Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
or
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
 
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

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.
4.30
 
current
rating
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!