Final IANT Meeting Today

By Susam Pal on 01 Oct 2021

Introduction

We have been reading the book Introduction to Analytic Number Theory (Apostol, 1976) since March 2021. It has been going consistently since then and the previous few posts on this blog provide an account of how this journey has been so far. After about seven months of reading this book together, we are having our final meeting for this book today. This is going to be the 120th meeting of our book discussion group. The meeting notes from all previous reading sessions are archived at IANT Notes. We will discuss the final two pages of this book today and complete reading this book.

In the meeting today, we will look at some applications of the recursion formula related to partition functions that we learnt earlier. Here is an excerpt from the book that shows a specific example that demonstrates the richness and beauty of concepts one can discover while studying analytic number theory:

Equation (24) becomes $np(n) = \sum_{k=1}^n \sigma(k) p(n - k).$ a remarkable relation connecting a function of multiplicative number theory with one of additive number theory.

Now what equation (24) contains is not important for this post. Of course, you can refer to the book if you really want to know what equation (24) is. We learnt to prove that equation in the penultimate meeting for this subject yesterday. In this post, I will emphasise how indeed this equation is remarkable.

The Divisor Sum Function

The divisor sum function $$\sigma(n)$$ represents the sum of all positive divisors of $$n.$$ Here are some examples: \begin{align*} \sigma(1) &= 1, \\ \sigma(2) &= 1 + 2 = 3, \\ \sigma(3) &= 1 + 3 = 4, \\ \sigma(4) &= 1 + 2 + 4 = 7, \\ \sigma(5) &= 1 + 5 = 6. \end{align*} We have spent a good amount of time with this function in the initial chapters of the book. However, for the purpose of this blog post, the definition and the examples above are good enough.

The Unrestricted Partition Function

The $$p(n)$$ function is the unrestricted partition function. It represents the number of ways $$n$$ can be written as a sum of positive integers $$\le n.$$ Further, we let $$p(0) = 1.$$ Here are some examples: \begin{align*} p(1) &= 1, \\ p(2) &= 2, \\ p(3) &= 3, \\ p(4) &= 4, \\ p(5) &= 7. \end{align*} Let me illustration the last value. The integer $$5$$ can be represented as a sum of positive integers $$\le 5$$ in 7 different ways. They are: $$5,$$ $$4 + 1,$$ $$3 + 2,$$ $$3 + 1 + 1,$$ $$2 + 2 + 1,$$ $$2 + 1 + 1 + 1,$$ and $$1 + 1 + 1 + 1 + 1.$$ Thus $$p(n) = 5.$$

The divisor sum function comes from multiplicative number theory. The partition function comes from additive number theory. Yet these two very different things get linked together in the formula mentioned in the excerpt included above. Here is the formula once again: $np(n) = \sum_{k=1}^n \sigma(k) p(n - k).$ How beautiful! How nicely the divisor sum function and the unrestricted partition function appear together elegantly in a single equation! Further, this equation provides a recursion formula for the partition function. Here is an illustration of this equation with $$n = 5$$: $5 \cdot p(5) = 5 \cdot 7 = 35.$ \begin{align*} \sum_{k=1}^5 \sigma(k) p(5 - k) &= \sigma(1) p(4) + \sigma(2) p(3) + \sigma(3) p(2) + \sigma(4) p(1) + \sigma(5) p(0) \\ &= (1)(5) + (3)(3) + (4)(2) + (7)(1) + (6)(1) \\ &= 5 + 9 + 8 + 7 + 6 \\ &= 35. \end{align*} We will go through this topic once more in the meeting today, so if you are interested to see this formula worked out in a step-by-step manner, do join our final meeting for this book.