Riemann zeta function

The zeta-function is a function very important to the Riemann Hypothesis. The function is $\zeta (x)=\displaystyle\sum_{i=1}^{\infty}\frac{1}{i^x}=1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+\cdots$ When $x$ is less than or equal to one, the sum equals infinity. Euler showed that when $x=2$ , the sum is equal to $\frac{\pi^2}{6}$ . Euler also found that since every number is the product of a certain combination of prime numbers, the zeta-function can also be expressed as $\zeta (x)=(\frac{1}{(2^0)^x}+\frac{1}{(2^1)^x}+\frac{1}{(2^2)^x}+\cdots)(\frac{1}{(3^0)^x}+\frac{1}{(3^1)^x}+\frac{1}{(3^2)^x}+\cdots)(\frac{1}{(5^0)^x}+\frac{1}{(5^1)^x}+\frac{1}{(5^2)^x}+\cdots)\cdots(\frac{1}{(p^0)^x}+\frac{1}{(p^1)^x}+\frac{1}{(p^2)^x}+\cdots)\cdots$

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Riemann zeta function