When is the derivative of an eta quotient another eta quotient?

J. Math. Anal. Appl. 480 (2019) 123366

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

When is the derivative of an eta quotient another eta quotient? ✩ Zafer Selcuk Aygin a , Pee Choon Toh b,∗ a

Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada b Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, 637616, Singapore

a r t i c l e

i n f o

Article history: Received 12 November 2018 Available online 23 July 2019 Submitted by S. Cooper Keywords: Eisenstein series Dedekind eta function Eta quotients Modular forms Derivatives Algorithms

a b s t r a c t In this paper we use techniques from the theory of modular forms to determine all eta quotients whose derivative is also an eta quotient of levels up to 36. In addition, we present an algorithm that determines all eta quotients in M2k (Γ0 (N )). We also discuss some applications of these results. In particular, we evaluate a number of integrals in terms of algebraic constants. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Let H denote the complex upper half plane and throughout the paper we let τ ∈ H and q = e2πiτ . Let Γ0 (N ) denote the congruence subgroup of level N defined by    a b Γ0 (N ) = : a, b, c, d ∈ Z, ad − bc = 1, c ≡ 0 (mod N ) . c d A set of representatives from each equivalence class of cusps for Γ0 (N ) is given by [13, Proposition 2.6]  a : 1 ≤ a ≤ N, gcd(a, c) = 1 and R(Γ0 (N )) = c c|N  a ≡ a (mod gcd(c, N/c)) iff a = a . (1.1) ✩ This research was partially supported by the Singapore Ministry of Education Academic Research Fund, Tier 2, project number MOE2014-T2-1-051, ARC40/14. * Corresponding author. E-mail addresses: [email protected] (Z.S. Aygin), [email protected] (P.C. Toh).

https://doi.org/10.1016/j.jmaa.2019.07.056 0022-247X/© 2019 Elsevier Inc. All rights reserved.

2

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

We use Mk (Γ0 (N )) to denote the space of all modular forms of weight k for Γ0 (N ). The Dedekind eta function is defined by the infinite product η(τ ) = eπiτ /12

∞

(1 − e2πinτ ) = q 1/24

n=1

∞

(1 − q n )

n=1

and an eta quotient of level N is defined to be of the form

f (τ ) =

η(δτ )rδ

0<δ|N

where δ runs over all positive divisors of the integer N and the exponents rδ are integers. The weight 1

attached to this eta quotient is k = rδ . We use 2 0<δ|N

ηN [r1 , . . . , rδ , . . . , rN ](τ ) =



η(δτ )rδ

(1.2)

0<δ|N

as a shorthand notation for an eta quotient of level N . Many interesting examples of modular forms or generating functions can be expressed as eta quotients. For example, 24

η(τ )

= η1 [24](τ ) = q

∞

(1 − q )

n 24

n=1

=

∞

τ(n)q n

n=1

is a modular form of weight 12 and level 1. The function τ(n) is called the Ramanujan tau function, in honor of Ramanujan who conjectured and proved [18] many properties satisfied by these coefficients. Deligne’s proof of Ramanujan’s conjectured bounds was an important breakthrough in the field. (See [16, p. 122] for more details.) A second example is η(τ )η(3τ )η(5τ )η(15τ ) = η15 [1, 1, 1, 1](τ ), a weight 2 modular form for Γ0 (15) that is associated with the elliptic curve y 2 + xy + y = x3 + x2 [17]. Our third and most important example for our purpose is the generating function for the number of representations of an integer as a sum of four squares.

2

2

2

2

q x1 +x2 +x3 +x4 =

(x1 ,x2 ,x3 ,x4 )∈Z4

η(2τ )20 η(τ )8 η(4τ )8

= η4 [−8, 20, −8](τ ). Jacobi [14] showed that the number of such representations of n is equal to eight times the sum of all the positive divisors of n which are not divisible by 4. In modern notation,

(x1 ,x2 ,x3 ,x4 )∈Z4

2

2

2

2

q x1 +x2 +x3 +x4 = 1 + 8

∞ ∞



nq n 4nq 4n − 8 n 1−q 1 − q 4n n=1 n=1

= 8 (E2 (τ ) − 4E2 (4τ )) , where

(1.3)

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

3

∞

E2 (τ ) = −

nq n 1 + 24 n=1 1 − q n

is the Eisenstein series of weight 2. (Remark: we follow [21] by normalizing the Eisenstein series such that the coefficient of q is 1.) It is straightforward to show that ∞

nq n 1 d − q log (η(τ )) = = −E2 (τ ). dq 24 n=1 1 − q n Thus it is possible to write Jacobi’s result above as q

d log dq



η(4τ )8 η(τ )8

 =

η(2τ )20 . η(τ )8 η(4τ )8

(1.4)

In other words, Jacobi’s identity can be expressed as an identity where the logarithmic derivative of an eta quotient equals another eta quotient. Another identity of the same type is d q log dq



η(9τ )3 η(τ )3

 =

η(3τ )10 . η(τ )3 η(9τ )3

(1.5)

This identity was discovered through symbolic computation by Borwein and Garvan [5], who subsequently deduced a proof with results from Ramanujan’s notebooks, and utilized it to produce a ninth order iteration to 1/π. Identities (1.4) and (1.5) involve eta quotients of levels 4 and 9 respectively. Other identities of levels 6, 8 and 12 were also studied by Fine [11, pp. 78-91]. We list below two of Fine’s identities: q q

d log (η8 [−4, 2, −2, 4](τ )) = η8 [−4, 6, 6, −4](τ ), dq

(1.6)

d log (η12 [−4, 4, 4, −4, −4, 4](τ )) = η12 [−4, 10, −4, −4, 10, −4](τ ). dq

(1.7)

An interesting question arises naturally. Does there exist infinitely many identities of these type? In this paper, we use the theory of modular forms to attempt to find a complete list of eta quotients whose logarithmic derivatives are constant multiples of eta quotients. In our search, we found a total of 203 distinct identities from level 4 to 36. (We shall explain in the next section what we mean by distinct.) For each of the levels our list is complete. No other distinct identities were found for other levels N < 100. We conjecture that our list is complete. Table 1 shows the number of distinct identities for each level.

Table 1 Number of distinct identities for each level. Level

4

6

8

9

12

16

18

20

24

36

No. of Identities

3

10

4

1

100

4

12

12

32

25

We shall explain in the next section our approach in finding the identities and how these identities are classified. In Section 3, we use elementary arguments to determine the sum of orders of an eta quotient. This will help us determine all eta quotients in M2k (Γ0 (N )). Section 4 is devoted to three algorithms based on the results of the two previous sections. The table listing all the 203 distinct identities for N ≤ 36 is given in Section 5. In the final section, we discuss applications of these identities and other related work.

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

4

2. Preliminaries We first explain what we mean by distinct identities in our tabulations in Table 1. Suppose f (τ ) is an eta quotient that is mapped to a constant multiple of another eta quotient under the differential operator 1 d d 2πi dτ = q dq . Then it is clear that any constant multiple or any integral power of f (τ ) will also be mapped to some constant multiple of another eta quotient under the differential operator. For this reason, we prefer the logarithmic derivative since df (τ )

q

d j log(f (τ )j ) = · dτ . dq 2πi f (τ )

Thus our identities with the logarithmic differential operator are unique up to constant multiples. For brevity, we shall henceforth omit the phrase “a constant multiple of” when referring to eta quotients. Secondly, any eta quotient of level N is automatically an eta quotient of level mN for any positive integer m. Hence we only consider identities which are primitive in the following sense. If f (τ ) and g(τ ) are eta quotients, and α a constant such that q

d log(f (τ )) = αg(τ ), dq

then except for constant multiples, there does not exist any eta quotients F (τ ), G(τ ) and integer m > 1 such that q

d log(F (τ )) = G(τ ), dq

f (τ ) = F (mτ ) and g(τ ) = G(mτ ).

There is a third sense in which identities can be equivalent. As discussed in Fine [11, p. 86], by applying Atkin-Lehner involutions, one may obtain (2.2) below from (2.1). d log (η6 [3, −3, −9, 9](τ )) = η6 [3, 3, −1, −1](τ ), dq d q log (η6 [−3, 3, 1, −1](τ )) = 3η6 [−1, −1, 3, 3](τ ). dq

q

(2.1) (2.2)

We have chosen to keep these cases as distinct in Table 2 to assist readers who may be searching for specific identities in the list. Identities (2.1) and (2.2) are labelled respectively as f6,4a and f6,4b in Table 2, as these form the fourth set of identities from level 6. The suffix means that these are equivalent up to some involution. Jacobi’s identity (1.4) is labelled f4,2 and the absence of suffixes means that it is invariant under Atkin-Lehner involutions. Finally, in addition to Atkin-Lehner involutions, it is also possible to obtain a new identity through the transformation q → −q because of the elementary property ∞

(1 − (−q)n ) =

n=1

∞

(1 − q 2n )3 . (1 − q n )(1 − q 4n ) n=1

(2.3)

Not only is every eta-quotient mapped to another eta-quotient, the same holds for Eisenstein series. ∞ ∞ ∞ ∞





n(−q)n nq n 6nq 2n 4nq 4n = − + − . n n 2n 1 − (−q) 1−q 1−q 1 − q 4n n=1 n=1 n=1 n=1

For example, applying the transformation q → −q to Jacobi’s identity (1.4) or f4,2 yields

(2.4)

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

q

5

d log (η4 [8, −24, 16](τ )) = η4 [8, −4, 0](τ ), dq

(2.5)

which is labelled as f4,1a in Table 2. We again chose to retain both cases and record these additional relations in Table 3. We use the notation − f4,1a ∼ f4,2 − to indicate the relation between Jacobi’s identity (1.4) and identity (2.5). Clearly, f4,2 ∼ f4,1a also holds but only one out of the two possible relations is recorded in Table 3. All together, 200 out of the 203 distinct identities were paired up to give 100 relations. The remaining three identities were invariant under the transformation. For example, we have

 η4 [−2, 3, −1](τ ) q→−q = η4 [2, −3, 1](τ ). − − − We recorded the above relation as f4,1b ∼ f4,1b and the other two are f12,18b ∼ f12,18b and f12,18d ∼ f12,18d . Suppose now that f (τ ) is an eta quotient of level N and weight k. We have

f (τ ) =



η rδ (δτ ) where

0<δ|N



rδ = 2k.

