Probabilities as Values of Modular Forms and Continued Fractions

Abstract

We consider certain probability problems which are naturally related to integer partitions. We show that the corresponding probabilities are values of classical modular forms. Thanks to this connection, we then show that certain ratios of probabilities are specializations of the Rogers-Ramanujan and Ramanujan- Selberg- Gordon-Göllnitz continued fractions. One particular evaluation depends on a result from Ramanujan's famous first letter to Hardy.

1. Introduction and Statement of Results

In a recent paper [1], Holroyd et al. defined probability models whose properties are linked to the number theoretic functions 𝑝_𝑘(𝑛), which count the number of integer partitions of 𝑛 which do not contain 𝑘 consecutive integers among its summands. Their asymptotic results for each 𝑘 lead to a nice application: the calculation of thresholds of certain two dimensional cellular automata (see [1, Theorem 4]).

In a subsequent paper [2], Andrews explained the deep relationship between the generating functions for 𝑝_𝑘(𝑛) and mock theta functions and nonholomorphic modular forms. Already in the special case of 𝑝₂(𝑛), one finds an exotic generating function whose unusual description requires both a modular form and a Maass form. This description plays a key role in the recent work of Bringmann and Mahlburg [3], who make great use of the rich properties of modular forms and harmonic Maass forms to obtain asymptotic results which improve on [1, Theorem 3] in the special case of 𝑝₂(𝑛).

It does not come as a surprise that the seminal paper of Holroyd et al. [1] has inspired further work at the interface of number theory and probability theory. (In addition to [2, 3], we point out the recent paper by Andrews et al. [4]) Here we investigate another family of probability problems which also turn out to be related to the analytical properties of classical partition generating functions.

These problems are relatives of the following standard undergraduate problem. Toss a fair coin repeatedly until it lands heads up. If one flips 𝑛 tails before the first head, what is the probability that 𝑛 is even? Since the probability of flipping 𝑛 tails before the first head is 1/2^𝑛+1, the solution is

12+12³+12⁵1+⋯=−11−1/2=21−1/43. (1.1)

Instead of continuing until the first head, consider the situation where a coin is repeatedly flipped: once, then twice, then three times, and so on. What is the probability of the outcome that each 𝑛th turn, where 𝑛 is odd, has at least one head?

More generally, let 0<𝑝<1, and let {𝐶₁,𝐶₂,…} be a sequence of independent events where the probability of 𝐶_𝑛 is given by

ℙ_𝑝(𝐶_𝑛)∶=1−𝑝^𝑛. (1.2) For each pair of integers 0≤𝑟<𝑡, we let

𝐴(𝑟,𝑡)∶={setofsequenceswhere𝐶_𝑛occursif𝑛≢±𝑟mod𝑡}. (1.3) In the case where 𝑝=1/2, one can think of 𝐶_𝑛 as the event where at least one of 𝑛 tosses of a coin is a head. Therefore, if 𝑝=1/2, the problem above asks for the probability of the outcome 𝐴(0,2).

Remark 1.1. Without loss of generality, we shall always assume that 0≤𝑟≤𝑡/2. In most cases 𝐴(𝑟,𝑡) is defined by two arithmetic progressions modulo 𝑡. The only exceptions are for 𝑟=0, and for even 𝑡 when 𝑟=𝑡/2.

It is not difficult to show that the problem of computing

ℙ_𝑝(𝐴(𝑟,𝑡))∶=“probabilityofA(r,t)” (1.4) involves partitions. Indeed, if 𝑝(𝑟,𝑡;𝑛) denotes the number of partitions of 𝑛 whose summands are congruent to ±𝑟(mod𝑡), then we shall easily see that the probabilities are computed using the generating functions

𝒫(𝑟,𝑡;𝑞)∶=^∞_𝑛=0𝑝(𝑟,𝑡;𝑛)𝑞^𝑛=^∞𝑛=01(1−𝑞^{𝑡𝑛+𝑟})(1−𝑞^{𝑡𝑛+𝑡−𝑟})𝑡,if0<𝑟<2_∞𝑛=1,1(1−𝑞^𝑡𝑛),if𝑟=0,_∞𝑛=01(1−𝑞^{𝑡𝑛+𝑡/2})𝑡,if𝑟=2. (1.5) This critical observation is the bridge to a rich area of number theory, one involving class field theory, elliptic curves, and partitions. In these areas, modular forms play a central role, and so it is natural to investigate the number theoretic properties of the probabilities ℙ_𝑟(𝐴(𝑟,𝑡)) from this perspective.

