+1 nℓi+bi
(3) p (nℓm+1+a)= ℓi p (ℓin−j)ℓm+1−i+a c (j) .
k+sℓm+1  k i 
i=0 j=0
X X (cid:0) (cid:1)
Since we are assumingthat there isa congruenceat(ℓr,k,a)for all1≤r≤m+1,
for each 0≤i≤m we have that for any 0≤j ≤nℓi+b ,
i
ℓip (ℓin−j)ℓm+1−i+a ≡0 (mod ℓm+1).
k
Combining this with (3(cid:0)) yields that (cid:1)
p (nℓm+1+a)≡0 (mod ℓm+1)
k+sℓm+1
for all n≥0, as desired.
(cid:3)
2.2. ConnectiontomodularformsonSL (Z). Recallthataholomorphicmod-
2
ularformofintegerweightkonthemodulargroupSL (Z)isaholomorphicfunction
2
fromthe upper half plane f :H→C, such that for all z ∈H, and a b ∈SL (Z),
c d 2
f az+b =(cz+d)kf(z). (cid:0) (cid:1)
cz+d
(cid:18) (cid:19)
In addition, f is holomorphic at ∞. I.e., f has a Fourier expansion of the form
f(z)= a(n)qn,
n≥0
X
where q =e2πiz. If a(0)=0, we say f is a cusp form. We denote by M (resp. S )
k k
the finite dimensional complex vector space of holomorphic modular (cusp) forms
ofweightk. For the readerunfamiliar withmodular forms, we recommend[Ser73],
[DS05], and [Ono04] for a nice introduction.
We now set q =e2πiz throughout. The delta function ∆(z), defined by
∞
∆(z):=q (1−qn)24,
n=1
Y
is a weight 12 cusp form. In addition, for even k ≥4 the classical Eisenstein series
E defined by
k
−2k
E (z):=1+ σ (n)qn,
k B k−1
k
n≥1
X
where B is the kth Bernoulli number, and σ (n)= dk−1, lies in M .
k k−1 d|n k
Foroddintegersk, M hasdimension0. Whenk ≥2iseven,M hasdimension
k k
P
given by
⌊k ⌋ if k ≡2 (mod 12),
(4) dim(M )= 12
k (⌊k ⌋+1 if k 6≡2 (mod 12).
12
8 OLEGLAZAREV,MATTHEWS.MIZUHARA,BENJAMINREID,ANDHOLLYSWISHER
A basis for M can be given in terms of the Eisenstein series E and E . The
k 4 6
following lemma is based on properties of Bernoulli numbers and can be found in
[Ono04] (Lemma 1.22).
Lemma 2.4. Let k ≥2 be even, and p prime. If k is divisible by p−1, then
E (z)≡1 (mod pordp(2k)+1),
k
where ord (2k) is the largest integer n for which pn |2k.
p
The proofofTheorem1.2relies onanelegantresultofChoie,Kohnen,andOno
foundin [CKO05]. To state this resultwe needsome additionalterminology. For a
function f(z) with Fourier expansion f(z)= a qn, define
n≥n0 n
const(f)=Pa0.
In addition, for a positive even integer k, define
1 if k ≡0 (mod 12),
E if k ≡2 (mod 12),
 14
(5) E˜k =EE46 iiff kk ≡≡46 ((mmoodd 1122)),,
E2 if k ≡8 (mod 12),
4
Theorem 2.5. (Choie, KohnEen4,EO6 noi)f Lke≡t f10∈ (Mm1o2dn+1124),.g ∈ Mk, and D(k) =
dim(M ). Then,
k
f ·g
const =0.
∆n+D(k)·E˜
(cid:18) k(cid:19)
Definition 2.6. Let k,m be positive integers. For all integers n ≥ 0 we define
τk,m(n) to be the nth Fourier coefficient of ∆(z)δk,ℓ,m. I.e.,
∞ ∞
(6) τ (n)qn =qδk,ℓ,m (1−qn)k(ℓ2m−1) =∆(z)δk,ℓ,m.
k,m
n=0 n=1
X Y
Note that τ (n)∈Z.
k,m
The reason we are able to use Theorem 2.5 to prove Theorem 1.2 is due to an
explicitrelationshipbetweenp (n)and∆(z). Noticethatifwefixpositiveintegers
k
k,m and a prime ℓ≥5, we have from (6) that
∞ ∞ ∞ kℓ2m
∆(z)δk,ℓ,m = τk,m(n)qn =qδk,ℓ,m pk(n)qn (1−qn) .
! !
n=0 n=0 n=1
X X Y
For any 1≤r≤2m,
∞ ∞ ∞ kℓ2m−r ℓr
(7) τ (n)qn = p (n−δ )qn (1−qn) .
k,m k k,ℓ,m
! ! 
n=0 n=0 n=1
X X Y
Applying Lemma 2.2 to ∞ (1−qn)kℓ2m−r, gives that 
n=1
∞ ∞
τ (n)qn = pQ(n−δ )qn f (qℓr)+ℓf (qℓr−1)+···+ℓrf (q) ,
k,m k k,ℓ,m 0 1 r
!
nX=0 nX=0 (cid:16) (cid:17)
A PROOF OF THE RAMANUJAN CONGRUENCES FOR MULTIPARTITIONS 9
for some f ,...,f ∈Z[[q]]. For 0≤i≤r, define coefficients c (n)∈Z by
0 r r,i
f (qℓr−i)= c (n)qnℓr−i.
i r,i
n≥0
X
Thus we have
∞ ∞
τ (n)qn = p (n−δ )qn ·
k,m k k,ℓ,m
!
n=0 n=0
X X
c (n)qnℓr +ℓ c (n)qnℓr−1 +···+ℓr c (n)qn .
r,0 r,1 r,r
 
