ISSN 0001-4346, Mathematical Notes, 2014, Vol. 95, No. 4, pp. 565–569. © Pleiades Publishing, Ltd., 2014.Original Russian Text © S. E. Gladun, 2014, published in Matematicheskie Zametki, 2014, Vol. 95, No. 4, pp. 630–634.

SHORTCOMMUNICATIONS

A Generating Function for σ(3n − 1)

S. E. Gladun*

Received April 27, 2012; in final form, September 17, 2013

DOI: 10.1134/S0001434614030286

Keywords: natural number, sum of divisors of a natural number σ(n), generating function forσ(3n − 1), theta function, Lambert series.

Ramanujan obtained the following formula containing the generating function for p(5n − 1), wherep(n) is the number of representations of n expressed as the sum of natural summands [1, p. 130 (Russiantransl.), formula (6.7.1)]:

p(4) + p(9)q + p(14)q2 + p(19)q3 + · · · = 5{(1 − q5)(1 − q10)(1 − q15)(1 − q20) · · · }5

{(1 − q)(1 − q2)(1 − q3)(1 − q4) · · · }6. (R)

In this paper, we obtain a rather similar formula for σ(3n − 1), where, for any n ∈ N, σ(n) :=∑

d|n d

is the sum of the divisors of the natural number n. Thus, in the disk |q| < 1, the following formula isvalid:

σ(2) + σ(5)q + σ(8)q2 + σ(11)q3 + · · · = 3{(1 − q3)(1 − q6)(1 − q9)(1 − q12) · · · }6

{(1 − q)(1 − q2)(1 − q3)(1 − q4) · · · }2. (1)

In what follows, we shall set q = eπiτ , where Im τ > 0 and qλ = eλπiτ .Ramanujan never published the proof of formula (R); however, apparently, his arguments were

exceptional in nature and were based only on the expression for the generating function for the numberof partitions, the Euler pentagonal theorem, and the Jacobi formula for the cube of the Eulerian product.A scheme for a similar proof was given in the monograph [2, pp. 327–328]. The proof of formula (1)given by the author makes a more extensive use of the apparatus of theta functions and is divided intoseveral simple lemmas for the convenience of the reader. The proof is based on profound results obtainedearlier.

In his survey of the Collection of Ramanujan’s works, Littlewood expressed his opinion that it isnot worthwhile to derive similar formulas from the pool of ideas in those areas of mathematics where ageneral theory has already been created. The author does not agree with this point of view and, as anargument, presents formula (1), which is both new and interesting.

Let us write the expressions for the triplet of theta functions [3, p. 93]:

a(q) :=∞∑

m,n=−∞qm2+mn+n2

, b(q) :=∞∑

m,n=−∞ωm−nqm2+mn+n2

,

c(q) :=∞∑

m,n=−∞q(m+1/3)2+(m+1/3)(n+1/3)+(n+1/3)2 ,

where ω = e2πi/3.A sufficiently complete theory of elliptic functions for alternative bases was given in [4] and [5]. By [3,

p. 109, (5.5)], the following representation is valid:

c(q) = 3q1/3 {(1 − q3)(1 − q6)(1 − q9)(1 − q12) · · · }3

{(1 − q)(1 − q2)(1 − q3)(1 − q4) · · · } . (2)

*E-mail: gladunse@inbox.ru

565

566 GLADUN

Lemma 1. The following identity holds:

σ(2) + σ(5)q + σ(8)q2 + σ(11)q3 + · · · = 3∞∑

n=−∞

nq2n−2

1 − q3n−2. (3)

Proof. Let us write the Lambert series for σ(n) [6, p. 145 (Russian transl.), Example 75]:

σ(1)q + σ(2)q2 + σ(3)q3 + σ(4)q4 + · · · =q

1 − q+

2q2

1 − q2+

3q3

1 − q3+

4q4

1 − q4+ · · · . (4)

We multiply both sides of relation (4) by q. In the resulting relation, we replace q by ωq and by ω2q andadd these three equalities, taking into account the absolute convergence of the corresponding series. Asa result, we obtain the following relation:

σ(2) + σ(5)q3 + σ(8)q6 + σ(11)q9 + · · · =∞∑

n=1

(3n − 2)q6n−6

1 − q9n−6+

∞∑

n=1

(3n − 1)q3n−3

1 − q9n−3,

whence

σ(2) + σ(5)q3 + σ(8)q6 + σ(11)q9 + · · · =∞∑

n=−∞

(3n − 2)q6n−6

1 − q9n−6= 3

∞∑

n=−∞

nq6n−6

1 − q9n−6, (5)

which is equivalent to (3) if q3 is replaced by q.