Here we explain some elegant examples of results which can be obtained in this way. Throughout, we let 𝜏∈ℍ, the upper-half of the complex plane, and we let 𝑞_𝜏∶=𝑒^{2𝜋𝑖𝜏}. The Dedekind eta-function is the weight 1/2 modular form (see [5]) defined by the infinite product

𝜂(𝜏)∶=𝑞_𝜏^∞1/24_𝑛=1(1−𝑞^𝑛_𝜏). (1.6) We also require some further modular forms, the so-called Siegel functions. For 𝑢 and 𝑣 real numbers, let 𝐵₂(𝑢)∶=𝑢²−𝑢+1/6 be the second Bernoulli polynomial, let 𝑧∶=𝑢−𝑣𝜏, and let 𝑞_𝑧∶=𝑒^{2𝜋𝑖𝑧}. Then the Siegel function 𝑔_𝑢,𝑣(𝜏) is defined by the infinite product

𝑔_𝑢,𝑣(𝜏)∶=−𝑞^𝐁2_𝜏^(𝑢)/2𝑒^{∞2𝜋𝑖𝑣(𝑢−1)/2}_𝑛=1(1−𝑞_𝑧𝑞^𝑛_𝜏)(1−𝑞_𝑧⁻¹𝑞^𝑛_𝜏). (1.7) These functions are weight 0 modular forms (e.g., see [5]).

For fixed integers 0≤𝑟≤𝑡/2, we first establish that ℙ_𝑝(𝐴(𝑟,𝑡)) is essentially a value of a single quotient of modular forms.

Theorem 1.2. If 0<𝑝<1 and 𝜏_𝑝∶=−log(𝑝)⋅𝑖/2𝜋, then ℙ_𝑝𝑞(𝐴(𝑟,𝑡))=^{𝜏(𝑡−1)/24𝑝}⋅𝜂(𝜏_𝑝)𝜂(𝑡𝜏_𝑝)𝑞,if𝑟=0,_{𝜏−(𝑡+2)/48𝑝}1−𝑞_𝜏^𝑡/2_𝑝⋅𝜂(𝜏_𝑝)𝜂(𝑡𝜏_𝑝)𝜂((𝑡/2)𝜏_𝑝)𝑡,if𝑟=2,−𝑞_𝜏^{−(2𝑡+1)/24}_𝑝𝑒^{−𝜋𝑖(𝑟/𝑡)}1−𝑞^𝑟_𝜏𝑝⋅𝜂(𝜏_𝑝)𝑔_{0,−𝑟/𝑡}(𝑡𝜏_𝑝),otherwise. (1.8)

One of the main results in the work of Holroyd et al. (see [1, Theorem  2]) was the asymptotic behavior of their probabilities as 𝑝→1. In Section 4 we obtain the analogous results for logℙ_𝑝(𝐴(𝑟,𝑡)).

Theorem 1.3. As 𝑝→1, one has −logℙ_𝑝𝜋(𝐴(𝑟,𝑡))∼²16(1−𝑝)(1−𝑡𝑡),if𝑟=0,2,𝜋²26(1−𝑝)(1−𝑡),otherwise. (1.9)

To fully appreciate the utility of Theorem 1.2, it is important to note that the relevant values of Dedekind's eta-function and the Siegel functions can be reformulated in terms of the real-analytic Eisenstein series (see [5])

𝐸_𝑢,𝑣(𝜏,𝑠)∶=_{(𝑚,𝑛)∈ℤ2(𝑚,𝑛)≠(0,0)}𝑒^{2𝜋𝑖(𝑚𝑢+𝑛𝑣)}𝑦^𝑠|𝑚𝜏+𝑛|^2𝑠,𝜏=𝑥+𝑖𝑦∈ℍ,Re(𝑠)>1. (1.10) One merely makes use of the Kronecker Limit Formulas (e.g., see [5, Part 4]). These limit formulas are prominent in algebraic number theory for they explicitly relate such values of modular forms to values of zeta-functions of number fields.