0<δ|N

Applying logarithmic differentiation, q

d log f (τ ) = − rδ δE2 (δτ ) dq 0<δ|N

= −2kE2 (τ ) +



rδ (E2 (τ ) − δE2 (δτ ))

0<δ|N

= −2kE2 (τ ) +



rδ Lδ (τ ),

(2.6)

1<δ|N

where we used the notation Lδ (τ ) = E2 (τ ) − δE2 (δτ ),

(2.7)

which is in M2 (Γ0 (N )) whenever δ is a positive divisor of N [21, Theorem 5.8]. We now prove an important lemma which will help us classify the eta quotients whose logarithmic derivatives are eta quotients. Lemma 2.1. Let f (τ ) =



η rδ (δτ ) be an eta quotient of level N and weight k. Then

1 d 2πi dτ f (τ )

0<δ|N

quotient only if k = 0. Furthermore, if k = 0, then

1 d 2πi dτ f (τ )

is an eta quotient if and only if

is an eta



rδ Lδ (τ )

1<δ|N

is an eta quotient. Proof. Let g(τ ) Then f (τ ) a2 M2 = c2

f (τ ) be as given. Assume that g(τ ) =

1 d 2πi dτ f (τ )

is an eta quotient of level  N2 and weight k2 . a b is an eta quotient of level lcm(N, N2 ). It is well known that for all M = ∈ Γ0 (N ), c d    b2 a3 b3 ∈ Γ0 (N2 ) and M3 = ∈ Γ0 (lcm(N, N2 )) we have d2 c3 d3

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

6

f (M τ ) = χM,f (cτ + d)k f (τ ), g(M2 τ ) = χM2 ,g (c2 τ + d2 )k2 g(τ ), g(M3 τ ) g(τ ) = χM3 ,g/f (c3 τ + d3 )k2 −k , f (M3 τ ) f (τ )

(2.8)

where χM,f , χM2 ,g and χM3 ,g/f are 24th roots of unity. By (2.6) we have

g(τ ) = −2kE2 (τ ) + rδ Lδ (τ ). f (τ )

(2.9)

1<δ|N

Then since M3 ∈ Γ0 (lcm(N, N2 )), Γ0 (N ) and Γ0 (N2 ) we have

g(M3 τ ) = −2kE2 (M3 τ ) + rδ Lδ (M3 τ ) f (M3 τ ) 1<δ|N



= −2k (c3 τ + d3 )2 E2 (τ ) − ⎛ = (c3 τ + d3 )2 ⎝−2kE2 (τ ) +

6ic3 (c3 τ + d3 ) π

 + (c3 τ + d3 )2

1<δ|N

= (c3 τ + d3 )2

rδ Lδ (τ )

1<δ|N

⎞ rδ Lδ (τ )⎠ + 2k



6ic3 (c3 τ + d3 ) π

6ic3 (c3 τ + d3 ) g(τ ) + 2k . f (τ ) π

This contradicts with (2.8), unless k = 0. 2 We combine (2.6) with Lemma 2.1 to summarize the above discussion as the following classification theorem. Theorem 2.1. Let f (τ ) =



η rδ (δτ ) be an eta quotient of level N and weight k. Then

0<δ|N

eta quotient if and only if k = 0 and



1 d 2πi dτ f (τ )

is an

rδ Lδ (τ ) is an eta quotient.

1<δ|N

Thus the problem of determining eta quotients whose derivatives are also eta quotients is reduced to that of finding eta quotients in M2 (Γ0 (N )) which are also in the space spanned by Lδ (τ ). We shall denote this space as

EN

⎧ ⎨

 ⎫  ⎬  = uδ Lδ (τ ) uδ ∈ Q . ⎩ ⎭  1<δ|N

(2.10)

Corollary 2.1. Let f (τ ) be an eta quotient in EN , i.e. f (τ ) =

0<δ|N

where uδ ∈ Q. Denote u1 = −

1<δ|N

η sδ (δτ ) =



uδ Lδ (τ )

1<δ|N

uδ and let c ∈ Z be such that cuδ ∈ Z for all δ | N . Then we have

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

7

1 d cuδ η (δτ ) = c η sδ +cuδ (δτ ). 2πi dτ 0<δ|N

0<δ|N

3. Eta quotients in M2k (Γ0 (N )) In this section we use elementary methods to deduce a formula for the sum of the order of vanishing of eta quotients at cusps of Γ0 (N ). If the eta quotient is holomorphic, the formula will give an upper bound for vanishing at each cusp. This upper bound will be used in the next section where we describe an algorithm to determine all the eta quotients in M2k (Γ0 (N )). This section is a refined version of Section 4.2 of Aygin’s unpublished PhD thesis [3]. We shall use φ(n) to denote the Euler totient function and all sums in this section are taken over positive divisors of N . We begin with several lemmas. Lemma 3.1. Let N ∈ Z+ and p be a prime such that p  N . If λ |

N p,

then for all c |

N p,

we have

N )) · gcd(cp, pλ) φ(gcd(cp, cp φ(gcd(c, Nc )) · gcd(c, λ) = N N gcd(c, c ) · lcm(c, λ) gcd(cp, cp ) · lcm(cp, pλ)

(3.1)

and N φ(gcd(cp, cp )) · gcd(cp, λ)

gcd(cp, Proof. Since c |

N p

N cp )

=

· lcm(cp, λ)

φ(gcd(c, Nc )) · gcd(c, pλ) . gcd(c, Nc ) · lcm(c, pλ)

(3.2)

N and p  N , we have gcd(c, p) = gcd(p, cp ) = 1. We can then conclude



N gcd cp, cp





N = gcd c, cp





N · gcd p, cp



    N N = gcd c, = gcd c, . cp c

Then (3.1) follows from gcd(cp, pλ) = p · gcd(c, λ) and lcm(cp, pλ) = p · lcm(c, λ). Equation (3.2) can be proved similarly by utilizing gcd(c, pλ) = gcd(c, λ) and lcm(c, pλ) = p · lcm(c, λ) whenever gcd(c, p) = 1. 2 Lemma 3.2. Let N ∈ Z+ and p be a prime such that ps  N for some s > 1. If λ | have

N p,

then for all c |

N ps ,

we

φ(gcd(c, Nc )) · gcd(c, λ) φ(gcd(c, Nc )) · gcd(c, pλ) = gcd(c, Nc ) · lcm(c, λ) gcd(c, Nc ) · lcm(c, pλ) +

N )) · gcd(cp, pλ) φ(gcd(cp, cp

(3.3)

N gcd(cp, cp ) · lcm(cp, pλ)

and N s φ(gcd(cps , cp s )) · gcd(cp , pλ) N s gcd(cps , cp s ) · lcm(cp , pλ)

=

s−1 , λ) φ(gcd(cps−1 , cpN s−1 )) · gcd(cp s−1 , λ) gcd(cps−1 , cpN s−1 ) · lcm(cp

+

N s φ(gcd(cps , cp s )) · gcd(cp , λ) N s gcd(cps , cp s ) · lcm(cp , λ)

.

(3.4)

Furthermore, for 1 ≤ i < s − 1, we have N i φ(gcd(cpi , cp i )) · gcd(cp , λ) N i gcd(cpi , cp i ) · lcm(cp , λ)

=

φ(gcd(cpi+1 , cpNi+1 )) · gcd(cpi+1 , pλ) gcd(cpi+1 , cpNi+1 ) · lcm(cpi+1 , pλ)

.

(3.5)

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

8

Proof. Since c |

N ps

N with s > 1 and ps  N , we have gcd(c, p) = 1 and gcd(p, cp ) = p. It then follows that

          N N N N N = gcd c, · gcd p, = p · gcd c, = p · gcd c, . gcd cp, cp cp cp cp c

(3.6)

   Also since gcd p, gcd c, Nc = 1, we have          N N N = φ p · gcd c, = (p − 1) · φ gcd c, . φ gcd cp, cp c c

(3.7)

Hence (3.3) follows from (3.6) and (3.7). The proofs of the other two identities are similar and hinge on the fact that if 1 ≤ i ≤ s, we have   i  p gcd(c, Nc ) i N gcd cp , i = cp ps−i gcd(c, N ) c

if i < 2s ;

(3.8)

if i ≥ 2s .

Furthermore, for all t ≥ 1, φ(pt ) p−1 φ(pt+1 ) = = . t+1 p p pt Combining the above observation with (3.8) proves (3.5). When i = s − 1,  gcd cps−1 ,

N cps−1





N = p · gcd c, c





N = p · gcd cp , s cp s

 .

Since the largest power of p that divides λ is at most s − 1, we have gcd(cps , λ) =

1 gcd(cps , pλ), p

lcm(cps , λ) = lcm(cps , pλ). The right side of (3.4) is then s−1 φ(gcd(cps−1 , cpN , λ) s−1 )) · gcd(cp s−1 , λ) gcd(cps−1 , cpN s−1 ) · lcm(cp

=

φ(p · gcd(c, Nc )) · p · gcd(c, Nc ) ·

1 p

1 p

gcd(cps , pλ)

lcm(cps , pλ)

+

+

N s φ(gcd(cps , cp s )) · gcd(cp , λ) N s gcd(cps , cp s ) · lcm(cp , λ)

φ(gcd(c, Nc )) ·

1 p

gcd(cps , pλ)

gcd(c, Nc ) · lcm(cps , pλ)

=

(p − 1)φ(gcd(c, Nc )) · gcd(cps , pλ) φ(gcd(c, Nc )) · gcd(cps , pλ) + p · gcd(c, Nc ) · lcm(cps , pλ) p · gcd(c, Nc ) · lcm(cps , pλ)

=

φ(gcd(c, Nc )) · gcd(cps , pλ) gcd(c, Nc ) · lcm(cps , pλ)

=

N s φ(gcd(cps , cp s )) · gcd(cp , pλ) N s gcd(cps , cp s ) · lcm(cp , pλ)

.

