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From the Fun Fact files, here is a Fun Fact at the Advanced level:

# Euler's Product Formula

Here is an amazing formula due to Euler:

SUMn=1 to infinity n-s = PRODp prime (1 - p-s)-1 .

What's interesting about this formula is that it relates an expression involving all the positive integers to one involving just primes!

And you can use it to prove there must be infinitely many primes. For, if there were only finitely many primes, then the right side of the expression is a finite product, and in particular for s=1. But for s=1, the left side of the equation is the harmonic series which we know must diverge! This is a contradiction, so there must be infinitely many primes.

Presentation Suggestions:
Interested students may wish to take a few terms on the right hand side, use a power expansion, and multiply them out... to get an idea of why the equality holds.

The Math Behind the Fact:
The left hand side, when s is viewed as a complex variable, is also known as the Riemann zeta function. Because of the above relationship, the study of zeta functions is closely related to the study of the distribution of primes!

How to Cite this Page:
Su, Francis E., et al. "Euler's Product Formula." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    infinite series, number theory, Riemann zeta function, distribution of primes
Subjects:    calculus, analysis, number theory
Level:    Advanced
Fun Fact suggested by:   Lesley Ward
Suggestions? Use this form.
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