We give two situations where one plainly sees the utility of these observations. Firstly, one can ask for a more precise limiting behavior than is dictated by Theorem 1.3. For example, consider the limiting behavior of the quotient 𝐿(𝑝)∶=ℙ_𝑝(𝐴(2,5))/ℙ_𝑝(𝐴(1,5)), as 𝑝→1. Table 1 is very suggestive.

Table 1 

Values of 𝐿(𝑝).

Theorem 1.4. The following limits are true: lim_𝑝→1ℙ_𝑝(𝐴(2,5))ℙ_𝑝=√(𝐴(1,5))−1+52,lim_𝑝→1ℙ_𝑝(𝐴(3,8))ℙ_𝑝√(𝐴(1,8))=−1+2. (1.11)

Remark 1.5. Notice that (1/2)(−1+√5)=−1+𝜙, where 𝜙 is the golden ratio1𝜙∶=2(1+√5)=1.61803…. (1.12)

Theorem 1.4 is a consequence of our second application which concerns the problem of obtaining algebraic formulas for all of these ratios, not just the limiting values. As a function of 𝑝, one may compute the ratios of these probabilities in terms of continued fractions. To ease notation, we let

𝑎₁𝑏₁⁺𝑎₂𝑏₂⁺𝑎₃𝑏₃^+⋯ (1.13) denote the continued fraction

𝑎₁𝑏₁+𝑎₂𝑏₂+𝑎₃𝑏₃.+⋯ (1.14) The following continued fractions are well known:

11⁺11⁺11^+⋯=111+11+=√1+⋯−1+52,12⁺12⁺12^+⋯=112+12+√2+⋯=−1+2. (1.15) Theorem 1.4 is the limit of the following exact formulas.

Theorem 1.6. If 0<𝑝<1 and 𝜏_𝑝∶=−log(𝑝)⋅(𝑖/2𝜋), then one has that ℙ_𝑝(𝐴(2,5))ℙ_𝑝1(𝐴(1,5))=(1⁺𝑞_𝜏𝑝1⁺𝑞²_𝜏𝑝1⁺𝑞³_𝜏𝑝1^+⋯ℙ),_𝑝(𝐴(3,8))ℙ_𝑝1(𝐴(1,8))=(1+𝑞_𝜏𝑝⁺𝑞²_𝜏𝑝1+𝑞³_𝜏𝑝⁺𝑞⁴_𝜏𝑝1+𝑞⁵_𝜏𝑝⁺𝑞⁶_𝜏𝑝1+𝑞⁷_𝜏𝑝^+⋯). (1.16)

Theorem 1.6 can also be used to obtain many further beautiful expressions, not just those pertaining to the limit as 𝑝→1. For example, we obtain the following simple corollary.

Corollary 1.7. For 𝑝₁=1/𝑒^2𝜋 and 𝑝₂=1/𝑒^𝜋, one has that ℙ_𝑝1(𝐴(2,5))ℙ_𝑝1(𝐴(1,5))=𝑒^−2𝜋/5(−𝜙+12(5+√ℙ5)),_𝑝2(𝐴(3,8))ℙ_𝑝2(𝐴(1,8))=𝑒^−𝜋/2(4+2√2−3+2√2). (1.17)

Remark 1.8. The evaluation of ℙ_𝑝(𝐴(2,5))/ℙ_𝑝(𝐴,1,5)) when 𝑝=1/𝑒^2𝜋 follows from a formula in Ramanujan's famous first letter to Hardy dated January 16, 1913.

Theorem 1.6 and Corollary 1.7 follow from the fact that the Fourier expansions of the relevant modular forms in Theorem 1.2 are realized as the 𝑞-continued fractions of Rogers-Ramanujan and Ramanujan-Selberg-Gordon-Göllnitz. We shall explain these results in Section 5.

