Hurwitz zeta function

zeta(s, q) = sum over k >= 0 of (q + k)^-s, by Euler-Maclaurin. It is the normalising constant of the discrete power law truncated below at q, which is why it is here; zeta(s, 1) is the Riemann zeta.

Usage

hurwitz_zeta(s, q = 1)

Arguments

s

Exponent, which must exceed 1 for the series to converge.

q

Lower limit, which must be positive.

Value

A numeric vector the length of s.

References

The Euler-Maclaurin expansion used here – direct terms to q + N, then the integral tail, then the Bernoulli-number corrections – is the standard evaluation; see Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions, Sec. 23.2. (Not in the local corpus; cited from the published reference. The implementation is checked against pi^2/6, pi^4/90, Apery's constant and the shift identity, which is stronger evidence than the citation.)

Examples

# The Riemann zeta at even integers has a closed form.
c(hurwitz_zeta(2), pi^2 / 6)
#> [1] 1.644934 1.644934
c(hurwitz_zeta(4), pi^4 / 90)
#> [1] 1.082323 1.082323

# Apery's constant.
hurwitz_zeta(3)
#> [1] 1.202057

# Shifting the lower limit removes exactly the leading term.
hurwitz_zeta(2.5, 3) - hurwitz_zeta(2.5, 4)
#> [1] 0.06415003
3^-2.5
#> [1] 0.06415003

# Outside the domain of convergence there is no value to return.
hurwitz_zeta(1)
#> [1] NA