Congruence Properties of Partitions: Part 2

Continuing our journey of partition congruences from the last post we now prove the congruences modulo $7$ and $11$.

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Congruence Properties of Partitions: Part 1

Introduction

We know that any positive integer greater than $1$ can be expressed as a product of prime numbers in a unique fashion ignoring the order of factors. This is one of the most basic results in number theory and is aptly called the fundamental theorem of arithmetic. This result also shows that prime numbers are the building blocks for all integers and this justifies the importance given to prime numbers in number theory.

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Proof of Chudnovsky Series for 1/π(PI)

In 1988 D. V. Chudnovsky and G. V. Chudnovsky (now famous as "Chudnovsky Brothers") established a general series for $\pi$ by extending Ramanujan's ideas (presented in this series of posts). It can be however shown that their general series can be derived using Ramanujan's technique. Chudnovsky's approach has the advantage that using class field theory the algebraic nature of parameters in the general series can be determined and this greatly aids in the empirical evaluation of the parameters and thereby providing an actual series consisting of numbers.

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Certain Lambert Series Identities and their Proof via Trigonometry: Part 2

In the last post we saw that using a trigonometric identity Ramanujan was able to express the functions $S_{2n - 1}(x)$ or equivalently $\Phi_{0, 2n - 1}(x)$ in terms of simpler functions $P(x), Q(x), R(x)$. Continuing our journey further we start with the equation: \begin{align}&\left(\frac{1}{4}\cot\frac{\theta}{2} + \frac{x\sin\theta}{1 - x} + \frac{x^{2}\sin 2\theta}{1 - x^{2}} + \frac{x^{3}\sin 3\theta}{1 - x^{3}} + \cdots\right)^{2}\notag\\ &\,\,\,\,= \left(\frac{1}{4}\cot\frac{\theta}{2}\right)^{2} + \frac{x\cos\theta}{(1 - x)^{2}} + \frac{x^{2}\cos 2\theta}{(1 - x^{2})^{2}} + \frac{x^{3}\cos 3\theta}{(1 - x^{3})^{2}} + \cdots\notag\\ &\,\,\,\,+\frac{1}{2}\left\{\frac{x(1 - \cos\theta)}{1 - x} + \frac{2x^{2}(1 - \cos 2\theta)}{1 - x^{2}} + \frac{3x^{3}(1 - \cos 3\theta)}{1 - x^{3}} + \cdots\right\}\tag{1}\end{align} which was established in the last post.

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Certain Lambert Series Identities and their Proof via Trigonometry: Part 1

Introduction

This is yet another post based on a paper of Ramanujan titled "On certain arithmetical functions" which appeared in Transactions of the Cambridge Philosophical Society in 1916. In this paper Ramanujan provided a lot of identities concerning Lambert series and thereby deduced many relations between various divisor functions. Apart from the amazing results proved in this paper, what I liked most is the very elementary approach followed by Ramanujan compared to the methods of modern authors who are seduced by the modular form.

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Elementary Approach to Modular Equations: Ramanujan's Theory 7

Continuing from previous post we proceed to derive further modular equations of degree $5$ in this post. Clearly in order to establish such equation we need to establish further theta function identities. This time we establish an identity concerning Ramanujan's $\psi$ function.

Identity Concerning $\psi(q)$ of Degree $5$

We will establish the following identity $$\psi^{2}(q^{2}) - q^{2}\psi^{2}(q^{10}) = \frac{\phi(-q^{10})f(-q^{10})}{\chi(-q^{2})}\tag{1}$$

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Elementary Approach to Modular Equations: Ramanujan's Theory 6

The Fundamental Formulas

In this post we will continue our journey of modular equations and derive a host of these mostly by using Lambert series for various theta functions. The following formula (see equation $ (14)$ of this post) will be of great help here: $$\phi^{2}(-ab)\,\frac{f(a, b)}{f(-a, -b)} = 1 + 2\sum_{n = 1}^{\infty}\frac{a^{n} + b^{n}}{1 + a^{n}b^{n}}\tag{1}$$

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Elementary Approach to Modular Equations: Ramanujan's Theory 5

Quintuple Product Identity

We first establish an identity similar to Jacobi's Triple Product which involves five factors and is quite useful in establishing various other identities involving q-series and products. This was first introduced in the mathematical literature by G. N. Watson in order to prove some of Ramanujan's theorems. The quintuple product identity is given by \begin{align}&\prod_{n = 1}^{\infty}(1 - q^{n})(1 - q^{n}z)(1 - q^{n - 1}z^{-1})(1 - q^{2n - 1}z^{2})(1 - q^{2n - 1}z^{-2})\notag\\ &\,\,\,\,\,\,\,\,= \sum_{n = -\infty}^{\infty}q^{n(3n + 1)/2}(z^{3n} - z^{-3n - 1})\notag\end{align}

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Elementary Approach to Modular Equations: Ramanujan's Theory 4

Lambert Series

In this post we will focus our attention on series of the form: $$\sum_{n = 0}^{\infty}a_{n}\cdot\frac{q^{b_n}}{1 \pm q^{c_n}}$$ which are more popularly known as Lambert Series. We will not deal with the general theorems concerning such series but will restrict ourselves to the Lambert series for the theta functions and study some identities involving these series.

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