We chose to focus on two particularly simple examples of ratios, namely,

ℙ_𝑝(𝐴(2,5))ℙ_𝑝,ℙ(𝐴(1,5))^𝑝(𝐴(3,8))ℙ_𝑝.(𝐴(1,8)) (1.18) One can more generally consider ratios of the form

ℙ_𝑝(𝐴(𝑟₁,𝑡))ℙ_𝑝(𝐴(𝑟₂.,𝑡)) (1.19) It is not the case that all such probabilities, for fixed 𝑟₁ and 𝑟₂, can be described by a single continued fraction. However, one may generalize these two cases, thanks to Theorem 1.2, by making use of the so-called Selberg relations [6] (see also [7]) which extend the notion of a continued fraction. Arguing in this way, one may obtain Theorem 1.6 in its greatest generality. Although we presented Theorem 1.4 as the limiting behavior of the continued fractions in Theorem 1.6, we stress that its conclusion also follows from the calculation of the explicit Fourier expansions at cusps of the modular forms in Theorem 1.2. In this way one may also obtain Theorem 1.4 in generality. Turning to the problem of obtaining explicit formulas such as those in Corollary 1.7, we have the theory of complex multiplication at our disposal. In general, one expects to obtain beautiful evaluations as algebraic numbers whenever 𝜏_𝑝 is an algebraic integer in an imaginary quadratic extension of the field of rational numbers. We leave these generalizations to the reader.

2. Combinatorial Considerations

Here we give a lemma which expresses the probability ℙ_𝑝(𝐴(𝑟,𝑡)) in terms of the infinite products in (1.5).

Lemma 2.1. If 0<𝑝<1 and 𝜏_𝑝∶=−log(𝑝)⋅(𝑖/2𝜋), then ℙ_𝑝(𝐴(𝑟,𝑡))=𝒫(𝑟,𝑡;𝑞_𝜏𝑝)⋅^∞_𝑛=1(1−𝑞^𝑛_𝜏𝑝)⋅ (2.1)

Proof. By the Borel-Cantelli lemma, with probability 1 at most finitely many of the 𝐶_𝑛's will fail to occur. Now, for each pair of integers 0≤𝑟≤𝑡/2, let 𝑆(𝑟,𝑡) be the countable set of binary strings 𝑎₁𝑎₂𝑎₃𝑎₄⋯∈{0,1}^ℕ in which 𝑎_𝑛=1 if 𝑛≢±𝑟mod𝑡, and 𝑎_𝑛=0 for at most finitely many 𝑛 satisfying 𝑛≡±𝑟mod𝑡 (with the appropriate modifications on the arithmetic progressions modulo 𝑡 for 𝑟=0, and for 𝑡 even with 𝑟=𝑡/2). Then the event 𝐴(𝑟,𝑡) can be written as the countable disjoint union 𝐴(𝑟,𝑡)=_{𝑎1𝑎2𝑎3⋯∈𝑆𝑟,𝑡𝑛∶𝑎𝑛=1}𝐶_𝑛∩_{𝑛∶𝑎𝑛=0}𝐶^𝑐_𝑛. (2.2) Using this, we find that ℙ_𝑝(𝐴(𝑟,𝑡))=_{𝑎1𝑎2𝑎3⋯∈𝑆𝑟,𝑡}ℙ_𝑝(_{𝑛∶𝑎𝑛=1}𝐶_𝑛∩_{𝑛∶𝑎𝑛=0}𝐶^𝑐_𝑛)=_{𝑎1𝑎2𝑎3⋯∈𝑆𝑟,𝑡𝑛∶𝑎𝑛=1}(1−𝑞^𝑛_𝜏𝑝)_{𝑛∶𝑎𝑛=0}𝑞^𝑛_𝜏𝑝=^∞_𝑖=1(1−𝑞^𝑛_𝜏𝑝)_{𝑎1𝑎2𝑎3⋯∈𝑆𝑟,𝑡𝑛∶𝑎𝑛=0}𝑞^𝑛_𝜏𝑝1−𝑞^𝑛_𝜏𝑝=^∞_𝑛=1(1−𝑞^𝑛_𝜏𝑝)⋅_{𝑎1𝑎2𝑎3⋯∈𝑆𝑟,𝑡𝑛∶𝑎𝑛=0}(𝑞^𝑛_𝜏𝑝+𝑞_𝜏^2𝑛_𝑝+𝑞_𝜏^3𝑛_𝑝=+⋯)_∞𝑛=1(1−𝑞^𝑛_𝜏𝑝)⋅𝒫(𝑟,𝑡;𝑞_𝜏𝑝). (2.3)

3. Proof of Theorem 1.2

We first note that 𝑞_𝜏𝑝=𝑒^{2𝜋𝑖𝜏𝑝}=𝑝, and by (1.6) we have

^∞_𝑛=1(1−𝑞^𝑛_𝜏𝑝)=𝑞_𝜏^−1/24_𝑝𝜂(𝜏_𝑝). (3.1)

