411 REFERENCE · Wikibooks
Riemann Hypothesis/Introduction to the Zeta function
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0 100% #FFFFFF none Definition 1
0 100% #FFFFFF none Theorem 1
Where denotes a product over the primes.
Proof
Note that,
And hence,
One will notice that every other term has been removed. It then follows that,
Subtracting,
One may notice that this process sieves the RHS, meaning that,
It therefore follows that,
0 100% #FFFFFF none 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.