Ramanujan's Theta Functions Research Articles

Ramanujan's theta functions and internal congruences modulo arbitrary powers of 3

In this work, we investigate internal congruences modulo arbitrary powers of 3 for two functions arising from Ramanujan's classical theta functions φ(q) and ψ(q). By letting∑n≥0ph3(n)qn:=φ(−q3)φ(−q)and∑n≥0ps3(n)qn:=ψ(q3)ψ(q), we prove that for any m≥1 and n≥0,ph3(32m−1n)≡ph3(32m+1n)(mod3m+2), andps3(32m−1n+32m−14)≡ps3(32m+1n+32m+2−14)(mod3m+2), thereby substantially generalizing the previous results of Bharadwaj et al. and Gireesh et al., respectively.

On string functions and double-sum formulas

String functions are important building blocks of characters of integrable highest modules over affine Kac--Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type $A_{1}^{(1)}$ in terms of Dedekind eta functions. We produce new relations between string functions by writing them as double-sums and then using certain symmetry relations. We evaluate the series using special double-sum formulas that express Hecke-type double-sums in terms of Appell--Lerch functions and theta functions, where we point out that Appell--Lerch functions are the building blocks of Ramanujan's classical mock theta functions.

Partition identities related to the minimal excludant

In this work, we study the newer statistic “minimal excludant” for integer partitions, and obtain several interesting partition identities. In particular, we find a new and simple refinement of Euler's celebrated partition identity. As an unexpected byproduct of this work, we establish a bivariate form of Ramanujan's special theta function φ(q).

Explicit Values for Ramanujan's Theta Function ϕ(q)

This paper provides a survey of particular values of Ramanujan's theta function $\varphi(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-\pi\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to evaluate theta functions are given. Second, classical values are briefly discussed. Third, certain values due to Ramanujan and later authors are given. Fourth, the methods that are used to determine these values are described. Lastly, an incomplete evaluation found in Ramanujan's lost notebook, but now completed and proved, is discussed with a sketch of its proof.

Quantum q-series identities

As analytic statements, classical $q$-series identities are equalities between power series for $|q|<1$. This paper concerns a different kind of identity, which we call a quantum $q$-series identity. By a quantum $q$-series identity we mean an identity which does not hold as an equality between power series inside the unit disk in the classical sense, but does hold on a dense subset of the boundary -- namely, at roots of unity. Prototypical examples were given over thirty years ago by Cohen and more recently by Bryson-Ono-Pitman-Rhoades and Folsom-Ki-Vu-Yang. We show how these and numerous other quantum $q$-series identities can all be easily deduced from one simple classical $q$-series transformation. We then use other results from the theory of $q$-hypergeometric series to find many more such identities. Some of these involve Ramanujan's false theta functions and/or mock theta functions.

THE ASYMPTOTIC DISTRIBUTION OF TRACES OF WEAK MAASS FORMS

We give an asymptotic formula with a power saving error term for traces of CM values of a generic class of weak Maass forms. We apply this result to study the asymptotic distribution of several arithmetic functions, including the partition function, the Andrews smallest parts function, and the coefficients of Ramanujan's mock theta function f ( q ) .

Bounds for coefficients of the f(q) mock theta function and applications to partition ranks

We compute effective bounds for α(n), the Fourier coefficients of Ramanujan's mock theta function f(q) utilizing a finite algebraic formula due to Bruinier and Schwagenscheidt. We then use these bounds to prove two conjectures of Hou and Jagadeesan on the convexity and maximal multiplicative properties of the even and odd partition rank counting functions.

New Fifth and Seventh Order Mock Theta Function Identities

We give simple proofs of Hecke-Rogers indefinite binary theta series identities for the two Ramanujan fifth order mock theta functions $\chi_0(q)$ and $\chi_1(q)$ and all three of Ramanujan's seventh order mock theta functions. We find that the coefficients of the three mock theta functions of order 7 are surprisingly related.

Some Hecke-Rogers type identities

Recently, a new partition function associated with Ramanujan's third order mock theta function ω(q) was discovered, and subsequently its overpartition analogue was introduced. In this paper, we prove an intriguing identity arising from the study of that analogue, which involves a double series of Hecke-Rogers type. In addition, two further identities will be given. Our proofs heavily rely on formulas from the work of Liu [12].

The umbral moonshine module for the unique unimodular Niemeier root system

We use canonically-twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan's fifth order mock theta functions appear as components.

MOCK THETA DOUBLE SUMS

AbstractWe prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.

