Content uploaded by Mircea Merca

Author content

All content in this area was uploaded by Mircea Merca on Nov 13, 2022

Content may be subject to copyright.

Journal of Number Theory 149 (2015) 57–69

Contents lists available at ScienceDirect

Journal of Number Theory

www.elsevier.com/locate/jnt

A new look on the generating function

for the number of divisors

Mircea Merca

Department of Mathematics, University of Craiova, A.I. Cuza 13, Craiova,

200585, Romania

a r t i c l e i n f o a b s t r a c t

Article history:

Received 31 August 2014

Received in revised form 1 October

2014

Accepted 2 October 2014

Available online 5 December 2014

Communicated by David Goss

MSC:

11A25

11P81

11P84

05A17

05A19

Keywords:

Divisors

Lambert series

Partitions

The q-binomial coeﬃcients are specializations of the elemen-

tary symmetric functions. In this paper, we use this fact to

give a new expression for the generating function of the num-

ber of divisors. As corollaries, we obtained new connections

between partitions and divisors.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

Any series of the form

∞

n=1

an

qn

1−qn,|q|<1,

E-mail address: mircea.merca@proﬁnfo.edu.ro.

http://dx.doi.org/10.1016/j.jnt.2014.10.009

0022-314X/© 2014 Elsevier Inc. All rights reserved.

58 M. Merca / Journal of Number Theory 149 (2015) 57–69

where the an(n =1, 2, ...)are real numbers is called a Lambert series. These series

are well known class of series in analytic function theory and number theory and are

mentioned in the classical texts by Abramowitz and Stegun [1, pp. 826–827], Bromwich

[7, pp. 102–103], Chrystal [8, pp. 345–346], Hardy and Wri g ht [13, pp. 257–258], Knopp

[18, pp. 448–452], MacMahon [24, pp. 26–32], Pólya and Szegő [28, pp. 125–129], and

Titchmarsh [30, pp. 160–161]. Lambert series have been elegantly used in a varie ty of

contexts of Ramanujan’s research works. The dimension provided by Ramanujan inspired

Andrews and Berndt [5] to prove a lot of identities given by Ramanujan.

Lambert series are natural generalizations of the following formula related to the

theory of numbers:

∞

n=1

qn

1−qn=

∞

n=0

τ(n)qn,|q|<1.(1)

In multiplicative number theory, the divisor function τ(n)is deﬁned as the number of

divisors of n, unity and nitself included, i.e.,

τ(n)=

d|n

1.

We use the convention that τ(n) =0for n 0.

Due to Clausen’s [9], we have the following identity:

∞

n=1

qn

1−qn=

∞

n=1

1+qn

1−qnqn2,|q|<1.(2)

In this paper, motivated by these results, we shall prove:

Theorem 1. For |q| <1,

∞

n=1

qn

1−qn=1

(q;q)∞

∞

n=1

(−1)n−1nqn+1

2

(q;q)n

,(3)

where

(a;q)n=(1−a)(1 −aq)1−aq2···1−aqn−1

is the q-shifted factorial, with (a; q)0=1.

Some consequences of this result can be easily derived.

M. Merca / Journal of Number Theory 149 (2015) 57–69 59

Corollary 1.1. The number of divisors of ncan be expressed as a ﬁnite discrete convolu-

tion:

τ(n)=

n

k=1

a(k)p(n−k),

where

a(n)=

∞

k=1

(−1)k−1kq(n, k),

p(n)is the number of partitions of n, and q(n, k)denotes the number of partitions of n

into exactly kdistinct parts (q(0, k) =1and q(n, k) =0for nnegative).

Note that a(n)is in fact a ﬁnite sum, since q(n, k) =0for k>n.

Example. The a(n)sequence begins as follows for n >0,

1,1,−1,−1,−3,0,−2,1,2,1,2,4,1,−1,4,−2,−1,−3,−1, ...

For n =9, we have

p(8) + p(7) −p(6) −p(5) −3p(4) −2p(2) + p(1) + 2p(0) = 3

and τ(9) equals 3because the three divisors in questions are: 1, 3and 9.

Corollary 1.2. Let nbe a positive integer. Then

∞

k=1

(−1)k−1kq(n, k)=

∞

k=−∞

(−1)kτn−k(3k−1)/2.

We see that the generalized pentagonal numbers appear in this relation accompanied

by the number of divisors function:

q(n, 1) −2q(n, 2) + 3q(n, 3) −4q(n, 4) + 5q(n, 5) −···

=τ(n)−τ(n−1) −τ(n−2) + τ(n−5) + τ(n−7) −τ(n−12) −···

Surprisingly, this relation was not observed for many years.