From Lemmas 3.1 and 3.2, we deduce the following.

2

(3.9)

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

Lemma 3.3. Let N ∈ Z+ and p be a prime divisor of N . If λ |

φ(gcd(c, N )) · gcd(c, λ) c|N

gcd(c,

c N c )

=

· lcm(c, λ)

N p,

9

we have

φ(gcd(c, N )) · gcd(c, pλ) c

c|N

gcd(c, Nc ) · lcm(c, pλ)

.

(3.10)

Lemma 3.4. Let N ∈ Z+ . If δ | N , we have

φ(gcd(c, N/c)) c|N

gcd(c, N/c) · c

φ(gcd(c, N/c)) · gcd(c, δ)

=

gcd(c, N/c) · lcm(c, δ)

c|N

.

(3.11)

Proof. Let δ = pi11 · · · pirr be a divisor of N . Then taking λ = δ/pj1 in Lemma 3.3 for j from 1 to i1 , we have

φ(gcd(c, N/c)) · gcd(c, δ) c|N

gcd(c, N/c) · lcm(c, δ)

=

φ(gcd(c, N/c)) · gcd(c, δ/p1 ) gcd(c, N/c) · lcm(c, δ/p1 )

c|N

=

φ(gcd(c, N/c)) · gcd(c, δ/p2 ) 1

gcd(c, N/c) · lcm(c, δ/p21 )

c|N

.. . =

φ(gcd(c, N/c)) · gcd(c, δ/pi1 ) 1

gcd(c, N/c) · lcm(c, δ/pi11 )

c|N

.

Inductively, we have

φ(gcd(c, N/c)) · gcd(c, δ) c|N

gcd(c, N/c) · lcm(c, δ)

=

φ(gcd(c, N/c)) · gcd(c, δ/pi1 ) 1

c|N

=

φ(gcd(c, N/c)) · gcd(c, δ/(pi1 pi2 )) 1

c|N

=

gcd(c, N/c) · lcm(c, δ/pi11 )

.. .

φ(gcd(c, N/c)) · gcd(c, δ/δ) c|N

2

gcd(c, N/c) · lcm(c, δ/(pi11 pi22 ))

gcd(c, N/c) · lcm(c, δ/δ)

.

2

We are now ready to prove an important theorem which allows us to determine all eta quotients in M2k (Γ0 (N )). Theorem 3.1. Let N, k ∈ Z+ . Suppose f (τ ) =



η(δτ )rδ is an eta quotient of level N and weight 2k. Let

0<δ|N

vr (f (τ )) be the order of vanishing of f (τ ) at the cusp r ∈ R(Γ0 (N )). Then we have

vr (f (τ )) =

r∈R(Γ0 (N ))

4kN φ(gcd(c, N/c)) . 24 gcd(c2 , N )

(3.12)

c|N

Furthermore, if f (τ ) ∈ M2k (Γ0 (N )), we have 0 ≤ vr (f (τ )) ≤

4kN φ(gcd(c, N/c)) . 24 gcd(c2 , N ) c|N

(3.13)

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

10

Proof. Note that, given r =

a c

with gcd(a, c) = 1,

vr (f (z)) =

gcd(δ, c)2 · rδ N , 24 gcd(c2 , N ) δ δ|N

depends only on the denominator of r. Using Lemma 3.3 we obtain

vr (f (z)) =

r∈R(Γ0 (N ))

r∈R(Γ0 (N ))

=

N 24

1≤δ|N

=

gcd(δ, c)2 · rδ N 24 gcd(c2 , N ) δ δ|N

r∈R(Γ0 (N ))

gcd(δ, c)2 · rδ gcd(c2 , N ) · δ

N φ(gcd(c, N/c)) · gcd(δ, c)2 rδ 24 gcd(c2 , N ) · δ δ|N

c|N

N φ(gcd(c, N/c)) rδ = 24 gcd(c2 , N ) δ|N

=

c|N

N φ(gcd(c, N/c))

rδ 24 gcd(c2 , N ) c|N

=

δ|N

4kN φ(gcd(c, N/c)) . 24 gcd(c2 , N ) c|N

When f (τ ) ∈ M2k (Γ0 (N )), then we have vr (f (z)) ≥ 0 for all r ∈ R(Γ(N )). Thus the inequality (3.13) follows. 2 4. Algorithms In this section we present three algorithms. Theorem 3.1 provides an upper bound for the orders of vanishing of eta quotients at all the cusps. The first algorithm uses this upper bound to determine all eta quotients in M2k (Γ0 (N )). We define the sum of the t-th powers of the divisors function, σt (n) =



dt .

0
Algorithm 4.1. This algorithm determines all eta quotients in M2k (Γ0 (N )). Input: N, k ∈ Z+ . (1) Compute: L(N ) =

4kN φ(gcd(c, N/c)) . 24 gcd(c2 , N ) c|N

(2) Set up the following system of σ0 (N ) linear equations for r = ac , c | N . vr =

gcd(δ, c)2 · rδ N . 24 gcd(c2 , N ) δ δ|N

(4.1)

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

11

(3) For all vr ∈ Z and 0 ≤ vr ≤ L(N ) such that

vr = L(N ),

r∈R(N )

 solve the above system to obtain rδ . If rδ ∈ Z for all δ | N , δ/p∈Z+ νp (δ)rδ is even for all prime p | N ,  and rδ = 4k, then save [r1 , . . . , rδ , . . . , rN ] in an array A. Here νp (δ) denotes the power of p that divides δ. Output: {ηN [r1 , . . . , rδ , . . . , rN ](z) : [r1 , . . . , rδ , . . . , rN ] ∈ A}. We can modify Algorithm 4.1 to find all eta quotients in M2k (Γ0 (N ), χ) where χ is a quadratic character. To do this one needs to let vr take half integer values, and put appropriate parity conditions for  δ/p∈Z+ νp (δ)rδ . For our purposes in this paper, we only require the case when k = 1. The second algorithm determines which of the eta quotients in M2 (Γ0 (N )) are in the space EN . We do this by attempting to express the first m Fourier coefficients of each eta quotient found by Algorithm 4.1 in terms of the Fourier coefficients of the basis elements Lδ (τ ) of EN . Here m is the Sturm bound [22]. Algorithm 4.2. This algorithm determines the eta quotients in EN . Input: N ∈ Z+ and array A from Algorithm 4.1 with inputs N and k = 1. (1) Put the divisors of N in an array Divs. (2) Set up an m × σ0 (N ) matrix M, whose (i + 1, j)th entry is ⎧ ⎪ ⎨ Divs[j]−1 , if i = 0, 24  M[i, j] = ⎪ (i) − Divs[j]σ σ ⎩ 1 1

i Divs[j]

 , if 1 ≤ i < m.

 a[j] (3) For η (Divs[j] · τ ) be an eta quotient with Fourier expansion f (τ ) =

each a in A, let f (τ ) = bn q n . Put b = [b0 , b1 , . . . , bm−1 ] and solve n≥0

M · x = b. If a solution x exists, save [a, x] in an array S. Output: Eta quotients in EN in terms of Lδ (τ ) where δ | N :

σ0 (N )

ηN S[i][1](τ ) =

S[i][2][l − 1]LDivs[l] (τ ).

l=1

Finally, the third algorithm implements the computation in Corollary 2.1 to determine the antiderivative of each eta quotient in EN . Algorithm 4.3. This algorithm determines the antiderivatives of the eta quotients in EN . Input: N ∈ Z+ and array S from Algorithm 4.2. (1) If all entries of S[i][2] ∈ Z then put C = 1, otherwise put C as the lowest common multiple of denominators of entries of S[i][2].

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

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(2) Define ⎡ T[i] = ⎣−C ·



σ0 (N )

⎤ S[i][2][l − 1], C · S[i][2][1], . . . , C · S[i][2][σ0 (N ) − 1]⎦ .

l=1

Output: Eta quotients whose logarithmic derivatives are eta quotients: 1 d log(ηN T[i](τ )) = CηN S[i][1](τ ). 2πi dτ 5. Results We used the algorithms given in Section 4 to find all eta quotients whose logarithmic derivatives are eta quotients in M2 (Γ0 (N )) for N ≤ 36. A total of 203 distinct eta quotients and their respective logarithmic derivatives are listed in Table 2. Note that in this list we chose c ∈ Z in Corollary 2.1 to be the smallest positive constant such that cuδ ∈ Z for all δ | N . We have suppressed the argument (τ ) in the eta quotients so as to improve readability. As mentioned in Section 2, we can apply transformation q → −q to all the 203 identities in Table 2. With the exception of three identities which are invariant, every other identity can be mapped to another (possibly of different level) in the list. Table 3 lists all the 103 relations. The three identities of level 4 are all equivalent up to Atkin-Lehner involution or the transformation q → −q. Each can be interpreted as series expansions of the product of four theta functions. Using Ramanujan’s notation for theta functions, ϕ(q) = 1 + 2

∞

qn

2

(5.1)

n=1

and ψ(q) =

∞

q n(n+1)/2 ,

(5.2)

n=0

identities f4,1a , f41,b and f4,2 correspond respectively to representations of ϕ(−q)4 , qψ(q 2 )4 and ϕ(q)4 . These representations are classical and known to Jacobi. The Eisenstein series representation for ϕ(q)4 was given in equation (1.3). The other two may be deduced from our identities. One may refer to [9, p. 196] for alternatives proofs of these representations. Remark: The anonymous referee pointed out that since 8

η4 [8, −24, 16](τ ) × (η4 [−2, 3, −1](τ )) = η4 [−8, 0, 8](τ ), we can recover another of Jacobi’s identities, namely ϕ(−q)4 + 16qψ(q 2 )4 = ϕ(q)4 . Identities equivalent to all ten entries of level 6 can be found in [9, pp. 375-378]. If we define za = η6 [6, −3, −2, 1](τ ), zb = η6 [−3, 6, 1, −2](τ ), zc = η6 [−2, 1, 6, −3](τ ) and