General Theorem for Explicit Evaluations and Reciprocity Theorems for Ramanujan-Göllnitz-Gordon Continued Fraction

In the paper A new parameter for Ramanujan's theta-functions and explicit values, Arab J. Math. Sc., 18 (2012), 105-119, Saikia studied the parameter involving Ramanujan's theta-functions and for any positive real numbers k and n and applied it to find explicit values of . As more application to the parameter , in this paper we prove a new general theorem for explicit evaluation of Ramanujan--Gordon continued fraction K(q) in terms of the parameter and give examples. We also find some new explicit values of the parameter and offer reciprocity theorems for the continued fraction K(q).

New congruences for Andrews' singular overpartitions

Recently, Andrews defined the combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function [Formula: see text] which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i ( mod k) may be overlined. He also proved that [Formula: see text]. Chen, Hirschhorn and Sellers then found infinite families of congruences modulo 3 and modulo powers of 2 for [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we find new congruences for [Formula: see text] modulo 4, 18 and 36, infinite families of congruences modulo 2 and 4 for [Formula: see text], congruences modulo 2 and 3 for [Formula: see text], [Formula: see text], and congruences modulo 2 for [Formula: see text] and [Formula: see text]. We use simple p-dissections of Ramanujan's theta functions.

New congruences modulo 5 for the number of 2-color partitions

Let pk(n) be the number of 2-color partitions of n where one of the colors appears only in parts that are multiples of k. In this paper, we find some interesting congruences modulo 5 for pk(n) for k∈{2,3,4} by employing Ramanujan's theta function identities and some identities for the Rogers–Ramanujan continued fraction. The congruence for p2(n) was earlier proved by Chen and Lin with the aid of modular forms.

On Ramanujan's general theta function and a generalization of the Borweins' cubic theta functions

The Borwein brothers have introduced and studied three cubic theta functions. Many generalizations of these functions have been studied as well. In this paper, we introduce a new generalization of these functions and establish general formulas that are connecting our functions and Ramanujan's general theta function. Many identities found in the literature follow as a special case of our identities. We further derive general formulas for certain products of theta functions.

Bell’s Ternary Quadratic Forms and Tunnel’s Congruent Number Criterion Revisited

Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.

Infinite families of arithmetic identities and congruences for bipartitions with 3-cores

Let A3(n) denote the number of bipartitions of n that are 3-cores. By employing Ramanujan's simple theta function identities, we prove that A3(2n+1)=13 σ(6n+5), where σ(n) denotes the sum of the positive divisors of n. We also find several infinite families of arithmetic identities and congruences for A3(n), which include generalizations of some recent results on A3(n) by B.L.S. Lin (2014) [6].

Bilateral series and Ramanujan’s radial limits

Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$ approaches an even order $2k$ root of unity $\zeta$, \[\lim_{q\to \zeta} \big(f(q) - (-1)^k (1-q)(1-q^3)(1-q^5)\cdots (1-2q + 2q^4 - \cdots)\big) = O(1),\] and hinted at the existence of similar statements for his other mock theta functions. Recent work of Folsom-Ono-Rhoades provides a closed formula for the implied constant in this radial limit of $f(q)$. Here, by different methods, we prove similar results for all of Ramanujan's 5th order mock theta functions. Namely, we show that each 5th order mock theta function may be related to a modular bilateral series, and exploit this connection to obtain our results. We further explore other mock theta functions to which this method can be applied.

New parity results for broken 11-diamond partitions

The notation of broken k-diamond partitions was introduced in 2007 by Andrews and Paule. For a fixed positive integer k, let Δk(n) denote the number of broken k-diamond partitions of n. Recently, Radu and Sellers established numerous congruence properties for (2k+1)-cores by using the theory of modular forms, where k=2,3,5,6,8,9,11. Employing their congruences for (2k+1)-cores, Radu and Sellers obtained a number of nice parity results for Δk(n). In particular, they proved that for n⩾0, Δ11(46n+r)≡0(mod2), where r∈{11,15,21,23,29,31,35,39,41,43,45}. In this paper, we derive several new infinite families of congruences modulo 2 for Δ11(n) by using an identity given by Chan and Toh, and the p-dissection of Ramanujan's theta function f1 due to Cui and Gu. For example, we prove that for n⩾0 and k,α⩾1, Δ11(23α−2×23kn+23α−2s×23k−1+1)≡0(mod2), where s∈{5,7,10,11,14,15,17,19,20,21,22}. This generalizes the parity results for Δ11(n) discovered by Radu and Sellers.

INFINITE FAMILIES OF ARITHMETIC IDENTITIES FOR DOUBLED DISTINCT t-CORES FOR t = 3, 4, …, 10

Relations between doubled distinct and self-conjugate t-cores for some small values of t are derived. By employing Ramanujan's simple theta function identities, several infinite families of arithmetic identities for doubled distinct t-cores for t = 3, 4, …, 10 are found.