Riemann Hypothesis/Introduction to the Zeta function

Definition 1
Theorem 1

Where denotes a product over the primes.


Note that,

And hence,

One will notice that every other term has been removed. It then follows that,


One may notice that this process sieves the RHS, meaning that,

It therefore follows that,

Theorem 2

Multiplying the integrand through ,

Writing as a power series,

Using the substitution

Using properties of infinite series',

By the definition of the function,

Which is true by definition.