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

Table 2 Logarithmic derivatives of eta quotients which are also eta quotients. Identity

Weight 0 eta quotient

Logarithmic derivative

f4,1a f4,1b f4,2

η4 [8, −24, 16] η4 [−2, 3, −1] η4 [−8, 0, 8]

η4 [8, −4, 0] 2η4 [0, −4, 8] η4 [−8, 20, −8]

f6,1a f6,1b f6,1c f6,1d f6,2a f6,2b f6,3a f6,3b f6,4a f6,4b

η6 [12, −48, 36, 0] η6 [−12, 3, 0, 9] η6 [−12, 0, −4, 16] η6 [0, −3, 4, −1] η6 [−1, 5, −5, 1] η6 [−5, 1, −1, 5] η6 [4, −8, −4, 8] η6 [−2, 1, 2, −1] η6 [3, −3, −9, 9] η6 [−3, 3, 1, −1]

η6 [12, −6, −4, 2] 2η6 [−6, 12, 2, −4] 3η6 [−4, 2, 12, −6] 6η6 [2, −4, −6, 12] η6 [7, −5, −5, 7] η6 [−5, 7, 7, −5] η6 [4, −2, 4, −2] 2η6 [−2, 4, −2, 4] η6 [3, 3, −1, −1] 3η6 [−1, −1, 3, 3]

f8,1a f8,1b f8,2 f8,3

η8 [4, −10, 2, 4] η8 [−2, −1, 5, −2] η8 [−4, 2, −2, 4] η8 [−2, 7, −7, 2]

η8 [4, −6, 10, −4] 2η8 [−4, 10, −6, 4] η8 [−4, 6, 6, −4] 2η8 [4, −2, −2, 4]

f9,1

η9 [−3, 0, 3]

η9 [−3, 10, −3]

f12,1a f12,1b f12,1c f12,1d f12,2a f12,2b f12,2c f12,2d f12,3a f12,3b f12,3c f12,3d f12,4a f12,4b f12,4c f12,4d f12,5a f12,5b f12,5c f12,5d f12,6a f12,6b f12,6c f12,6d f12,7a f12,7b f12,7c f12,7d f12,8a f12,8b f12,8c f12,8d f12,9a f12,9b f12,9c f12,9d f12,10a f12,10b f12,11a f12,11b f12,11c f12,11d f12,12a f12,12b f12,12c f12,12d f12,13a f12,13b

η12 [10, −36, 18, 8, 0, 0] η12 [−18, 0, −10, 0, 36, −8] η12 [4, −18, 0, 5, 0, 9] η12 [0, 0, −4, −9, 18, −5] η12 [−2, 4, 6, 0, −16, 8] η12 [6, −16, −2, 8, 4, 0] η12 [−4, 8, 0, −3, −2, 1] η12 [0, −2, −4, 1, 8, −3] η12 [2, 0, −6, −8, 12, 0] η12 [6, −12, −2, 0, 0, 8] η12 [−4, 0, 0, 1, 6, −3] η12 [0, −6, 4, 3, 0, −1] η12 [6, −12, −18, 24, 0, 0] η12 [−6, 0, 2, 0, −4, 8] η12 [−12, 6, 0, −3, 0, 9] η12 [0, 0, −4, 3, 2, −1] η12 [9, −30, 9, 12, 0, 0] η12 [−3, 0, −3, 0, 10, −4] η12 [12, −30, 0, 9, 0, 9] η12 [0, 0, −4, −3, 10, −3] η12 [−8, −2, −8, 0, 26, −8] η12 [8, −26, 8, 8, 2, 0] η12 [4, −13, 0, 4, 1, 4] η12 [0, −1, −4, −4, 13, −4] η12 [−4, 6, 12, 4, −30, 12] η12 [−2, −3, −6, 2, 15, −6] η12 [12, −30, −4, 12, 6, 4] η12 [−6, 15, −2, −6, −3, 2] η12 [−1, −3, −9, −5, 27, −9] η12 [−10, −6, −18, −2, 54, −18] η12 [18, −54, 10, 18, 6, 2] η12 [9, −27, 1, 9, 3, 5] η12 [−6, 6, 18, 18, −54, 18] η12 [−9, −3, −9, 3, 27, −9] η12 [−3, 9, −3, −3, −1, 1] η12 [6, −18, −2, 6, 2, 6] η12 [−12, −12, −36, −12, 108, −36] η12 [12, −36, 4, 12, 4, 4] η12 [−1, 6, −9, 4, 0, 0] η12 [−9, 0, −1, 0, 6, 4] η12 [−4, −6, 0, 1, 0, 9] η12 [0, 0, −4, 9, −6, 1] η12 [7, −21, 3, 8, 3, 0] η12 [8, −21, 0, 7, 3, 3] η12 [−3, −3, −7, 0, 21, −8] η12 [0, −3, −8, −3, 21, −7] η12 [−1, 4, −1, −4, 2, 0] η12 [1, −2, 1, 0, −4, 4]

η12 [10, −7, −6, 1, 9, −3] 3η12 [−6, 9, 10, −3, −7, 1] 4η12 [1, −7, −3, 10, 9, −6] 12η12 [−3, 9, 1, −6, −7, 10] η12 [−2, 7, 6, −3, −5, 1] η12 [6, −5, −2, 1, 7, −3] 4η12 [1, −5, −3, 6, 7, −2] 4η12 [−3, 7, 1, −2, −5, 6] η12 [2, 5, 2, −3, −3, 1] 3η12 [2, −3, 2, 1, 5, −3] 4η12 [−3, 5, 1, 2, −3, 2] 12η12 [1, −3, −3, 2, 5, 2] η12 [6, 3, −2, −3, −1, 1] 3η12 [−2, −1, 6, 1, 3, −3] 4η12 [−3, 3, 1, 6, −1, −2] 12η12 [1, −1, −3, −2, 3, 6] η12 [9, −6, −3, 3, 2, −1] 3η12 [−3, 2, 9, −1, −6, 3] 4η12 [3, −6, −1, 9, 2, −3] 12η12 [−1, 2, 3, −3, −6, 9] η12 [−8, 16, 8, −6, −8, 2] η12 [8, −8, −8, 2, 16, −6] 2η12 [2, −8, −6, 8, 16, −8] 2η12 [−6, 16, 2, −8, −8, 8] η12 [−4, 14, 4, −6, −6, 2] 2η12 [−6, 14, 2, −4, −6, 4] 3η12 [4, −6, −4, 2, 14, −6] 6η12 [2, −6, −6, 4, 14, −4] η12 [−9, 23, 3, −10, −9, 6] η12 [−10, 23, 6, −9, −9, 3] 3η12 [6, −9, −10, 3, 23, −9] 3η12 [3, −9, −9, 6, 23, −10] η12 [−6, 21, 2, −9, −7, 3] η12 [−9, 21, 3, −6, −7, 2] 3η12 [3, −7, −9, 2, 21, −6] 3η12 [2, −7, −6, 3, 21, −9] η12 [−12, 30, 4, −12, −10, 4] 3η12 [4, −10, −12, 4, 30, −12] η12 [9, −4, −3, −1, 0, 3] 3η12 [−3, 0, 9, 3, −4, −1] 4η12 [−1, −4, 3, 9, 0, −3] 12η12 [3, 0, −1, −3, −4, 9] η12 [7, −7, −5, 4, 9, −4] 2η12 [4, −7, −4, 7, 9, −5] 3η12 [−5, 9, 7, −4, −7, 4] 6η12 [−4, 9, 4, −5, −7, 7] η12 [5, −2, 1, −1, −2, 3] η12 [1, −2, 5, 3, −2, −1]

13

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14

Table 2 (Continued) Identity

Weight 0 eta quotient

Logarithmic derivative

f12,13c f12,13d f12,14a f12,14b f12,14c f12,14d f12,15a f12,15b f12,15c f12,15d f12,16a f12,16b f12,16c f12,16d f12,17a f12,17b f12,18a f12,18b f12,18c f12,18d f12,19a f12,19b f12,20a f12,20b f12,21a f12,21b f12,21c f12,21d f12,22a f12,22b f12,22c f12,22d f12,23a f12,23b f12,24a f12,24b f12,25a f12,25b f12,25c f12,25d f12,26a f12,26b f12,26c f12,26d f12,27 f12,28a f12,28b f12,28c f12,28d f12,29a f12,29b f12,30

η12 [0, −2, 4, 1, −4, 1] η12 [−4, 4, 0, −1, 2, −1] η12 [5, −12, −3, 4, 6, 0] η12 [−3, 6, 5, 0, −12, 4] η12 [−4, 12, 0, −5, −6, 3] η12 [0, −6, −4, 3, 12, −5] η12 [−7, 0, −3, 1, 12, −3] η12 [−1, 0, 3, 7, −12, 3] η12 [3, −12, −1, 3, 0, 7] η12 [−3, 12, −7, −3, 0, 1] η12 [−1, 3, 3, −8, 3, 0] η12 [−8, 3, 0, −1, 3, 3] η12 [−3, −3, 1, 0, −3, 8] η12 [0, −3, 8, −3, −3, 1] η12 [−1, −2, −5, −1, 14, −5] η12 [5, −14, 1, 5, 2, 1] η12 [−1, 3, 3, −2, −9, 6] η12 [−2, 3, 6, −1, −9, 3] η12 [−3, 9, 1, −6, −3, 2] η12 [−6, 9, 2, −3, −3, 1] η12 [−3, 6, 9, −3, −18, 9] η12 [−3, 6, 1, −3, −2, 1] η12 [−4, −1, 0, 0, 1, 4] η12 [0, −1, 4, −4, 1, 0] η12 [4, −9, 0, 2, −3, 6] η12 [−2, 9, −6, −4, 3, 0] η12 [−6, 3, −2, 0, 9, −4] η12 [0, −3, 4, 6, −9, 2] η12 [4, −6, −12, 8, 6, 0] η12 [−4, 3, 0, −2, −3, 6] η12 [−12, 6, 4, 0, −6, 8] η12 [0, −3, −4, 6, 3, −2] η12 [12, −33, 0, 12, 9, 0] η12 [0, −3, −4, 0, 11, −4] η12 [1, 0, −3, −1, 0, 3] η12 [−3, 0, 1, 3, 0, −1] η12 [−1, 1, −1, −1, −1, 3] η12 [2, −2, −6, 2, 2, 2] η12 [−1, −1, −1, 3, 1, −1] η12 [−6, 2, 2, 2, −2, 2] η12 [−2, 6, 6, −10, −6, 6] η12 [−5, 3, 3, −1, −3, 3] η12 [−3, 3, 5, −3, −3, 1] η12 [−6, 6, 2, −6, −6, 10] η12 [−4, 4, 4, −4, −4, 4] η12 [3, −7, −1, 2, 1, 2] η12 [−1, 1, 3, 2, −7, 2] η12 [−2, −1, −2, 1, 7, −3] η12 [−2, 7, −2, −3, −1, 1] η12 [−3, 2, 1, −1, −2, 3] η12 [−1, 2, 3, −3, −2, 1] η12 [−2, 5, 2, −2, −5, 2]