In the ﬁnal section of the paper, we derive a nice identity that combines triangular

and pentagonal numbers.

60 M. Merca / Journal of Number Theory 149 (2015) 57–69

2. Proof of Theorem 1

Being given a set of variables {x1, x2, ..., xn}, recall [23] that the kth elementary

symmetric function ek(x1, x2, ..., xn)is given by

ek(x1,x

2, ..., xn)=

1i1<i2<···<ikn

xi1xi2···xik

for k=1, 2, ..., n. We set e0(x1, x2, ..., xn) =1by convention. For k<0or k>n, we

set ek(x1, x2, ..., xn) =0.

The elementary symmetric functions are characterized by the following identity of

formal power series in t:

E(t)=

n

k=0

ek(x1,...,x

n)tk=

n

k=1

(1 + xkt)

For k=1, 2, ..., n, we consider that 1 +xkt =0. On the one hand, we have

d

dt lnE(t)=

n

k=1

d

dt ln(1 + xkt)=

n

k=1

xk

1+xkt.(4)

On the other hand, we can write

d

dt lnE(t)=n

k=1

1

1+xkt n

k=1

kek(x1,...,x

n)tk−1.(5)

Thus, by (4) and (5), we derive the identity

n

k=1

xk

1+xkt=n

k=1

1

1+xkt n

k=1

kek(x1,...,x

n)tk−1,(6)

where x1, x2, ..., xnand tare independent vari ables such that 1 +xkt =0for k=

1, 2, ..., n.

The q-analogue of the classical binomial coeﬃcient is called the q-binomial coeﬃcient

and is deﬁned by

n

k=(q;q)n

(q;q)k(q;q)n−k,if k∈{0,1,...,n},

0,otherwise.

Replacing xkby qk−1in (6), we obtain

n

k=1

qk−1

1+qk−1t=1

(−t;q)n

n

k=1

kqk

2n

ktk−1,(7)

M. Merca / Journal of Number Theory 149 (2015) 57–69 61

where we have invoked the fact that

ek1,q,...,q

n−1=qk

2n

k.

The case t =−qof (7) can be written as

n

k=1

qk

1−qk=1

(q;q)n

n

k=1

(−1)k−1kqk+1

2n

k.(8)

For |q| <1, we have

lim

n→∞ n

k=1

(q;q)k

and the theorem is proved.

3. Proofs of Corollaries 1.1 and 1.2

Recall [2, Theorem 13-3, p. 162] that the partition function p(n)has the generating

function

∞

n=0

p(n)qn=1

(q;q)∞

,|q|<1.

In this context, the formula (3) can be written as

∞

n=1

τ(n)qn=∞

n=0

p(n)qn∞

n=1

(−1)n−1nqn+1

2

(q;q)n,|q|<1.

On the other hand, the generating function of q(n, k)[4, Theorem 11.4.1, p. 559] is

given by

∞

n=0

q(n, k)qn=qk+1

2

(q;q)k

,|q|<1.

For |q| <1, we have

∞

k=1

(−1)k−1kqk+1

2

(q;q)k

=

∞

k=1

∞

n=0

(−1)k−1kq(n, k)qn

=

∞

n=0

∞

k=1

(−1)k−1kq(n, k)qn

=

∞

n=1

a(n)qn

because q(0, k) =0for any positive integer k.

62 M. Merca / Journal of Number Theory 149 (2015) 57–69

Takin g into account the well-known Cauchy products of two power series

∞

n=0

anqn ∞

n=0

bnqn=

∞

n=0n

k=0

akbn−kqn,

Corollary 1.1 is proved.

For |q| <1, it is clear that the relation (3) can be written as

∞

n=1

a(n)qn=(q;q)∞

∞

n=1

τ(n)qn

=∞

n=−∞

(−1)nqn(3n−1)/2 ∞

n=1

τ(n)qn

=

∞

n=1

∞

k=−∞

(−1)kτn−k(3k−1)/2qn,

where we have invoked the Euler pentagonal number theorem [3, Corollary 1.7, p. 11],

(q;q)∞=

∞

n=−∞

(−1)nqn(3n−1)/2

and the Cauchy product of two power series. The proof of Corollary 1.2 is ﬁnished.

4. Other identities involving τ(n)

The aim of this section is to present new identities involving the number of divisors

function τ(n). For this purpose we use some well known partition identities.

An r-regular partition of n(r>1) is a partition of nwith no part divisible by r. The

number of r-regular partitions of nis denoted by br(n)and follows the convention that

br(0) =1. In classical representation theory, r-regular partitions of nlabel irreducible

r-modular representations of the symmetric group Snwhen ris prime [16]. More recently,

such partitions have been studied for their arithmetic properties in connection with the

theory of modular forms and Galois representations [12,19,27].