Now we prove each of the three relevant cases in turn. First, suppose that 𝑟=0. By Lemma 2.1 we need to show

𝑞_𝜏^−1/24_𝑝𝜂(𝜏_𝑝)^∞_𝑛=1(1−𝑞_𝜏^𝑡𝑛_𝑝)⁻¹=𝑞_𝜏^{(𝑡−1)/24}_𝑝𝜂(𝜏_𝑝)𝜂(𝑡𝜏_𝑝). (3.2) This follows from the identity

^∞_𝑛=1(1−𝑞_𝜏^𝑡𝑛_𝑝)⁻¹=^∞_𝑛=1(1−𝑞^𝑛_𝑡𝜏𝑝)⁻¹=𝑞^1/24_𝑡𝜏𝑝𝜂(𝑡𝜏_𝑝)⁻¹. (3.3)

Next, suppose that 𝑡 is even and 𝑟=𝑡/2. By Lemma 2.1 we need to show

𝑞_𝜏^−1/24_𝑝𝜂(𝜏_𝑝)^∞_𝑛=0(1−𝑞_𝜏^{𝑡𝑛+(𝑡/2)}_𝑝)⁻¹=𝑞_𝜏^{−(𝑡+2)/48}_𝑝1−𝑞_𝜏^𝑡/2_𝑝𝜂(𝜏_𝑝)𝜂(𝑡𝜏_𝑝)𝜂((𝑡/2)𝜏_𝑝). (3.4) Observe that

^∞_𝑛=1(1−𝑞_𝜏^{𝑡𝑛+(𝑡/2)}_𝑝)⁻¹=^∞_𝑛=1(1−𝑞^2𝑛+1_{(𝑡/2)𝜏𝑝})⁻¹=^∞_𝑛=1(1−𝑞^2𝑛_{(𝑡/2)𝜏𝑝})(1−𝑞^𝑛_{(𝑡/2)𝜏𝑝})⁻¹=^∞_𝑛=1(1−𝑞^𝑛_𝑡𝜏𝑝)(1−𝑞^𝑛_{(𝑡/2)𝜏𝑝})⁻¹. (3.5) Then (3.4) follows from the identities

^∞_𝑛=1(1−𝑞^𝑛_𝑡𝜏𝑝)=𝑞^−1/24_𝑡𝜏𝑝𝜂(𝑡𝜏_𝑝),^∞_𝑛=1(1−𝑞^𝑛_{(𝑡/2)𝜏𝑝})⁻¹=𝑞^1/24_{(𝑡/2)𝜏𝑝}𝑡𝜂(2𝜏_𝑝)⁻¹. (3.6)

Finally, suppose that 0<𝑟<𝑡/2. By Lemma 2.1 we need to show

𝑞_𝜏^−1/24_𝑝𝜂(𝜏_𝑝)^∞_𝑛=0(1−𝑞_𝜏^{𝑡𝑛+𝑟}_𝑝)⁻¹(1−𝑞_𝜏^{𝑡𝑛+𝑡−𝑟}_𝑝)⁻¹𝑞=−^{𝜏−(2𝑡+1)/24𝑝}𝑒^{−𝜋𝑖(𝑟/𝑡)}1−𝑞^𝑟_𝜏𝑝𝜂(𝜏_𝑝)𝑔_{0,−𝑟/𝑡}(𝑡𝜏_𝑝). (3.7) Observe that

