Two Mathematical Identities

This page discusses two mathematical equations. Both of these equations were discovered by the great mathematician Leonhard Euler, whose extensive mathematical publications have served as the foundation for much of present-day mathematics. One has been presented to nonmathematicians as something profound, but is perhaps not quite as profound as claimed, and the other is more genuinely beautiful, but has been known mostly only to mathematicians.

The first one is:

  i * pi
e        = -1

and the second one is:

               1      1      1      1
zeta(x) = 1 + ---- + ---- + ---- + ---- + ... =
                x      x      x      x
               2      3      4      5

     1              1              1              1
------------ * ------------ * ------------ * ------------ * ...
       1              1              1              1
( 1 - ---- )   ( 1 - ---- )   ( 1 - ---- )   ( 1 - ---- )
        x              x              x              x
       2              3              5              7

which is true when (the real part of x) is bigger than 1.

• e to i times pi equals minus one
• The Riemann Zeta Function

If your browser were to support MathML, (at the present time, the major browsers do not; a freely-available browser, Amaya, does; however, it does not display frames conventionally, and it does not maintain bookmarks for you; for these and other reasons, one can't really use it as one's principal browser at this time: however, it is primarily intended for use as an HTML editor rather than a browser in any case) the following version will be readable for you:

The first one is:

$e^{i\pi }=-1$

and the second one is:

$\zeta \left(x\right)=1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+\frac{1}{5^x}+...=\sum _{i=1}^{\infty }\frac{1}{i^x}$

$=\frac{1}{(1-\frac{1}{2^x})}\times \frac{1}{(1-\frac{1}{3^x})}\times \frac{1}{(1-\frac{1}{5^x})}\times \frac{1}{(1-\frac{1}{7^x})}\times ...=\underset{p\in P}{\Π}\frac{1}{(1-\frac{1}{p^x})}$

which is true when (the real part of x) is bigger than 1.