The Golden Key

Figure 1: this is the book that mentions the
Golden Key, a description of its contents is
included at the end of this post

The sets of infinite natural numbers and infinite prime numbers are related by a formula given by Euler, which, famously known as the Golden Key, is given by:$$

\prod_{ p} \frac{1}{1-\displaystyle \frac{1}{p^{\,s}}}=\sum_n \frac{1}{n^{\, s}} \text{ where } s>1

$$where the left-hand-side products are carried over all the prime numbers \(p\) and the right-hand-side sum is carried over all the natural numbers \(n\).

For the range of numbers from 1 to 100,000 (with \(k\)=100,000), the results are as follows:$$
\prod_{p=2}^k \frac{1}{1-\displaystyle \frac{1}{p^{\, s}}}=1.64493274720203 \text{ and } \sum_{n=1} ^{k} \frac{1}{n^s}=1.64492406679823

$$The right hand side of Euler's formula is of course the Riemann Zeta function and so the equation can be rewritten as:$$

\zeta(s)=\sum_{n \geq 1}n^{-s}=\prod_p (1-p^{-s})^{-1}

$$

Figure 2

which is an easier form to remember (s can be any complex number with ℜs>1). The relationship at first seems strange, linking as it does a sum involving the reciprocals of the natural numbers and a product involving the reciprocals of the prime numbers. However, as this blog post points out, the formula is nothing but a fancy way of writing out the Sieve of Eratosthenes. The post goes on to derive the formula. I've just taken a screenshot of the working (Figure 2) rather than type it all out using LaTeX (lazy I know). In fact, this post is only the first in a long series of posts (from September 2013 to May 2017) dealing with Understanding the Riemann Hypothesis.

I came across the Golden Key when perusing Kumar Asok Mallik's book The Story of Numbers during his introduction to prime numbers on page 23. This is quite an interesting book that I've added to my Calibre library. The description of the book in the metadata is as follows:

This book is more than a mathematics textbook. It discusses various kinds of numbers and curious interconnections between them. Without getting into hardcore and difficult mathematical technicalities, the book lucidly introduces all kinds of numbers that mathematicians have created. Interesting anecdotes involving great mathematicians and their marvellous creations are included. The reader will get a glimpse of the thought process behind the invention of new mathematics. 

Starting from natural numbers, the book discusses integers, real numbers, imaginary and complex numbers and some special numbers like quaternions, dual numbers and p-adic numbers. Real numbers include rational, irrational and transcendental numbers. Iterations on real numbers are shown to throw up some unexpected behaviour, which has given rise to the new science of "Chaos". Special numbers like e, pi, golden ratio, Euler's constant, Gauss's constant, amongst others, are discussed in great detail.The origin of imaginary numbers and the use of complex numbers constitute the next topic. 

It is shown why modern mathematics cannot even be imagined without imaginary numbers. Iterations on complex numbers are shown to generate a new mathematical object called 'Fractal', which is ubiquitous in nature. Finally, some very special numbers, not mentioned in the usual textbooks, and their applications, are introduced at an elementary level.The level of mathematics discussed in this book is easily accessible to young adults interested in mathematics, high school students, and adults having some interest in basic mathematics. The book concentrates more on the story than on rigorous mathematics.

If I can read an entry a day from this book, I'll soon be a wiser man mathematically. Here is a link to a very useful series of slides explaining the importance of the Riemann zeta function and also mentioning the Golden Key.

on January 16th 2021

mainly improving the look of the mathematical expressions