Euler product formula

The Euler product formula, considered one of the elegant equations in all of mathematics, is as follows:

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ as prime}}}{\frac {1}{1-p^{-s}}}$

The Riemann zeta function is the left side of the equation, and is the basis for the greatest unsolved math problem today, the Riemann Hypothesis. The Riemann zeta function can be expanded as follows:

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=1+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{4^{s}}}+{\frac {1}{5^{s}}}+\ldots$

while the right side of the equation is a product over all prime numbers represented by p:

$\prod _{p{\text{ as prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdots {\frac {1}{1-p^{-s}}}\cdots$

Euler developed the foregoing product formula that carries his name by using integers. Bernhard Riemann extended it to complex numbers.

See also

• Fermat's Last Theorem Blogspot