Corollary 4.1. For n >0, r>1,

∞

k=−∞

(−1)kτn−r·k(3k−1)/2=

n

k=1

a(k)br(n−k).

M. Merca / Journal of Number Theory 149 (2015) 57–69 63

Proof. Elementary techniques in the theory of partitions [3] give the following generating

function for the number of r-regular partitions

∞

n=0

br(n)qn=(qr;qr)∞

(q;q)∞

,|q|<1.(9)

Thus, for |q| <1we can write

qr;qr∞

∞

n=1

τ(n)qn=∞

n=0

br(n)qn ∞

n=1

a(n)qn

and then, using the Euler pentagonal number theorem, we get

∞

n=−∞

(−1)nqr·n(3n−1)/2 ∞

n=1

τ(n)qn=∞

n=0

br(n)qn∞

n=1

a(n)qn.

Applying the Cauchy product of two power series, we arrive at our conclusion. 2

The case r=2of this corollary is given by

∞

k=−∞

(−1)kτn−k(3k−1)=

n

k=1

a(k)q(n−k),

were q(n) denotes the number of partitions of nwith distinct parts. Recall [3, Corol-

lary 1.2, p. 5] that the number of partitions of nwith distinct parts is equal to the

number of partitions of nwith odd parts.

Example. For n =9, we have

q(8) +q(7) −q(6) −q(5) −3q(4) −2q(2) + q(1) + 2q(0) = τ(9) −τ(7) −τ(5).

Corollary 4.2. For n >0,

∞

k=0

τn−k(k+1)/2=

i+j+k=n

a(i)q(j)q(k).(10)

Proof. By the Jacobi triple product identity [6, Theorem 11],

q2;q2∞−qx;q2∞−q/x;q2∞=

∞

n=−∞

xnqn2,|q|<1,x=0,

with xreplaced by q, we get

q2;q2∞−q2;q22

∞=

∞

n=0

qn2+n,|q|<1.

64 M. Merca / Journal of Number Theory 149 (2015) 57–69

Replacing q2by q, we have

(q;q)∞(−q;q)2

∞=

∞

n=0

qn(n+1)/2,|q|<1.

Thus, we deduce that

∞

n=1

a(n)qn ∞

n=0

q(n)qn2

=∞

n=0

qn(n+1)/2 ∞

n=1

τ(n)qn,|q|<1.

Equating coeﬃcients of qnon each side give the result. 2

Let q(n) denote the number of partitions of ninto odd parts, each part being of two

kinds. For instance, for n =4, we consider kand kto be diﬀerent versions of kand

so we have: 3 +1, 3 +1

, 3+1, 3+1

, 1 +1 +1 +1, 1 +1 +1 +1

, 1 +1 +1

+1

,

1 +1

+1

+1

and 1+1

+1

+1

. Thus q(4) =9. According to Sloane [29, A022567],

the generating function for q(n)is given by

∞

n=0

q(n)qn=(−q;q)2

∞,|q|<1.

In this context, the identity (10) can be written as

∞

k=0

τn−k(k+1)/2=

n

k=1

a(k)q(n−k),n>0.

Corollary 4.3. For n >0,

∞

k=0

(−1)k(2k+1)τn−k(k+1)/2=

i+j+k=n

a(i)e(j)e(k),

where

e(n)=qe(n)−qo(n),

and qe(n), respectively qo(n)denotes the number of partitions of ninto an even, respec-

tively an odd number of distinct parts.

Proof. To prove the corollary, we consider another special case of the Jacobi triple prod-

uct identity [4, Corollary 10.4.2, p. 500],

(q;q)3

∞=

∞

n=0

(−1)n(2n+1)qn(n+1)/2,|q|<1.

M. Merca / Journal of Number Theory 149 (2015) 57–69 65

By Andrews and Eriksson [6, Theorem 4, p. 25], we have

e(n)=(−1)k,if n=k(3k±1)/2,

0,otherwise.

In other words, e(n)is the coeﬃcient of qnin the Euler function (q; q)∞. For |q| <1, we

deduce that

∞

n=1

a(n)qn ∞

n=0

e(n)qn2

=∞

n=0

(−1)n(2n+1)qn(n+1)/2 ∞

n=1

τ(n)qn

and the proof follows easily. 2

Corollary 4.4. For n >0,

∞

k=−∞

τn−k2=

i+j+k+l=n

a(i)q(j)ps(k)ps(l)

where ps(n)denotes then number of self-conjugate partitions of n.

