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==Relation to the Prime Numbers==

==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

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 (calculus)|derivative]] of the zeta function, divided by the zeta function itself, could be expressed in a power series

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]

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]