Euler’s Product Formula: The Hidden Connection Between Prime Numbers and Infinite Series

As many have rightly noted, the previous fact about the connection between the coprimality of two numbers and π appeared on this channel quite recently — though that hardly makes it any less astonishing. But that last post was inspired by the very foundation underlying the link between primality and π: the Euler product formula.

In 1737, Leonhard Euler discovered a remarkable connection between prime numbers and infinite series. He showed that the sum of all fractions whose denominators are raised to a power can be expressed as a product that runs exclusively over the prime numbers.

The Core Idea Behind the Euler Product Formula

Prime numbers are the building blocks of all natural numbers. Euler found a way to “sieve” the natural numbers — eliminating composites — and express an infinite series as a product over primes. This is not merely beautiful; it is profound. Prime numbers seem to encode every property of the integers themselves.

How to Prove the Euler Product Formula

Euler used a sieving argument to progressively remove from the series all terms divisible by primes.

• First, subtract every fraction whose numerator is divisible by 2. What remains contains only terms with no factor of two in the denominator.
• Next, eliminate multiples of 3, then 5, then 7, and so on — until the series retains only fractions tied to the primes themselves.
• At each step, a product of factors associated with prime numbers emerges. The more primes we account for, the more precise the result. In the limit, passing through all primes, we arrive at Euler’s product formula.

Why the Riemann Zeta Function Makes This Formula So Significant

The formula reveals a fundamental connection between addition (series) and multiplication (products). This bridge opens a path to a deeper understanding of numbers and plays a central role in research on prime numbers, analytic number theory, and even one of the hardest unsolved problems in mathematics — the Riemann hypothesis.

The Riemann zeta function, which sits at the heart of Euler’s product formula, continues to be one of the most studied objects in all of mathematics precisely because that connection between primes and infinite sums runs so deep.

Feel how prime numbers surprise us again and again? Mathematics is always more than it appears at first glance.

For a detailed and engaging walkthrough, see the video by Dr. Trefor Bazett.