^∞_𝑛=0(1−𝑞_𝜏^{𝑡𝑛+𝑟}_𝑝)(1−𝑞_𝜏^{𝑡𝑛+𝑡−𝑟}_𝑝)=(1−𝑞^𝑟_𝜏𝑝)(1−𝑞_𝜏^𝑡−𝑟_𝑝)^∞_𝑛=1(1−𝑞_𝜏^{𝑡𝑛+𝑟}_𝑝)(1−𝑞_𝜏^{𝑡𝑛+𝑡−𝑟}_𝑝)=(1−𝑞^𝑟_𝜏𝑝)(1−𝑞_𝜏^𝑡−𝑟_𝑝)^∞_𝑛=1(1−𝑞_𝑟𝜏𝑝𝑞_𝜏^𝑡𝑛_𝑝)(1−𝑞⁻¹_𝑟𝜏𝑝𝑞_𝜏^𝑡(𝑛+1)_𝑝)=(1−𝑞^𝑟_𝜏𝑝)(1−𝑞⁻¹_𝑟𝜏𝑝𝑞_𝑡𝜏𝑝)(1−𝑞⁻¹_𝑟𝜏𝑝𝑞_𝑡𝜏𝑝)^∞_𝑛=1(1−𝑞_𝑟𝜏𝑝𝑞^𝑛_𝑡𝜏𝑝)(1−𝑞⁻¹_𝑟𝜏𝑝𝑞^𝑛_𝑡𝜏𝑝)=(1−𝑞^𝑟_𝜏𝑝)^∞_𝑛=1(1−𝑞_𝑟𝜏𝑝𝑞^𝑛_𝑡𝜏𝑝)(1−𝑞⁻¹_𝑟𝜏𝑝𝑞^𝑛_𝑡𝜏𝑝) (3.8) It follows from the definition of the Siegel function 𝑔_𝑢,𝑣(𝜏) that

^∞_𝑛=1(1−𝑞_𝑟𝜏𝑝𝑞^𝑛_𝑡𝜏𝑝)(1−𝑞⁻¹_𝑟𝜏𝑝𝑞^𝑛_𝑡𝜏𝑝)=−𝑞^1/12_𝑡𝜏𝑝𝑒^{𝜋𝑖(𝑟/𝑡)}𝑔_{0,−𝑟/𝑡}(𝑡𝜏_𝑝), (3.9) from which we obtain

^∞_𝑛=0(1−𝑞_𝜏^{𝑡𝑛+𝑟}_𝑝)⁻¹(1−𝑞_𝜏^{𝑡𝑛+𝑡−𝑟}_𝑝)⁻¹=−𝑞_𝜏^−𝑡/12_𝑝𝑒^{−𝜋𝑖(𝑟/𝑡)}(1−𝑞^𝑟_𝜏𝑝)⁻¹𝑔_{0,−𝑟/𝑡}(𝑡𝜏_𝑝)⁻¹. (3.10) Substitute the preceding identity into the left-hand side of (3.7) to obtain the result.

4. Proof of Theorem 1.3

Recall that the partition generating function is defined by

𝐺(𝑥)∶=^∞_𝑛=1(1−𝑥^𝑛)⁻¹,0<𝑥<1. (4.1) We now prove each of the relevant cases in turn. First, suppose that 𝑟=0. Then 𝒫(0,𝑡;𝑞_𝜏𝑝)=𝐺(𝑞^𝑡_𝜏𝑝), and by Lemma 2.1,

−logℙ_𝑝(𝐴(0,𝑡))=log𝐺(𝑞_𝜏𝑝)−log𝐺(𝑞^𝑡_𝜏𝑝). (4.2) Make the change of variables 𝑥=𝑒^−𝑤. Then a straightforward modification of the analysis in [8, pages 19–21] shows that, for each integer 𝛼≥1,

log𝐺(𝑒^−𝛼𝑤𝜋)=²+16𝛼𝑤2log(1−𝑒^−𝛼𝑤)+𝐶_𝛼+𝑂(𝑤)as𝑤→0 (4.3) for some constant 𝐶_𝛼. It follows from (4.3) that

log𝐺(𝑞_𝜏𝑝)−log𝐺(𝑞^𝑡_𝜏𝑝𝜋)∼²16(1−𝑝)(1−𝑡)as𝑝→1. (4.4)

Next, suppose that 𝑡 is even and 𝑟=𝑡/2. Then

𝑡𝒫(2,𝑡;𝑞_𝜏𝑝1)=1−𝑞_𝜏^𝑡/2_𝑝^∞_𝑛=1(1−𝑞^𝑛_{(𝑡/2)𝜏𝑝})⁻¹(1−𝑞^𝑛_𝑡𝜏𝑝)=𝐺(𝑞_𝜏^𝑡/2_𝑝)𝐺(𝑞^𝑡_𝜏𝑝)⁻¹1−𝑞_𝜏^𝑡/2_𝑝, (4.5) and by Lemma 2.1 and (4.3),