4η12 [3, −2, −1, 1, −2, 5] 4η12 [−1, −2, 3, 5, −2, 1] η12 [5, −4, 1, 3, 0, −1] 3η12 [1, 0, 5, −1, −4, 3] 4η12 [3, −4, −1, 5, 0, 1] 12η12 [−1, 0, 3, 1, −4, 5] η12 [−7, 14, 5, −3, −6, 1] η12 [−3, 14, 1, −7, −6, 5] 3η12 [1, −6, −3, 5, 14, −7] 3η12 [5, −6, −7, 1, 14, −3] η12 [3, 5, −1, −4, −3, 4] 2η12 [−4, 5, 4, 3, −3, −1] 3η12 [−1, −3, 3, 4, 5, −4] 6η12 [4, −3, −4, −1, 5, 3] η12 [−7, 16, 5, −7, −8, 5] η12 [5, −8, −7, 5, 16, −7] η12 [−1, 5, 3, 0, −3, 0] 2η12 [0, 5, 0, −1, −3, 3] 3η12 [3, −3, −1, 0, 5, 0] 6η12 [0, −3, 0, 3, 5, −1] η12 [−3, 12, 1, −3, −4, 1] 3η12 [1, −4, −3, 1, 12, −3] 2η12 [−2, −2, 6, 6, −2, −2] 2η12 [6, −2, −2, −2, −2, 6] 2η12 [2, −4, 2, 6, 0, −2] 2η12 [6, −4, −2, 2, 0, 2] 6η12 [−2, 0, 6, 2, −4, 2] 6η12 [2, 0, 2, −2, −4, 6] η12 [4, 2, −4, −2, 6, −2] 2η12 [−2, 2, −2, 4, 6, −4] 3η12 [−4, 6, 4, −2, 2, −2] 6η12 [−2, 6, −2, −4, 2, 4] 2η12 [6, −6, −2, 6, 2, −2] 6η12 [−2, 2, 6, −2, −6, 6] η12 [1, 2, −3, 1, 6, −3] 3η12 [−3, 6, 1, −3, 2, 1] η12 [−1, 1, −5, 2, 13, −6] η12 [2, 1, −6, −1, 13, −5] η12 [−5, 13, −1, −6, 1, 2] η12 [−6, 13, 2, −5, 1, −1] η12 [−2, 11, −2, −5, 3, −1] η12 [−5, 11, −1, −2, 3, −2] 3η12 [−1, 3, −5, −2, 11, −2] 3η12 [−2, 3, −2, −1, 11, −5] η12 [−4, 10, −4, −4, 10, −4] η12 [3, −5, −1, 4, 7, −4] η12 [−1, 7, 3, −4, −5, 4] 2η12 [−4, 7, 4, −1, −5, 3] 2η12 [4, −5, −4, 3, 7, −1] η12 [−3, 4, 1, 1, 4, −3] η12 [1, 4, −3, −3, 4, 1] 2η12 [2, −2, 2, 2, −2, 2]

f16,1a f16,1b f16,2 f16,3

η16 [2, −5, 2, −1, 2] η16 [−2, 1, −2, 5, −2] η16 [−2, 1, 0, −1, 2] η16 [−2, 5, 0, −5, 2]

η16 [2, −5, 8, 1, −2] 2η16 [−2, 1, 8, −5, 2] η16 [−2, 1, 6, 1, −2] 2η16 [2, −5, 10, −5, 2]

f18,1a f18,1b f18,1c f18,1d f18,2a f18,2b f18,2c f18,2d f18,3a f18,3b f18,4a f18,4b

η18 [3, −6, −2, 2, 3, 0] η18 [−6, 3, 2, −2, 0, 3] η18 [−3, 0, 2, −2, −3, 6] η18 [0, −3, −2, 2, 6, −3] η18 [−1, 2, 2, −6, 3, 0] η18 [2, −1, −6, 2, 0, 3] η18 [−3, 0, −2, 6, 1, −2] η18 [0, −3, 6, −2, −2, 1] η18 [−2, 1, 1, −1, −1, 2] η18 [−1, 2, 1, −1, −2, 1] η18 [1, −2, 0, 0, −1, 2] η18 [−2, 1, 0, 0, 2, −1]

η18 [3, −3, 2, 4, −1, −1] 2η18 [−3, 3, 4, 2, −1, −1] 3η18 [−1, −1, 2, 4, 3, −3] 6η18 [−1, −1, 4, 2, −3, 3] η18 [1, 1, 6, −4, −3, 3] 2η18 [1, 1, −4, 6, 3, −3] 3η18 [−3, 3, 6, −4, 1, 1] 6η18 [3, −3, −4, 6, 1, 1] η18 [−2, 1, 3, 3, 1, −2] η18 [1, −2, 3, 3, −2, 1] η18 [1, −2, 2, 4, 1, −2] 2η18 [−2, 1, 4, 2, −2, 1]

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

Table 2 (Continued) Identity

Weight 0 eta quotient

Logarithmic derivative

f20,1a f20,1b f20,1c f20,1d f20,2a f20,2b f20,2c f20,2d f20,3a f20,3b f20,3c f20,3d

η20 [−7, 1, 1, −5, 15, −5] η20 [−1, −1, 7, 5, −15, 5] η20 [5, −15, 5, −1, −1, 7] η20 [−5, 15, −5, −7, 1, 1] η20 [−1, 4, −8, 5, 0, 0] η20 [−8, 4, −1, 0, 0, 5] η20 [−5, 0, 0, 1, −4, 8] η20 [0, 0, −5, 8, −4, 1] η20 [7, −20, 8, 5, 0, 0] η20 [8, −20, 7, 0, 0, 5] η20 [−5, 0, 0, −7, 20, −8] η20 [0, 0, −5, −8, 20, −7]

η20 [−7, 16, −5, 3, −4, 1] η20 [−5, 16, −7, 1, −4, 3] 5η20 [1, −4, 3, −5, 16, −7] 5η20 [3, −4, 1, −7, 16, −5] η20 [5, 1, −2, −1, −1, 2] 4η20 [−2, 1, 5, 2, −1, −1] 5η20 [−1, −1, 2, 5, 1, −2] 20η20 [2, −1, −1, −2, 1, 5] η20 [7, −5, 2, −3, 5, −2] 4η20 [2, −5, 7, −2, 5, −3] 5η20 [−3, 5, −2, 7, −5, 2] 20η20 [−2, 5, −3, 2, −5, 7]

f24,1a f24,1b f24,1c f24,1d f24,2a f24,2b f24,2c f24,2d f24,3a f24,3b f24,3c f24,3d f24,4a f24,4b f24,4c f24,4d f24,5a f24,5b f24,5c f24,5d f24,6a f24,6b f24,6c f24,6d f24,7a f24,7b f24,7c f24,7d f24,8a f24,8b f24,8c f24,8d

η24 [−1, 5, 3, −11, −3, 4, 3, 0] η24 [4, −11, 0, 5, 3, −1, −3, 3] η24 [−3, 3, 1, −3, −5, 0, 11, −4] η24 [0, −3, −4, 3, 11, −3, −5, 1] η24 [−2, 5, 6, −4, −15, 4, 6, 0] η24 [6, −15, −2, 6, 5, 0, −4, 4] η24 [−4, 4, 0, −5, −6, 2, 15, −6] η24 [0, −6, −4, 15, 4, −6, −5, 2] η24 [1, 2, −3, −10, 6, 4, 0, 0] η24 [−3, 6, 1, 0, 2, 0, −10, 4] η24 [−4, 10, 0, −2, 0, −1, −6, 3] η24 [0, 0, −4, −6, 10, 3, −2, −1] η24 [2, −1, −6, −2, 3, 4, 0, 0] η24 [−6, 3, 2, 0, −1, 0, −2, 4] η24 [−4, 2, 0, 1, 0, −2, −3, 6] η24 [0, 0, −4, −3, 2, 6, 1, −2] η24 [−1, 2, 3, −2, −6, 4, 0, 0] η24 [3, −6, −1, 0, 2, 0, −2, 4] η24 [−4, 2, 0, −2, 0, 1, 6, −3] η24 [0, 0, −4, 6, 2, −3, −2, 1] η24 [−1, 1, 3, 1, −3, −4, 3, 0] η24 [−4, 1, 0, 1, 3, −1, −3, 3] η24 [−3, 3, 1, −3, −1, 0, −1, 4] η24 [0, −3, 4, 3, −1, −3, −1, 1] η24 [−2, 5, 6, −10, −3, 4, 0, 0] η24 [4, −10, 0, 5, 0, −2, −3, 6] η24 [−6, 3, 2, 0, −5, 0, 10, −4] η24 [0, 0, −4, 3, 10, −6, −5, 2] η24 [−2, 1, 6, 8, −15, −4, 6, 0] η24 [−4, 8, 0, 1, 6, −2, −15, 6] η24 [−6, 15, 2, −6, −1, 0, −8, 4] η24 [0, −6, 4, 15, −8, −6, −1, 2]