n≥0 n≥0 n≥0
X X X
The coefficient ofqnℓr in the term 
∞
p (n−δ )qn ℓi c (n)qnℓr−i
k k,ℓ,m r,i
! 
n=0 n≥0
X X
 
is
nℓi
ℓi p (nℓi−j)ℓr−i−δ c (j).
k k,ℓ,m r,i
j=0
X (cid:0) (cid:1)
Thus we have for any 1≤r≤2m,
r nℓi
τ (nℓr)= ℓi p (nℓi−j)ℓr−i−δ c (j)
k,m k k,ℓ,m r,i
i=0 j=0
X X (cid:0) (cid:1)
r−1 nℓi
(8) ≡ ℓi p (nℓi−j)ℓr−i−δ c (j) (mod ℓr).
k k,ℓ,m r,i
i=0 j=0
X X (cid:0) (cid:1)
If we rewrite (7) as
∞ ∞ ∞ kℓ2m−r ℓr
1
p (n−δ )qn = τ (n)qn
k k,ℓ,m k,m ! (1−qn)! 
n=0 n=0 n=1
X X Y
 
and let
∞ kℓ2m−r ℓr
1
=
 (1−qn)! 
n=1
Y
 b (n)qnℓr +ℓ b (n)qnℓr−1 +···+ℓr b (n)qn,
r,0 r,1 r,r
n≥0 n≥0 n≥0
X X X
again using Lemma 2.2, then similarly we get that
r nℓi
p (nℓr−δ )= ℓi τ ((nℓi−j)ℓr−i)b (j)
k k,ℓ,m k,m r,i
i=0 j=0
X X
r−1 nℓi
(9) ≡ ℓi τ ((nℓi−j)ℓr−i)b (j) (mod ℓr).
k,m r,i
i=0 j=0
X X
10 OLEGLAZAREV,MATTHEWS.MIZUHARA,BENJAMINREID,ANDHOLLYSWISHER
We will use these facts in our proof of the following lemma, which generalize a
proposition of Lachterman, Schayer,and Younger in [LSY08].
Lemma 2.7. Let ℓ≥5 be prime, k,m and B be positive integers, and 1≤r≤m.
There is a congruence at (ℓs,k,−δ ) for each 1≤s≤r if and only if
k,ℓ,m
τ (ℓsn)≡0 (mod ℓs)
k,m
for all n≥0, for each 1≤s≤r.
Moreover if there is a congruence at (ℓs,k,−δ ) for each 1 ≤ s ≤ r, and
k,ℓ,m
p (ℓr+1n−δ )≡0 (mod ℓr+1) for all n<B, then τ (ℓr+1n)≡0 (mod ℓr+1)
k k,ℓ,m k,m
for all n<B.
Proof. Suppose there is a congruence at (ℓs,k,−δ ) for each 1≤s≤r, and fix
k,ℓ,m
1≤s≤r, n≥0. Then for each 0≤i≤s−1, and 0≤j ≤nℓi we have that
p (nℓi−j)ℓs−i−δ ≡0 (mod ℓs−i).
k k,ℓ,m
Since 1≤s≤m, by (8(cid:0)) it follows that (cid:1)
τ (nℓs)≡0 (mod ℓs)
k,m
for all n≥0.
The final sentence of the theorem follows from (8) as well, by considering r+1,
and restricting values of n to n<B.
The reverse implication follows from a similar argument utilizing (9). (cid:3)
3. The proof of Theorem 1.2
For each n∈Z we define the integers
w (n)=12(nℓm−δ )+2,
k,ℓ,m k,ℓ,m
when k,m are positive integers, and ℓ≥5 is prime.
Proposition 3.1. Fix a prime ℓ≥5, and positive integers k,m such that k ≡−4
(mod ℓm−1). Then for any integer n, we have
w (n)≡0 (mod ℓm−1).
k,ℓ,m
Proof. Since k ≡ −4 (mod ℓm−1), we have k(ℓ2m−1)≡ 4 (mod ℓm−1), and since
2 is relatively prime to ℓ,
k(ℓ2m−1)
12δ = ≡2 (mod ℓm−1).
k,ℓ,m
2
Thus for any integer n,
w (n)≡−12δ +2≡0 (mod ℓm−1).
k,ℓ,m k,ℓ,m
(cid:3)
In light of Proposition 3.1, for each n ∈ Z, there is an integer C such that
n
w (n) = C ℓm−1. Moreover, C must be even since w (n) is even, and
k,ℓ,m n n k,ℓ,m
ℓ is odd. Thus we define
K ∈{0,4,6,...,ℓ−3,ℓ+1}
n
when ℓ≥7, and K ∈{0,4,6} when ℓ=5, so that
n
K ≡C (mod ℓ−1).
n n