−logℙ_𝑝𝑡(𝐴(2,𝑡))=log(1−𝑞_𝜏^𝑡/2_𝑝)+log𝐺(𝑞_𝜏𝑝)−log𝐺(𝑞_𝜏^𝑡/2_𝑝)+log𝐺(𝑞^𝑡_𝜏𝑝)∼𝜋²26(1−𝑝)(1−𝑡+1𝑡)as𝑝→1. (4.6)

Finally, suppose that 0<𝑟<𝑡/2. Then by Lemma 2.1,

logℙ_𝑝(𝐴(𝑟,𝑡))=log𝐺(𝑞_𝜏𝑝)−log(^∞_𝑛=0(1−𝑞_𝜏^{𝑡𝑛+𝑟}_𝑝)⁻¹)−log(^∞_𝑛=0(1−𝑞_𝜏^{𝑡𝑛+𝑡−𝑟}_𝑝)⁻¹). (4.7) One can show in a manner similar to (4.3) that for each integer 𝛽≥1,

log(^∞_𝑛=0(1−𝑒^{−(𝛼𝑛+𝛽)𝑤})⁻¹)=−log(1−𝑒^−𝛽𝑤)+^∞_𝑘=1𝑒^{−𝛽𝑤𝑘}𝛼𝑤𝑘²+12log(1−𝑒^{−(𝛼+𝛽)𝑤})+𝐶_𝛼,𝛽+𝑂(𝑤)as𝑤→0 (4.8) for some constant 𝐶_𝛼,𝛽. It follows from (4.3) and (4.8) that

log𝐺(𝑞_𝜏𝑝)−log(^∞_𝑛=0(1−𝑞_𝜏^{𝑡𝑛+𝑟}_𝑝)⁻¹)−log(^∞_𝑛=0(1−𝑞_𝜏^{𝑡𝑛+𝑡−𝑟}_𝑝)⁻¹)∼𝜋²26(1−𝑝)(1−𝑡)as𝑝→1. (4.9)

5. Proof of Theorem 1.6 and Corollary 1.7

Here we prove Theorem 1.6 and Corollary 1.7. These results will follow from well-known identities of Rogers-Ramanujan, Selberg, and Gordon-Göllnitz (e.g., see [9]). We require the celebrated Rogers-Ramanujan continued fraction

𝑅(𝑞)=𝑞^1/5(11⁺𝑞1⁺𝑞²1⁺𝑞³1^+⋯), (5.1) and the Ramanujan-Selberg-Gordon-Göllnitz continued fraction

𝐻(𝑞)=𝑞^1/2(11+𝑞⁺𝑞²1+𝑞³⁺𝑞⁴1+𝑞⁵⁺𝑞⁶1+𝑞^7+⋯). (5.2) It turns out that these 𝑞-continued fractions satisfy the following identities (e.g., see [10]):

𝑅(𝑞)=𝑞^∞1/5_𝑛=1(1−𝑞^5𝑛−1)(1−𝑞^5𝑛−4)(1−𝑞^5𝑛−2)(1−𝑞^5𝑛−3)=𝑞^1/5𝒫(2,5;𝑞),𝒫(1,5;𝑞)𝐻(𝑞)=𝑞^∞1/2_𝑛=1(1−𝑞^8𝑛−1)(1−𝑞^8𝑛−7)(1−𝑞^8𝑛−3)(1−𝑞^8𝑛−5)=𝑞^1/2𝒫(3,8;𝑞).𝒫(1,8;𝑞) (5.3) Theorem 1.6 now follows easily from Theorem 1.2.

To prove Corollary 1.7, notice that if 𝑝=1/𝑒^2𝜋 (resp., 𝑝=1/𝑒^𝜋), then 𝜏_𝑝=𝑖 (resp., 𝜏_𝑝=𝑖/2). In particular, we have that 𝑞_𝜏𝑝=𝑒^−2𝜋 (resp., 𝑞_𝜏𝑝=𝑒^−𝜋). Corollary 1.7 now follows from the famous evaluations (e.g., see [11] and [12, page xxvii] which is Ramanujan's first letter to Hardy)

𝑅(𝑒^−2𝜋)=−𝜙+12(5+√5),𝐻(𝑒^−𝜋)=4+2√2−3+2√2. (5.4)

Acknowledgment

Ken Ono thanks the support of the NSF, the Hilldale Foundation and the Manasse family.