η24 [−1, 9, −1, −6, −1, 2, 4, −2] 2η24 [2, −6, −2, 9, 4, −1, −1, −1] 3η24 [−1, −1, −1, 4, 9, −2, −6, 2] 6η24 [−2, 4, 2, −1, −6, −1, 9, −1] η24 [−2, 9, 2, −2, −5, −1, 4, −1] 3η24 [2, −5, −2, 4, 9, −1, −2, −1] 4η24 [−1, −2, −1, 9, 4, −2, −5, 2] 12η24 [−1, 4, −1, −5, −2, 2, 9, −2] η24 [1, 6, 1, −5, −4, 2, 5, −2] 3η24 [1, −4, 1, 5, 6, −2, −5, 2] 4η24 [2, −5, −2, 6, 5, 1, −4, 1] 12η24 [−2, 5, 2, −4, −5, 1, 6, 1] η24 [2, 3, −2, 0, 1, −1, 2, −1] 3η24 [−2, 1, 2, 2, 3, −1, 0, −1] 4η24 [−1, 0, −1, 3, 2, 2, 1, −2] 12η24 [−1, 2, −1, 1, 0, −2, 3, 2] η24 [1, 4, 1, 1, −2, −2, −1, 2] 3η24 [1, −2, 1, −1, 4, 2, 1, −2] 4η24 [−2, 1, 2, 4, −1, 1, −2, 1] 12η24 [2, −1, −2, −2, 1, 1, 4, 1] η24 [−1, 7, −1, 0, 1, −2, −2, 2] 2η24 [−2, 0, 2, 7, −2, −1, 1, −1] 3η24 [−1, 1, −1, −2, 7, 2, 0, −2] 6η24 [2, −2, −2, 1, 0, −1, 7, −1] 2η24 [2, 4, −2, −3, 2, 1, −1, 1] 4η24 [1, −3, 1, 4, −1, 2, 2, −2] 6η24 [−2, 2, 2, −1, 4, 1, −3, 1] 12η24 [1, −1, 1, 2, −3, −2, 4, 2] 2η24 [−2, 10, 2, −5, −4, 1, 1, 1] 4η24 [1, −5, 1, 10, 1, −2, −4, 2] 6η24 [2, −4, −2, 1, 10, 1, −5, 1] 12η24 [1, 1, 1, −4, −5, 2, 10, −2]

f36,1 f36,2a f36,2b f36,3a f36,3b f36,4 f36,5 f36,6a f36,6b f36,7a f36,7b f36,8a f36,8b f36,9a f36,9b f36,10a f36,10b f36,11a f36,11b f36,12a f36,12b f36,12c f36,12d f36,13a f36,13b

η36 [−1, 1, 0, −1, 0, 1, 0, −1, 1] η36 [−3, 3, 2, −3, −4, −3, 2, 9, −3] η36 [3, −9, −2, 3, 4, 3, −2, −3, 3] η36 [−1, 1, 2, −1, 0, 3, 2, −9, 3] η36 [−3, 9, −2, −3, 0, 1, −2, −1, 1] η36 [−2, 5, 0, −2, 0, 2, 0, −5, 2] η36 [3, −9, 0, 3, 0, −3, 0, 9, −3] η36 [−3, 6, 10, 0, −24, −3, 8, 6, 0] η36 [0, −6, −8, 3, 24, 0, −10, −6, 3] η36 [2, −5, −1, 2, 2, 1, −1, −1, 1] η36 [−1, 1, 1, −1, −2, −2, 1, 5, −2] η36 [3, −3, −10, 3, 6, 3, −2, −3, 3] η36 [−3, 3, 2, −3, −6, −3, 10, 3, −3] η36 [−6, 3, 2, 0, 0, 0, −2, −3, 6] η36 [0, −3, 2, 6, 0, −6, −2, 3, 0] η36 [−3, 6, 2, 0, 0, −3, −8, 6, 0] η36 [0, −6, 8, 3, 0, 0, −2, −6, 3] η36 [−2, 5, 6, −2, −16, 0, 6, 3, 0] η36 [0, −3, −6, 0, 16, 2, −6, −5, 2] η36 [−6, 3, 4, 0, −6, −6, 2, 15, −6] η36 [0, −3, −2, 6, 6, 6, −4, −15, 6] η36 [−6, 15, 4, −6, −6, −6, 2, 3, 0] η36 [6, −15, −2, 6, 6, 0, −4, −3, 6] η36 [6, −15, −2, 6, 4, 0, −2, 3, 0] η36 [0, −3, 2, 0, −4, −6, 2, 15, −6]

η36 [−1, 1, −2, −1, 10, −1, −2, 1, −1] η36 [−3, 6, −2, −3, 10, 1, −2, −4, 1] 3η36 [1, −4, −2, 1, 10, −3, −2, 6, −3] η36 [−1, 4, −6, −1, 14, 3, −6, −6, 3] 3η36 [3, −6, −6, 3, 14, −1, −6, 4, −1] 2η36 [2, −5, −4, 2, 14, 2, −4, −5, 2] η36 [3, −9, −10, 3, 30, 3, −10, −9, 3] 3η36 [−1, 3, 6, −2, −6, −1, 4, 3, −2] 12η36 [−2, 3, 4, −1, −6, −2, 6, 3, −1] η36 [2, −5, −3, 2, 12, −1, −3, 1, −1] η36 [−1, 1, −3, −1, 12, 2, −3, −5, 2] 3η36 [1, 0, −6, −1, 12, 1, −2, 0, −1] 3η36 [−1, 0, −2, 1, 12, −1, −6, 0, 1] 6η36 [−1, 0, 0, 1, 4, 1, 0, 0, −1] 6η36 [1, 0, 0, −1, 4, −1, 0, 0, 1] 3η36 [1, −3, 2, 2, 6, 1, −4, −3, 2] 12η36 [2, −3, −4, 1, 6, 2, 2, −3, 1] 2η36 [−1, 4, 4, −1, −6, −3, 4, 6, −3] 6η36 [−3, 6, 4, −3, −6, −1, 4, 4, −1] 6η36 [−2, 3, 0, −1, 4, 2, 0, −3, 1] 6η36 [−1, 3, 0, −2, 4, 1, 0, −3, 2] 6η36 [2, −3, 0, 1, 4, −2, 0, 3, −1] 6η36 [1, −3, 0, 2, 4, −1, 0, 3, −2] 2η36 [3, −6, −4, 3, 14, 1, −4, −4, 1] 6η36 [1, −4, −4, 1, 14, 3, −4, −6, 3]

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Table 3 Equivalent eta quotients under q → −q. − f4,1a ∼ f4,2

− f4,1b ∼ f4,1b

− f6,1a ∼ f12,10a − f6,2a ∼ f12,17a − f6,4a ∼ f12,19a

− f6,1b ∼ f12,23a − f6,2b ∼ f12,17b − f6,4b ∼ f12,19b

− f8,1a ∼ f8,2

− f8,1b ∼ f8,3

− f6,1c ∼ f12,10b − f6,3a ∼ f12,27

− f6,1d ∼ f12,23b − f6,3b ∼ f12,30

− f9,1 ∼ f36,5 − f12,1a ∼ f12,8b − f12,2a ∼ f12,25b − f12,3a ∼ f12,26a − f12,4a ∼ f12,9a − f12,5a ∼ f12,9b − f12,6a ∼ f12,6b − f12,7a ∼ f12,22a − f12,8a ∼ f12,11a − f12,12a ∼ f12,15a − f12,13a ∼ f12,25c − f12,15b ∼ f12,16a − f12,18a ∼ f12,24a − f12,21a ∼ f12,22b − f12,28a ∼ f12,29a

− f12,1b ∼ f12,8c − f12,2b ∼ f12,25d − f12,3b ∼ f12,26d − f12,4b ∼ f12,9d − f12,5b ∼ f12,9c − f12,6c ∼ f12,20a − f12,7b ∼ f12,21b − f12,8d ∼ f12,11b − f12,12b ∼ f12,16b − f12,13b ∼ f12,25a − f12,15c ∼ f12,16c − f12,18b ∼ f12,18b − f12,21d ∼ f12,22d − f12,28b ∼ f12,29b

