Riemann zeta function
The functional equation was established by Riemann in his 1859 paper "" and used to construct the analytic continuation in the first place. An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function (the alternating zeta function):
Riemann also found a symmetric version of the functional equation applying to the xi-function:
These two conjectures opened up new directions in the investigation of the Riemann zeta function.
The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.
The critical line theorem asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is 1.)What is the probability that two numbers selected at random are relatively prime?
For sums involving the zeta function at integer and half-integer values, see rational zeta series.
in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part of s is greater than one, we have
holds true, which may be used for a numerical evaluation of the zeta function.
This can be used recursively to extend the Dirichlet series definition to all complex numbers.