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I'm not sure if this is too detailed or not - I think anybody who has taken two semesters or more of calculus should be able to understand. The reference is to a page I personally wrote on ProofWiki
Two thing are noticible from the last equation, the function is zero at negative even integers (the trivial zeros) and that the function has dependence on the values on either side of the [[real line]] 1/2.
Two thing are noticible from the last equation, the function is zero at negative even integers (the trivial zeros) and that the function has dependence on the values on either side of the [[real line]] 1/2.
[[Category:Mathematics]]
==Relation to the Prime Numbers==
The zeta function is frequently said to "encode" the prime numbers in its structure. This was first noticed in conjunction with the [[Prime Number Theorem]], with the realization that the [[derivative]] of the zeta function, divided by the zeta function itself, could be expressed in a power series
<math>\frac{\zeta'(z)}{\zeta(z)} = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^z}
\ </math>
where
<math>\Lambda(n) = \begin{cases} \log p, & \mbox{if } n = p^m, m \in \mathbb{N} \\ 0, & \mbox{otherwise } \end{cases} </math>
and <math>p \ </math> is a prime number.
A more specific relation was later proved, that the number of prime numbers less than some value <math>x \ </math>, <math>\pi(x) \ </math>, is precisely related to a sum of sums of functions involving the zeroes of the zeta function. [http://www.proofwiki.org/wiki/Exact_Form_of_Prime-Counting_Function]