− f16,1a ∼ f16,2

− f16,1b ∼ f16,3

− f18,1a ∼ f36,2a − f18,2a ∼ f36,3a − f18,3a ∼ f36,7a

− f18,1b ∼ f36,13a − f18,2b ∼ f36,11a − f18,3b ∼ f36,7b

− f18,1c ∼ f36,2b − f18,2c ∼ f36,3b − f18,4a ∼ f36,1

− f18,1d ∼ f36,13b − f18,2d ∼ f36,11b − f18,4b ∼ f36,4

− f20,1a ∼ f20,3a − f20,2b ∼ f20,3b

− f20,1b ∼ f20,2a − f20,2d ∼ f20,3d

− f20,1c ∼ f20,2c

− f20,1d ∼ f20,3c

f24,1a f24,2a f24,3c f24,5a

− ∼ f24,3a − ∼ f24,4a − ∼ f24,5c − ∼ f24,6a

− f36,6a ∼ f36,8a − f36,9b ∼ f36,12b

f24,1b f24,2b f24,3d f24,5b

− ∼ f24,6b − ∼ f24,4b − ∼ f24,5d − ∼ f24,6c

− f36,6b ∼ f36,10b − f36,12a ∼ f36,12c

f12,1c f12,2c f12,3c f12,4c

∼ ∼ ∼ ∼

− f12,11c − f12,13d − f12,14c − f12,5c

f12,1d f12,2d f12,3d f12,4d

∼ ∼ ∼ ∼

− f12,11d − f12,13c − f12,14d − f12,5d

− f12,6d ∼ f12,20b − f12,7c ∼ f12,22c

− f12,7d ∼ f12,21c

− f12,12c ∼ f12,15d − f12,14a ∼ f12,26b

− f12,12d ∼ f12,16d − f12,14b ∼ f12,26c

− f12,18c ∼ f12,24b

− f12,18d ∼ f12,18d

− f12,28c ∼ f12,28d

f24,1c f24,2c f24,4c f24,7a

− ∼ f24,3b − ∼ f24,8b − ∼ f24,7b − ∼ f24,8a

− f36,8b ∼ f36,10a

f24,1d f24,2d f24,4d f24,7c

− ∼ f24,6d − ∼ f24,8d − ∼ f24,7d − ∼ f24,8c

− f36,9a ∼ f36,12d

zd = η6 [1, −2, −3, 6](τ ), then the ten entries for level 6 in Table 2 correspond respectively to Eisenstein series representations for za2 , zb2 , zc2 , zd2 , za zd , zb zc , za zc , zb zd , za zb and zc zd . The series representations can be found in Theorems 6.17 and 6.19 of [9]. (Remark: there is a minor misprint in the identity for zc2 .) One should consult Chapter 6 of the excellent treatise by Cooper [9] to understand the rich interplay between these ten weight 2 eta quotients of level 6. Equivalent statements for f6,1a , f6,3a , f6,3b , f6,4a and f6,4b can also be found in [11]. Under the transformation q → −q, identities of level N are mapped to level 4N/ gcd(4, N ). Thus, all the level 6 identities are equivalent to some level 12 identities. The four identities of level 8 in Table 2 are all equivalent up to Atkin-Lehner involutions or the transformation q → −q. Equivalent statements can be found in Theorem 8.16 of [9]. In particular, f8,1a , f8,1b , f8,2 , and f8,3 correspond to Equations (8.38), (8.41), (8.39) and (8.42) respectively. As mentioned previously, f8,2 can also be found in [11]. There are a total of 100 distinct identities of level 12 and the relationships between these identities are worth investigating. From Table 3, we can see that the twelve identities labelled f12,1∗ , f12,8∗ and f12,11∗ are

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17

all equivalent. In total, there are six groups of 12 equivalent identities accounting for a total of 72. There are another ten identities equivalent to the ten from level 6. The remaining 18 out of the 100 identities can be classified into three groups of six equivalent identities each. It is interesting to note that identities f12,18b and f12,18d are invariant under q → −q. As mentioned in Section 2, f12,27 can be found in [11]. It is equivalent to the representations of ϕ(q)2 ϕ(q 3 )2 and the result was known to Ramanujan [4, p. 223]. (See also [9, Theorem 3.27].) Equivalent forms of identities f12,24a , f12,29a , f12,18a and f12,28a can be found in [9], where they correspond respectively to the second, fourth, fifth and sixth equations in the statement of Theorem 12.8. In [23], Williams studied the problem of finding eta quotients in M2 (Γ0 (12)) whose Fourier coefficients can be represented by divisor sums. In other words, he attempted to determine eta quotients in E12, which by Lemma 2.1 is equivalent to the problem of finding eta quotients whose derivatives are eta quotients. Williams found 126 eta quotients, each of which can be easily mapped to the identities in Table 2. The number 126 came about because he did not restrict his search to primitive identities (see Section 2). With reference to Table 1, the 3 identities from level 4 and the 10 identities from level 6, are each counted twice at level 12. Together with the 100 primitive identities at level 12 totals the 126 eta quotients listed in [23]. The four identities of level 16 in Table 2 are all equivalent up to Atkin-Lehner involutions or the transformation q → −q. Identity f16,2 appeared in [6, p. 608]. An application of this identity to the theory of partitions is described in the next section. Each of the 12 identities of level 18 are equivalent to some level 36 identities under the transformation q → −q. We are not aware of any previous appearances of these identities. The 12 identities of level 20 in Table 2 are all equivalent up to Atkin-Lehner involutions or the transformation q → −q. We are not aware of any previous appearances of these identities. The 32 identities of level 24 can be divided into two groups, each of which consisting of 16 equivalent identities. In [1], Alaca, Alaca and Aygin attempted to determine eta quotients in E24 . Among the eta quotients that they found, they listed 32 which do not arise directly from level 12 [1, Theorem 5.1]. These 32 can be easily mapped to the 32 level 24 identities in Table 2. For the more general problem of determining which modular forms space is spanned by eta quotients, see [20]. Finally, there are 25 identities of level 36. One is equivalent to the unique level 9 identity, while another 12 are each equivalent to an identity from level 18. The remaining 12 identities can be divided into two groups, each of which consisting of 6 equivalent identities. 6. Applications In his lost notebook [19], Ramanujan recorded eight evaluations of integrals of theta functions. Four of these are equivalent to identities in Table 2. For example, Entry 14.3.1 [2, p. 314] is as follows: For 0 < q < 1, ⎛

⎞

q

3

ϕ(−q ) = exp ⎝2 ϕ(−q)

ψ(t)2 ψ(t3 )2 dt⎠ .

(6.1)

0

Identity (6.1) is equivalent to f6,3b from Table 2, namely q

d log (η6 [−2, 1, 2, −1](τ )) = 2η6 [−2, 4, −2, 4](τ ). dq

(6.2)

Similarly, Entries 14.3.2, 14.3.3 and 14.3.4 of [2] correspond respectively to f12,27 , f4,1b and f4,2 . The last of which is simply Jacobi’s identity (1.4). Other analogues of (6.1) may be deduced from Table 2. For example, the identity labelled f24,2c is equivalent to the following.

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Theorem 6.1. For 0 < q < 1, ⎛ ϕ(q 6 )3 = exp ⎝4 ϕ(−q)2 ϕ(q 2 )

q

⎞ ψ(t2 )3 ψ(t3 )ϕ(−t4 )2 ⎠ dt . ψ(−t)ϕ(t6 )

(6.3)

0

A second application is to the evaluation of certain integrals in terms of algebraic constants. Fine evaluated the following three integrals [11, Eq. (33.43), (33.53), (33.63)]: e−π ∞ 0 e

(6.4)

√ −π/ 2

√ ∞ (1 − q 2n )8 (1 − q 4n )4 2 , dq = n 8 (1 − q ) 8 n=1

0 e

1 (1 − q 2n )20 , dq = n )16 (1 − q 16 n=1

(6.5)

√ −π/ 3

0

∞ 1 (1 − q 2n )14 (1 − q 6n )6 dq = . n )8 (1 − q 4n )8 (1 − q 3 n=1

(6.6)

The proof of each boils down to integrating the identities f4,2 , f8,2 and f12,27 respectively, and subsequently evaluating the resulting eta quotients. Cooper [8] employed a similar technique to the identity associated with f9,1 , i.e. the identity discovered by Borwein and Garvan (1.5), to obtain e−2π/3 ∞ 0

1 (1 − q 3n )10 dq = √ . n )6 (1 − q 3 3 n=1

(6.7)

By combing through our list of identities, we are able to obtain the following 15 evaluations. We organize the results according to level. Theorem 6.2. Let j ∈ Z+ . The following evaluations hold: √

e−2π/

6

q

j−1

0 e

√ −2π/ 6

q j−1 0

 j ∞ 1 1 (1 − q 2n )7+j (1 − q 3n )7−j √ dq = , (1 − q n )5+5j (1 − q 6n )5−5j j 6 2 n=1

 j ∞ 3 1 (1 − q n )3+3j (1 − q 2n )3−3j √ dq = . − 1 (1 − q 3n )1+9j (1 − q 6n )1−9j j 2 2 n=1

Proof. From the identity associated with f6,2b in Table 2, we can deduce that q

d j j (η6 [−5, 1, −1, 5](τ )) = j (η6 [−5, 1, −1, 5](τ )) · η6 [−5, 7, 7, −5](τ ). dq

Thus √

e−2π/

6

1 j · (η6 [−5, 1, −1, 5](τ )) · η6 [−5, 7, 7, −5](τ )dq q 0

(6.8)

(6.9)

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19

 1 j  = (η6 [−5, 1, −1, 5](τ ))  √ j q=e−2π/ 6  j  1 η(6τ )5 η(2τ )  = ·  j η(τ )5 η(3τ )  √ τ =i/ 6

1 = j

!

#j " √ η(i 6)5 η(i 2/3) √ " · . η(i/ 6)5 η(i 3/2)

(6.10)

Using the well known inversion formula η(−1/τ ) =

"

τ /i · η(τ )

(6.11)

 j " √ 1 1 √ with τ = i/ 6 and τ = i 3/2, the right side of (6.10) becomes which proves (6.8). The proof j 6 2 of (6.9) follows in a similar manner from the identity f6,4a in Table 2. 2 Theorem 6.3. Let s ∈ Z, s = 0. The following evaluation holds: √

e−π/

2

2 0

∞ % (1 − q n )4−2s (1 − q 8n )4+2s 1 $ s/4 2 dq = − 1 . (1 − q 2n )2−7s (1 − q 4n )2+7s s n=1

(6.12)

Proof. Use the identity f8,3 in Table 2 and the inversion formula (6.11). 2 Theorem 6.4. Let j ∈ Z+ and s ∈ Z, s = 0. The following evaluations hold: √

e−π/

3

2q

∞

1 (1 − q 3n )6 (1 − q 4n )6 dq = n 2+4j 2n 2+j 6n 2−j 12n 2−4j (1 − q ) (1 − q ) (1 − q ) (1 − q ) j n=1

2j−1

0 √

e−π/

3

(1 − q n )6 (1 − q 12n )6 1 2q dq = 2n 2+s 3n 2−4s 4n 2+4s 6n 2−s (1 − q ) (1 − q ) (1 − q ) (1 − q ) s n=1

0 e

∞

√ −π/ 3

q

0 e

√ −π/ 3

2 0

4 3 · 31/4

j

s

,

(6.13)

1 − , (6.14) s

(6.15)

 s ∞ 2 1 1 (1 − q n )1−s (1 − q 2n )4+2s (1 − q 6n )4−2s (1 − q 12n )1+s √ dq = − , 3n 3−3s 4n 3+3s (1 − q ) (1 − q ) s s 3 n=1

