## Euler’s Product Formula

The great mathematician Leonard Euler (1707 – 1783) was comfortable working in any branch of mathematics (and he even invented some new branches, such as topology). This post is about one of his many results in number theory. I will describe a  formula that he discovered in 1737  which involves fractions formed from prime numbers. The formula is an odd and surprising result, and its derivation illustrates Euler’s remarkable ingenuity.

For more than one hundred years, this formula was just another curiosity among Euler’s many results. Then, in 1859 Bernard Reimann used it as the starting point in his landmark paper on prime numbers.
Euler began with the familiar geometric series:
As shown in high school math classes, if r < 1, this infinite series converges to the value:
(in note 1 below, I show the derivation of this formula). Repeating decimal numbers are a familiar example of this type of series. For example
For reasons that will become apparent, Euler imagines geometric series that have a very special type of r value:
Where p is one of the prime numbers, and s is an integer that is greater than 1. The first three such series would look like this:
Now Euler does something bazaar: he multiplies all of these series together. When we multiply the right hand sides, the resulting product will be another infinite series, with each term formed by multiplying one term from each of the original series.
Note that the denominators are the integers from 1 to infinity, with each raised to the s power. Now we see the point of using only prime numbers for the original series; each of the integers can be produced from some combination of the prime factors in the original series.
In summary, the product is
This is the result, and there is hint of something deep here. The left hand side involves the prime numbers, and the right hand side has reciprocals of the integers. Also, the derivation is  typical of Euler’s audacity; he has an infinite product equal to an infinite sum, with the infinite sum being the infinite  product of other infinite sums.
Now, consider the particular case of s=2:
As a test, I computed the left side through prime number 57:
Product =  1.6414
The sum through n = 50 gives:
Sum=1.6251
This particular series happens to have a known sum: pi ^ 2 / 6. Euler discovered this early in his career, and I have described his discovery in another post.
Note 1 – the geometric series formula:
We want to find
First, multiply this by r:
Note 2 – Reimann begins a revolution
Reimann’s 1859 paper began by presenting a generalized version of the Euler formula:
Here, s can  be a complex number, not just an integer, and the result is now called the Reimann Zeta function. Reimann showed that the Zeta function has profound connections to the distribution of the prime numbers, and these connections are contingent on a particular postulate about Zeta being true.  The quest to prove this hypothesis (the “reimann Hypothesis”) is generally considered the greatest unsolved problem in mathematics.

For more than one hundred years, this formula was just another curiosity among Euler’s many results. Then, in 1859 Bernhard Reimann used it as his starting point in his landmark paper on prime numbers.

Euler began with the familiar geometric series:

$S=1+r+r^2+r^3+\dots$

As shown in high school math classes, if $$\mid r \mid < 1$$, this infinite series converges to the value:

$S=\frac{1}{1-r}$

(in note 1 below, I show the derivation of this formula). A familiar example of this type of series is a repeating decimal number.  For example

$0.121212\ldots = \frac{12}{100}\left( 1+\left( \frac{1}{100} \right) +\left( \frac{1}{100}\right) ^2+\ldots\right)=\frac{12}{100} \dot \frac{1}{1-\frac{1}{100}}=\frac{4}{33}$

For reasons that will become apparent, Euler imagines geometric series that have a very special type of r value:

$r=\frac{1}{p^s}$

where $$p$$ is one of the prime numbers, and $$s$$ is an integer that is greater than 1. The first three such series would look like this:

\begin{align*}\frac{1}{1 - 1 / 2^s}= 1+\frac{1}{2^s}+\frac{1}{2^{2s}}+\frac{1}{2^{3s}}+\ldots \\ \frac{1}{1 - 1 / 3^s}=1+\frac{1}{3^s}+\frac{1}{3^{2s}}+\frac{1}{3^{3s}}+\ldots \\ \frac{1}{1 - 1 / 5^s}=1+\frac{1}{5^s}+\frac{1}{5^{2s}}+\frac{1}{5^{3s}}+\ldots \end{align*}

Now Euler does something bizarre: he multiplies all of these series together. When we multiply the right hand sides, the resulting product will be another infinite series, with each term formed by multiplying one term from each of the original series. The result is:

$1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\ldots$

Note that the denominators are the integers from 1 to infinity, with each raised to the $$s$$ power. Now we see the point of using only prime numbers for the original series; each of the integers can be produced from some combination of the prime numbers in the original series.