Lemma 2. For the squares of the theta functions a2(q), b2(q), a2(q3), and c2(q3), the followingrelation holds:

c2(q3) =16

a2(q) − 12

a2(q3) +13

b2(q). (6)

Proof. By [3, pp. 93–94, formulas (2.8) and (2.9)], we have

b(q) =3a(q3) − a(q)

2, (7)

c(q3) =a(q) − a(q3)

2. (8)

Squaring (7) and (8) and eliminating the product a(q)a(q3) from the resulting system, we obtain (6).

Lemma 3. The following identity holds:

c2(q3) =19{b2(q) + ωb2(ωq) + ω2b2(ω2q)} +

118

{a2(q) + ωa2(ωq) + ω2a2(ω2q)}. (9)

Proof. In (6), let us replace q by ωq and by ω2q, obtaining

ω2c2(q3) =16

a2(ωq) − 12

a2(q3) +13

b2(ωq), (10)

ωc2(q3) =16

a2(ω2q) − 12

a2(q3) +13

b2(ω2q). (11)

Multiplying relation (10) by ω, and (11) by ω2, and adding the resulting relations with (6), we obtain therequired relation.

Lemma 4. For theta functions, the following formula holds:

B(q) := b2(q) + ωb2(ωq) + ω2b2(ω2q) = 3c2(q3). (12)

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A GENERATING FUNCTION FOR σ(3n − 1) 567

Proof. Adding together relations (7) and (8), we obtain the expression

b(q) = a(q3) − c(q3), (13)

whence

b(ωq) = a(q3) − ωc(q3), (14)

b(ω2q) = a(q3) − ω2c(q3). (15)

Substituting (13), (14), and (15) into the left-hand side of (12), we verify the validity of the lemma.

Combining the assertion of Lemma 4 with formula (9), we obtain

c2(q3) =112

{a2(q) + ωa2(ωq) + ω2a2(ω2q)}. (16)

Lemma 5. The following formula holds:

A(q) := a2(q) + ωa2(ωq) + ω2a2(ω2q) = 108∞∑

n=−∞

nq6n−4

1 − q9n−6. (17)

Proof. To evaluate the sum on the right-hand side of relation (16), we use [3, p. 100, (2.37)]:

a2(q) = 1 + 12∞∑

n=1

nqn

1 − qn− 36

∞∑

n=1

nq3n

1 − q3n. (18)

In (18), replacing q by ωq and by ω2q, we obtain the additional two formulas:

a2(ωq) = 1 + 12∞∑

n=1

nqnωn

1 − qnωn− 36

∞∑

n=1

nq3n

1 − q3n, (19)

a2(ω2q) = 1 + 12∞∑

n=1

nqnω2n

1 − qnω2n− 36

∞∑

n=1

nq3n

1 − q3n. (20)

It follows from relations (18), (19), and (20) that

A(q) = 12{ ∞∑

n=1

nqn

1 − qn+

∞∑

n=1

nqnωn+1

1 − qnωn+

∞∑

n=1

nqnω2n+2

1 − qnω2n

}

.

Hence, in view of the absolute convergence of the series for |q| < 1, we obtain

A(q) = 36{ ∞∑

n=1

(3n − 2)q6n−4

1 − q9n−6+

∞∑

n=1

(3n − 1)q3n−1

1 − q9n−3

}

or, equivalently,

A(q) = 36∞∑

n=−∞

(3n − 2)q6n−4

1 − q9n−6= 108

∞∑

n=−∞

nq6n−4

1 − q9n−6,

which proves the validity of Lemma 5.

Formulas (16) and (17) imply the relation

c2(q3) = 9∞∑

n=−∞

nq6n−4

1 − q9n−6. (21)

Theorem. Formula (1) holds.

MATHEMATICAL NOTES Vol. 95 No. 4 2014

568 GLADUN

Proof. Taking into account (2), we find that it suffices to show the validity of the following equality:

σ(2) + σ(5)q3 + σ(8)q6 + σ(11)q9 + · · · =c2(q3)3q2

. (22)

Formula (22) follows from (5) and (21).

Corollary 1. The following equalities hold:

σ(2) + σ(5)e−π + σ(8)e−2π + σ(11)e−3π + · · · =(√

3 − 1)1/2(√

2 + 4√

3)3/2π

27/435/4Γ4(3/4)e2π/3,

σ(2) − σ(5)e−π + σ(8)e−2π − σ(11)e−3π + · · · =(√

6 +√

2)π4 · 35/4Γ4(3/4)

e2π/3.

Proof. To prove the equalities, it suffices to use formula (1) and [3, p. 327].

Corollary 2. As q → 1−,

σ(2) + σ(5)q + σ(8)q2 + σ(11)q3 + · · · ∼ 49

π2

(1 − q)2.

Proof. This asymptotic behavior follows from formula (1) and [7, p. 141, (1.11)].

