Riemann Zeta Function - The Functional Equation

The Functional Equation

The Riemann zeta function satisfies the functional equation (also called the Riemann('s) functional equation)

\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s) \!,

where Γ(s) is the gamma function, which is an equality of meromorphic functions valid on the whole complex plane. This equation relates values of the Riemann zeta function at the points s and 1 − s. The functional equation (owing to the properties of sin) implies that ζ(s) has a simple zero at each even negative integer s = −2n — these are known as the trivial zeros of ζ(s). For s an even positive integer, the product sin(πs/2)Γ(1−s) is regular and the functional equation relates the values of the Riemann zeta function at odd negative integers and even positive integers.

The functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude 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 (alternating zeta function)

\eta(s)= \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} = (1-{2^{1-s}})\zeta(s).

Incidentally, this relation is interesting also because it actually exhibits ζ(s) as a Dirichlet series (of the η-function) which is convergent (albeit non-absolutely) in the larger half-plane σ > 0 (not just σ > 1), up to an elementary factor.

Riemann also found a symmetric version of the functional equation, given by first defining

The functional equation is then given by

(Riemann defined a similar but different function which he called ξ(t).)

Read more about this topic:  Riemann Zeta Function

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