In summary, multiplying the series together gives:

$\Pi_p \frac{1}{1-1 / p^s}=1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\cdots \tag{1}$

This is the result, Euler’s Product Formula, with $$\Pi_p$$ denoting multiplication over the primes.

There is hint of something deep here. The left hand side involves the prime numbers, and the right hand side has reciprocals of the integers. Just to look at the formula, is would be far from obvious how the two sides could be equal.

Also, note that the derivation is  typical of Euler’s audacity; he has an infinite product equal to an infinite sum, with the infinite sum being the infinite  product of other infinite sums!

To make the formula seem less abstract, consider the particular case of $$s=2$$:

$\Pi_p \frac{1}{1-1 / p^2}=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^4}+\ldots$

As a test, I computed the left side through the prime number 57:

$\frac{1}{1 - 1 / 2^2} \cdot \frac{1}{1 - 1 / 3^2} \cdot \frac{1}{1 - 1 / 5^2} \ldots \frac{1}{1 - 1 / 57^2} = 1.6414$

The first 50 terms of the right hand side give:

$1 + \frac{1}{2^2} + \frac{1}{3^2} + \ldots \frac{1}{50^2} =1.6251$

This is looking reasonable: it is easy to believe that the infinite product will become equal to the infinite sum.

This particular series happens to have a known sum: $$\pi^ 2 / 6 \approx 1.64493$$. Euler himself discovered this early in his career, and I have described his discovery in another post.

#### Note 1 – the geometric series formula:

We want to find

$s=1+r^2+r^3+\ldots$

First, multiply this by r:

$rs=r+r^2+r^3+\ldots = s-1$

From $$rs = s-1$$, we can just solve for s: $$s={1 \over 1-r}$$.

#### Note 2 – Reimann begins a revolution

In 1859, Bernhard Reimann published a 10 page paper concerning the distribution of prime numbers. It may be the most influential mathematical article of the past 200 years. Riemann  begins the paper with the Euler Product Formula:

$\zeta(s) = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\ldots = \Pi_p \frac{1}{1-1 / p^2}$

Now however, $$s$$ can  be a complex number, not just an integer, and the result is now called the Reimann Zeta function. Reimann showed that the Zeta function has profound connections to the distribution of the prime numbers, but these connections are contingent on a particular postulate about Zeta being true.  Reimann did not offer a proof of the postulate in his paper (though he hinted that he was close to having one). The quest to prove this (the “Reimann Hypothesis”) is generally considered the greatest unsolved problem in mathematics.

#### Note 3 – There are an infinite number of primes, added July 10, 2012

In an earlier post, I showed how Euclid proved that there are an infinite number of primes. In the current post I failed to mention that Euler’s Product Formula provides another proof of this important result. In fact, it was the first new proof since Euclid, a gap of about 2,000 years.

To see how this works, let’s write the product formula with $$s=1$$:

$\Pi_p \frac{1}{1-1 / p}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$

The infinite series on the right, called the Harmonic Series, can be shown to diverge; it goes off to infinity. (I gave a proof for this in another post). Thus, the left side is infinite, and this can only happen if there are an infinite number of factors in the product on the left – that is, an infinite number of primes.

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### 3 Responses to “Euler’s Product Formula”

1. prdxp Says:

I like the articles in this blog, especially mathematics and astronomy.
They are well explained.
Thanks for the writing such article. And hope to read more of such articles…!

2. BJ Sinon Says:

Thanks, but a minor correction…. When you write, “Now Euler does something bazaar: he multiplies all of these series together…” you mean he does something “bizarre.” There’s enough poor grammar in common use (think of the number of “invites” you receive or how one is described as “messaging” someone) without ading to it.

3. curiousCharacter Says:

Good point about my use of the word “bazaar” vs. “bizarre”, and I made a correction. The words are very different, and this is an example of how spell checking can produce a false sense of security that all is well.