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Riemann Zeta Function

Source: R/zeta_functions.R
zeta.Rd

Computes the Riemann zeta function zeta(z), one of the most important functions in analytic number theory.

Mathematical Definition:

The Riemann zeta function is defined by the series: $$\zeta(z) = \sum_{n=1}^\infty \frac{1}{n^z}$$

for Re(z) > 1, and by analytic continuation elsewhere.

Special Values:

  • zeta(2) = pi^2/6 (Basel problem)

  • zeta(4) = pi^4/90

  • zeta(0) = -1/2

  • zeta(-1) = -1/12

  • Closed forms exist for all even positive integers and all negative integers

  • For odd positive integers > 1, values are computed numerically

Implementation:

The function uses different computational strategies depending on the argument:

  • For 0 < z < 1: Rational approximation form

  • For 1 < z < 4: Rational approximation around nearby integers

  • For z > 4: Simple rational approximation series

  • Reflection formula for negative arguments

  • Pre-computed cached values for positive odd integers

  • Specialised rational approximations for standard floating-point precisions

Applications:

The Riemann zeta function appears in number theory (distribution of primes), physics (quantum field theory, statistical mechanics), and probability theory. The famous Riemann Hypothesis concerns the non-trivial zeros of this function.

Usage

zeta(z)

Arguments

z

Real number input

Value

The value of the Riemann zeta function zeta(z).

See also

Boost Documentation for more details on the mathematical background.

Examples

# Riemann Zeta Function
zeta(2)  # Should return pi^2 / 6 ~= 1.6449340668
#> [1] 1.644934
zeta(3)  # Apery's constant ~= 1.2020569032
#> [1] 1.202057
zeta(4)  # pi^4 / 90 ~= 1.0823232337
#> [1] 1.082323