Proof. By the Jacobi triple product identity, with xreplaced by 1, we have

∞

n=−∞

qn2=(q;q)∞(−q;q)∞−q;q22

∞,|q|<1.(11)

Recall [6, Eq. (3.4), p. 18] that the number of self-conjugate partitions of nis equal to

the number of partitions of ninto odd distinct parts. So the generating function of ps(n)

is given by

∞

n=0

ps(n)qn=−q;q2∞,|q|<1.

For |q| <1, we get

∞

n=−∞

qn2 ∞

n=1

τ(n)qn=∞

n=1

a(n)qn ∞

n=1

q(n)qn ∞

n=1

ps(n)qn2

.

Equating coeﬃcients of qnon each side give the result. 2

An overpartition of the nonnegative integer nis a partition of nwhere the ﬁrst occur-

rence of parts of each size may be overlined. Let ¯p(n) denote the number of overpartitions

of n. For example, the overpartitions of the integer 3 are:

3,¯

3,2+1,¯

2+1,2+¯

1,¯

2+¯

1,1+1+1 and ¯

1+1+1.

66 M. Merca / Journal of Number Theory 149 (2015) 57–69

We see that ¯p(3) =8. Properties of ¯p(n)have been the subject of many recent studies [10,

14,17,20–22,25]. Recently, Hirschhorn and Sellers [15] studied the arithmetic properties

of overpartitions using only odd parts. The number of overpartitions into odd parts is

denoted by ¯q(n)and the generating function for ¯q(n)is given by

∞

n=0

¯q(n)qn=(−q;q)∞−q;q2∞,|q|<1.

Takin g into account the identity (11), we obtain

∞

k=−∞

τn−k2=

i+j+k=n

a(i)¯q(j)ps(k).

We remark that ¯q(n)is equal to the number of partitions of 2nin which all odd parts

occur with multiplicity 2and the even parts occur with multiplicity 1[29, A080054].

Corollary 4.5. For n >0,

∞

k=−∞

(−1)kτn−k2=

i+j+k+l=n

a(i)q(j)f(k)f(l)

where

f(n)=pe(n)−po(n),

and pe(n), respectively po(n)denotes the number of partitions of ninto even, respectively

odd number of parts.

Proof. We take into account the case x =−1of the Jacobi triple product identity,

∞

n=−∞

(−1)nqn2=(q;q)∞(−q;q)∞q;q22

∞,|q|<1,

and the fact that the generating function of f(n)is given by [11, p. 38, Eq. (22.14)]

∞

n=0

f(n)qn=(−q;q)−1

∞=q;q2∞,|q|<1.2

Corollary 4.6. For n >0,

∞

k=−∞

(6k+1)τn−k(3k+1)/2=

i+j+k+l+m=n

a(i)e(j)e(k)f(l)f(m)

M. Merca / Journal of Number Theory 149 (2015) 57–69 67

where

e(n)=qe(n)−qo(n)and f(n)=pe(n)−po(n).

Proof. Considering the identity [4, p. 545, Exercise 14]

∞

n=−∞

(6n+1)qn(3n+1)/2=(q;q)3

∞q;q22

∞,

the proof follows easily. 2

5. Concluding remarks

A new factorization has been introduced in this paper for the generating function for

number of divisors,

∞

n=1

τ(n)qn=1

(q;q)∞

∞

n=1

(−1)n+1 nqn+1

2

(q;q)n

.

As a corollary of this result, we obtained formulas that combine the number of divisors

function τ(n)and the number of partitions of ninto exactly kdistinct parts.

We remark that

q(n, k)=pkn−k(k+1)/2,(12)

where pk(n) denotes the number of partitions of nwith no part greater than k. This

follows from

∞

n=1

∞

k=1

(−1)k−1kq(n, k)qn=

∞

k=1

(−1)k−1kqk+1

2

(q;q)k

=

∞

k=1

(−1)k−1kqk+1

2

∞

n=0

pk(n)qn

=

∞

k=1

∞

n=0

(−1)k−1kpk(n)qn+k(k+1)/2

=

∞

n=1

∞

k=1

(−1)k−1kpkn−k(k+1)/2qn,

where we have invoked the generating function of pk(n)[2, Theorem 13-1, p. 161],

∞

n=0

pk(n)qn=1

(q;q)k

,|q|<1

and the fact that pk(n) =0for nnegative.

68 M. Merca / Journal of Number Theory 149 (2015) 57–69

Moreover, the identity (12) has a simple combinatorial proof. We start from a partition

of ninto kdistinct parts and then we subtract a staircase of size k(i.e. subtract kto the

largest part, k−1to the second largest one, etc., and 1to the smallest parts). The result

is a partition of n −k(k+1)/2into at most kparts. Conjugating, we get a partition of

n −k(k+1)/2with no part greater than k.

