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 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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This entry was posted on Saturday, May 19th, 2012 at 4:42 pm and is filed under Euler's Greatest Hits, Mathematics History, Number Theory, Thrilling Math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.