Harmonic series

A harmonic series is a form of the zeta function : $\zeta (x) = 1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+...$ .

When $\ x$ has a value less than or equal to one the function outputs infinity. Euler found that when $\ x=2$ , the zeta function outputs $\frac{\pi^2}{6}$ . Euler also realized that since every number is the multiplication of some order of primes, then the zeta function is equal to $(1+\frac{1}{2^x}+\frac{1}{4^x}+...)(1+\frac{1}{3^x}+\frac{1}{9^x}+...)...(1+\frac{1}{p^x}+\frac{1}{(p^2)^x}+...)...$

Riemann found that when complex numbers are the input to the zeta function, the resulting graph is that which aids in the finding of the exact value of $\ \pi (n)$ or the number of primes less than or equal to $\ n$ .

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Harmonic series

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