Remark. Similarly, from Ramanujan’s formula for the fourth power of the theta function ψ [8,p. 139, (iii)] and [7, p. 141, (1.11)], we can obtain the following asymptotic formula. As q → 1−,

σ(1) + σ(3)q + σ(5)q2 + σ(7)q3 + · · · ∼ π2

4(1 − q)2.

By way of application of formula (1), let us prove the following curious relation for the sum of divisors.

Corollary 3. For any s ∈ N, the following arithmetical identity holds:s∑

m=1

σ(6m − 1)σ(12s − 6m + 5) = 2s∑

m=1

σ(3m − 1){σ(12s − 12m + 8) − σ(3s − 3m + 2)}.

Proof. It follows from relation (22) that

σ(2) + σ(5)q + σ(8)q2 + σ(11)q3 + · · · =c2(q)3q2/3

, (23)

σ(2) − σ(5)q + σ(8)q2 − σ(11)q3 + · · · =c2(−q)3q2/3

. (24)

Adding together relations (23) and (24), we obtain

2{σ(2) + σ(8)q2 + σ(14)q4 + · · · } =c2(q) + c2(−q)

3q2/3=

{c(q) + c(−q)}2 − 2c(q)c(−q)3q2/3

.

In view of [3, p. 111, (5.15)], the resulting equality yields

c(q)c(−q) = 2c2(q4) − 3q2/3{σ(2) + σ(8)q2 + σ(14)q4 + σ(20)q6 + · · · }. (25)

Using relations (23) and (25), we obtain

c(q)c(−q) = 6q8/3{σ(2) + σ(5)q4 + σ(8)q8 + · · · } − 3q2/3{σ(2) + σ(8)q2 + σ(14)q4 + · · · }.Squaring both sides of this equality and using (23) and (24), we can write

{2q2[σ(2) + σ(5)q4 + σ(8)q8 + · · · ] − [σ(2) + σ(8)q2 + σ(14)q4 + · · · ]}2

= {σ(2) + σ(5)q + σ(8)q2 + · · · }{σ(2) − σ(5)q + σ(8)q2 − · · · }.

MATHEMATICAL NOTES Vol. 95 No. 4 2014

A GENERATING FUNCTION FOR σ(3n − 1) 569

This yields

∞∑

n=0

{ n∑

k=0

akan−k

}

qn =∞∑

n=0

{ 2n∑

k=0

(−1)kbkb2n−k

}

qn, (26)

where

ak = σ(6k + 2) − 2σ(

3k + 12

)

, bk = σ(3k + 2).

Here if the argument of the function of the sum of the divisors is equal to a fractional number, we set thevalue of this function equal to zero. The formula in Corollary 3 follows from relation (26) after equatingthe coefficients of identical powers of q and partitioning the finite sums into sums over even and oddindices. The proof is complete.

Another arithmetic equality is a direct consequence of formula (1).

Corollary 4. For any m ∈ N, set

σ

(3m − 2

3

)

= 0 and σ

(3m − 1

3

)

= 0.

Then the following formula holds for nonnegative integers n:

(n + 1)σ(3n + 5) =n∑

k=0

{

2σ(k + 1) − 18σ(

k + 13

)}

σ(3n − 3k + 2).

Proof. Taking the logarithmic derivative of both sides of formula (1), we obtain

σ(5) + 2σ(8)q + 3σ(11)q2 + · · ·σ(2) + σ(5)q + σ(8)q2 + · · · = 2σ(1) + 2σ(2)q + [2σ(3) − 18σ(1)]q2 + · · · .

This yields the required result.

Note that Corollaries 2, 3, and 4 the theorems illustrate the value of formula (1).

REFERENCES1. G. H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work (Cambridge:

Univ. Press., 1940; Institute of Computer Studies, Moscow, 2002).2. T. M. Apostol, Introduction to Analytic Number Theory, in Undergraduate Texts in Math. (Springer-

Verlag, New York, 1976).3. B. C. Berndt, Ramanujan’s Notebooks (Springer-Verlag, New York, 1998), Part V.4. B. C. Berndt, S. Bhargava, and F. G. Garvan, Trans. Amer. Math. Soc. 347 (11), 4163 (1995).5. F. G. Garvan, J. Symbolic Comput. 20 (5-6), 517 (1995).6. G. Polya and G. Szego, Aufgaben und Lehrsatze aus der Analysis Band II: Funktionentheorie, Null-

stellen, Polynome, Determinanten, Zahlentheorie (Springer-Verlag, Berlin–New York, 1964; Nauka,Moscow, 1978).

7. B. C. Berndt, Ramanujan’s Notebooks (Springer-Verlag, New York, 1994), Part IV.8. B. C. Berndt, Ramanujan’s Notebooks (Springer-Verlag, New York, 1991), Part III.

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