Module zeta 

Riemann zeta function

Implements the Riemann zeta function ζ(s) with analytic continuation to the entire complex plane except for a simple pole at s=1.

Mathematical Background

The Riemann zeta function is defined for Re(s) > 1 as: ζ(s) = Σ(n=1 to ∞) 1/n^s

It extends to the entire complex plane via analytic continuation, with a simple pole at s=1 with residue 1. The functional equation relates ζ(s) to ζ(1-s), enabling evaluation across all complex values.

The zeta function is central to number theory, appearing in the prime number theorem and the Riemann hypothesis.

Performance

Uses Euler-Maclaurin acceleration (50 terms) for 200x speedup vs direct summation (10,000 terms). Achieves 14-digit accuracy.

Functions

zeta

Riemann zeta function ζ(s)

zeta_numerical

Numerical evaluation of Riemann zeta function