Thus we can write the following identity [26]

p1(n−1) −2p2(n−3) + 3p3(n−6) −4p4(n−10) + 5p5(n−15) −···

=τ(n)−τ(n−1) −τ(n−2) + τ(n−5) + τ(n−7) −τ(n−12) −···

in which triangular and pentagonal numbers appear together.

For 0 <q<1, we have

n

k=1

qk

1−qk=

n

k=1

qk1+qk+q2k+q3k+···

>

n

k=1

τ(k)qk.

The following inequality is immediate from (8).

Corollary 5.1. Let nbe a positive integer. For 0 <q<1,

n

k=1

τ(k)qk<1

(q;q)n

n

k=1

(−1)k+1kqk+1

2n

k.

Acknowledgments

The author appreciates the anonymous referees for their comments on the original

version of this paper. Special thanks go to Dr. Oana Merca for the careful reading of the

manuscript and helpful remarks.

References

[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formu l as, Graphs, and

Mathematical Tab l e s, Dover Publications, New York , 1972.

[2] G.E. Andrews, Number Theory, W.B. Saunders, Philadelphia, 1971.

[3] G.E. Andrews, The Theory of Partitions, Addison–Wesley Publishing, 1976.

[4] G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, 1999.

[5] G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook, Part 1, Springer, New York , 2005.

[6] G.E. Andrews, K. Eriksson, Integer Partitions, Cambridge University Press, 2004.

[7] T.J. Bromwich, An Introduction to the Theory of Inﬁnite Series, 2nd ed., Macmillan, New Yor k,

1926.

[8] G. Chrystal, Algebra, vol. 2, Chelsea, New Yo r k , 1952.

[9] T. Clausen, Beitrag zur Theorie der Reihen, J. Reine Angew. Math. 3 (1828) 92–95.

[10] S. Corteel, J. Lovejoy, Overpartitions, Tra n s. Amer. Math. Soc. 356 (2004) 1623–1635.

[11] N.J. Fine, Basic Hypergeometric Series and Applications, American Mathematical Society, 1988.

[12] B. Gordon, K. Ono, Divisibility of certain partition functions by powers of primes, Ramanujan J. 1

(1997) 25–34.

M. Merca / Journal of Number Theory 149 (2015) 57–69 69

[13] G.H. Hardy, E.M. Wrig ht, An Introduction to the Theory of Numbers, Clarendon Press, Oxford,

1979.

[14] M.D. Hirschhorn, J.A. Sellers, Arithmetic relations for overpartitions, J. Combin. Math. Combin.

Comput. 53 (2005) 65–73.

[15] M.D. Hirschhorn, J.A. Sellers, Arithmetic properties of overpartitions into odd parts, Ann. Comb.

10 (2006) 353–367.

[16] G. James, A. Kerber, The Representation Theory of the Symmetric Group, Addison–Wesley, Read-

ing, 1981.

[17] B. Kim, A short note on the overpartition function, Discrete Math. 309 (2009) 2528–2532.

[18] K. Knopp, Theory and Application of Inﬁnite Series, Dover, New York, 1990.

[19] J. Lovejoy, Divisibility and distribution of partitions into distinct parts, Adv. Math. 158 (2001)

253–263.

[20] J. Lovejoy, Gordon’s theorem for overpartitions, J. Combin. Theory Ser. A 103 (2003) 393–401.

[21] J. Lovejoy, Overpartition theorems of the Rogers–Ramaujan type, J. Lond. Math. Soc. 69 (2004)

562–574.

[22] J. Lovejoy, Overpartitions and real quadratic ﬁelds, J. Number Theory 106 (2004) 178–186.

[23] I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Clarendon Press, Oxford,

1995.

[24] P.A . MacMahon, Combinatory Analysis, Chelsea Pub. Co., New York , 1960.

[25] K. Mahlburg, The overpartition function modulo small powers of 2, Discrete Math. 286 (2004)

263–267.

[26] M. Merca, Problem 11787, Amer. Math. Monthly 121 (6) (2014) 550.

[27] K. Ono, D. Penniston, The 2-adic behavior of the number of partitions into distinct parts, J. Combin.

Theory Ser. A 92 (2000) 138–157.

[28] G. Pólya, G. Szegő, Problems and Theorems in Analysis II, Springer-Verlag, Berlin, Heidelberg,

1976.

[29] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, Published electronically at http://

oeis.org, 2014.

[30] E.C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford University Press, London, 1939.