(6.16)

0 √



1 12 · 31/4

j  ∞ 1 (1 − q 2n )4+2j (1 − q 3n )1+j (1 − q 4n )1−j (1 − q 6n )4−2j 1 √ dq = , (1 − q n )3+3j (1 − q 12n )3−3j j 2 3 n=1

j−1

e−π/



3

∞ % (1 − q n )2−2s (1 − q 3n )2+2s (1 − q 4n )2−2s (1 − q 12n )2+2s 1 $ s/4 dq = − 1 . 3 (1 − q 2n )2−5s (1 − q 6n )2+5s s n=1

(6.17)

Proof. These five evaluations can be proved by respectively utilizing the identities f12,20a , f12,20b , f12,29a , f12,29b and f12,30 in Table 2 and the inversion formula (6.11). 2

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Theorem 6.5. Let s ∈ Z, s = 0. The following evaluation holds: e−π/2

2 0

∞ % 1 $ s/2 (1 − q n )2−2s (1 − q 4n )10 (1 − q 16n )2+2s dq = 2 −1 . 2n 5−5s 8n 5+5s (1 − q ) (1 − q ) s n=1

(6.18)

Proof. Use identity f16,3 in Table 2 and the inversion formula (6.11). 2 Theorem 6.6. Let j ∈ Z+ . The following evaluations hold: e−

√

2π/3

q

j−1

0 √

e−2

j  ∞ 1 (1 − q 2n )1+j (1 − q 3n )3+j (1 − q 6n )3−j (1 − q 9n )1−j 1 √ dq = , (1 − q n )2+2j (1 − q 18n )2−2j j 6 n=1

2π/3

q

j−1

0

∞ 1 (1 − q n )1+j (1 − q 3n )2 (1 − q 6n )4 (1 − q 9n )1−j dq = 2n 2+2j 18n 2−2j (1 − q ) (1 − q ) j n=1

& #j 2 1− . 3

(6.19)

!

(6.20)

Theorem 6.7. Let j ∈ Z+ and s ∈ Z, s = 0. The following evaluations hold: e−π/3

q

j−1

0

∞

1 F1 (q n )(1 − q 2n )1+j (1 − q 18n )1−j dq = n )1+j (1 − q 4n )1+j (1 − q 9n )1−j (1 − q 36n )1−j (1 − q j n=1



1 √ 3

j (6.21)

where F1 (q n ) = e−π/3

2 0

(1 − q 6n )10 ; (1 − q 3n )2 (1 − q 12n )2

∞ % 1 $ s/2 F2 (q n )(1 − q n )2−2s (1 − q 4n )2−2s (1 − q 9n )2+2s (1 − q 36n )2+2s dq = 3 −1 2n 5−5s 18n 5+5s (1 − q ) (1 − q ) s n=1

(6.22)

where F2 (q n ) = e−π/3

q

j−1

0

(1 − q 6n )14 ; (1 − q 3n )4 (1 − q 12n )4

j  ∞ 1 1 F3 (q n )(1 − q n )3+3j (1 − q 4n )3+3j (1 − q 9n )3−3j (1 − q 36n )3−3j √ dq = (1 − q 2n )9+9j (1 − q 18n )9−9j j 3 3 n=1

(6.23)

where F3 (q n ) = e−π/3

6q 0

6j−1

(1 − q 6n )30 ; (1 − q 3n )10 (1 − q 12n )10

j  ∞ 1 F4 (q n )(1 − q 2n )3j (1 − q 3n )2j (1 − q 12n )−2j (1 − q 18n )−3j 1 √ dq = (1 − q n )1+6j (1 − q 36n )1−6j j 12 3 n=1

where F4 (q n ) = (1 − q 4n )(1 − q 6n )4 (1 − q 9n ).

(6.24)

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366

21

Proof. These four evaluations can be proved by respectively utilizing the identities f36,1 , f36,4 , f36,5 and f36,9a in Table 2 and the inversion formula (6.11). 2 Our 15 evaluations in Theorems 6.2 to 6.7 relied on evaluating eta quotients with the inversion formula (6.11). In principle, all the 203 identities in Table 2 can be evaluated at special values. Doyle and Williams [10] gave a very well written account of how to use the Chowla-Selberg formula for fundamental discriminants and its generalizations to evaluate such integrals. In their work, they proved generalizations of (6.4), (6.5) and (6.6). In addition, they also gave two other evaluations by using the identities f12,3c and f16,2 . Our third application comes from the theory of partitions. Let p(n) denote the partition function, then it is known that for primes , p(n + δ ) ≡ 0 (mod )

(6.25)

holds if and only if  = 5, 7 or 11. Thus it is somewhat surprising that Kim and Toh [15] showed that for every odd integer m, the function c(n) satisfies c(mn + δm ) ≡ 0 (mod m)

(6.26)

where 4δm ≡ 1 (mod m). Here c(n) counts the number of cubic partition pairs of n, weighted by the parity of the crank. The generating function of c(n) is ∞

n=0

c(n)q n =

∞ (1 − q n )6 (1 − q 2n )2 . (1 − q 4n )4 n=1

(6.27)

The proof of (6.26) relies on the fact that the generating function of c(n) is essentially an eta quotient that happens to be the derivative of some power series in q with integer coefficients. The precise identity responsible is f16,2 in Table 2. Thus our list of identities paves the way to look for other partition functions with similar properties. We shall illustrate this with identity f16,1b . It is equivalent to q

d η16 [−2, 1, −2, 5, −2](τ ) = 2η16 [−4, 2, 6, 0, 0](τ ). dq

(6.28)

In other words, suppose d(n) has the generating function, ∞

d(n)q n =

n=0

=

∞ (1 − q 2n )2 (1 − q 4n )6 (1 − q n )4 n=1 ∞ ∞ ∞ (1 − q 4n )4 (1 − q 2n )2 (1 − q 4n )2 · · . (1 − q n ) n=1 (1 − q n ) n=1 (1 − q n )2 n=1

(6.29)

We can interpret d(n) as the number of four-colored partitions, where the first color consists of 4-core partitions [12], the second color consists of 2-core partitions and the third and fourth are partitions where no parts divisible by 4 appears. Now observe that η16 [−2, 1, −2, 5, −2](τ ) = 1 + 2

∞

an q n

n=1

where an ∈ Z for every positive integer n. Identity (6.28) is then equivalent to

(6.30)

22

Z.S. Aygin, P.C. Toh / J. Math. Anal. Appl. 480 (2019) 123366 ∞

n=0

d(n)q n+1 =

∞

d(n − 1)q n =

n=1

∞

nan q n .

(6.31)

n=1

In other words, for every n, m ∈ Z+ where m > 1, the following congruence holds d(mn − 1) ≡ 0

(mod m).

(6.32)

We end by discussing a related work. Choi, Kim and Lim [7] also consider the problem of determining eta quotients which are mapped to other eta quotients by the differential operator. Their approach to the problem is combinatorial and quite different from our computational approach. They found that the only results for square-free levels are, up to Atkin-Lehner involutions, the ten level 6 identities in Table 2, which corroborates with our findings. It will be interesting if their methods can be used to show that there are no other identities for levels N > 36. Acknowledgments The authors thank the anonymous referee for many helpful comments which greatly improved the paper. The authors also thank Byungchan Kim for sharing the preprint [7] and providing helpful comments. Research of the first named author was partially supported by a PIMS postdoctoral fellowship and a postdoctoral fellowship at Nanyang Technological University, Singapore, where the majority of the work was done. References [1] A. Alaca, Ş. Alaca, Z.S. Aygin, Theta products and eta quotients of level 24 and weight 2, Funct. Approx. Comment. Math. 57 (2017) 205–234. [2] G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook, Part I, Springer, 2005. [3] Z.S. Aygin, Eisenstein Series, Eta Quotients and Their Applications in Number Theory, Doctoral dissertation, Carleton University, Ottawa, Canada, 2016. [4] B.C. Berndt, Ramanujan’s Notebook, Part III, Springer, 1991. [5] J.M. Borwein, F.G. Garvan, Approximations to π via the Dedekind Eta Function, CMS Conf. Proc., vol. 20, American Mathematical Society, Providence, RI, 1997, pp. 89–115. [6] S.H. Chan, Generalized Lambert series identities, Proc. Lond. Math. Soc. 91 (2005) 598–622. [7] D. Choi, B. Kim, S. Lim, Pairs of eta-quotients with dual weights and their applications, preprint. [8] S. Cooper, A simple proof of an expansion of an eta-quotient as a Lambert series, Bull. Aust. Math. Soc. 71 (2005) 353–358. [9] S. Cooper, Ramanujan’s Theta Functions, Springer International Publishing, 2017. [10] G. Doyle, K.S. Williams, Evaluation of some q-integrals in terms of the Dedekind eta function, Analysis 38 (2018) 63–79. [11] N.J. Fine, Basic Hypergeometric Series and Applications, Mathematical Surveys and Monographs, vol. 27, American Mathematical Society, Providence, RI, 1988. [12] F. Garvan, D. Kim, D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990) 1–17. [13] H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. [14] C.G.J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Sumptibus fratrum Bornträger, 1829. [15] B. Kim, P.C. Toh, On the crank function of cubic partition pairs, Ann. Comb. 22 (2018) 803–818. [16] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, 2nd edition, Graduate Texts in Mathematics, vol. 97, Springer, 1993. [17] Y. Martin, K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (11) (1997) 3169–3176. [18] S. Ramanujan, On certain arithmetical functions, Trans. Camb. Philos. Soc. 22 (1916) 159–184. [19] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988. [20] J. Rouse, J.J. Webb, On spaces of modular forms spanned by eta-quotients, Adv. Math. 272 (2015) 200–224. [21] W.A. Stein, Modular Forms, A Computational Approach, Graduate Studies in Mathematics, vol. 79, Amer. Math. Soc., 2007. [22] J. Sturm, On the Congruence of Modular Forms, Lect. Notes in Math., vol. 1240, Springer, 1984, pp. 275–280. [23] K.S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (4) (2012) 993–1004.