Function zeta 

pub fn zeta(s: &Expression) -> Expression

Riemann zeta function ζ(s)

The Riemann zeta function extends the series Σ 1/n^s to the entire complex plane via analytic continuation.

Mathematical Properties

• ζ(2) = π²/6 (Basel problem)
• ζ(4) = π⁴/90
• ζ(6) = π⁶/945
• ζ(8) = π⁸/9450
• ζ(10) = π¹⁰/93555
• ζ(0) = -1/2
• ζ(-1) = -1/12 (famous result used in string theory)
• ζ(-2n) = 0 for positive integers n (trivial zeros)
• ζ(-3) = 1/120
• ζ(-5) = -1/252
• ζ(-7) = 1/240
• Pole at s=1 with residue 1
• Functional equation: ζ(s) = 2^s π^(s-1) sin(πs/2) Γ(1-s) ζ(1-s)

Arguments

• s - Expression argument to evaluate zeta function at

Examples

use mathhook_core::functions::special::zeta;
use mathhook_core::{Expression, Number};

let zeta_2 = zeta(&Expression::Number(Number::Integer(2)));