TCplx RiemannZeta(const TCplx &Z, int n = 64);
Riemann Zeta function.
| # | Name | Type | Description |
|---|---|---|---|
| 1 | Z | const TCplx & | complex value at which the Riemann zeta function is evaluated. |
| 2 | n = 64 | int | number of terms used in the complex series approximation (default 64). |
Returns: the Riemann zeta function evaluated at the complex point Z.
The Riemann zeta function of a complex variable z is initially defined by the series
zeta(z) = sum_(n=1)^(inf)1/n^z ,
for Re(z) > 1, and is analytically continued to all complex z != 1 through the functional equation
zeta(z) = 2^z pi^(z-1) sin((pi z)/2) Gamma(1-z) zeta(1-z) .
It is related to the Dirichlet lambda and eta functions by
zeta(z)/2^z = lambda(z)/(2^z-1) = eta(z)/(2^z-2) .
Domain: any complex Z with z != 1. Defined behavior: it is computed as zeta(z)=eta(z) 2^z/(2^z-2) from the accelerated Probabilities::DirichletEta; at z=1 (the simple pole) the result is complex infinity. On the real axis the imaginary part of the result is zero, and the function is strictly decreasing for real z > 1. Known values: zeta(2)=pi^2/6, zeta(4)=pi^4/90.