Hurwitz zeta function

Computes the Hurwitz zeta function \(\zeta(s, q) = \sum_{k \ge 0} (q + k)^{-s}\) for a vector of exponents s and one lower limit q. It is the normalising constant of the discrete power law truncated below at q, which is why it is here (see hilltlidx()); zeta(s, 1) is the Riemann zeta.

Usage

hurwzeta(s, q = 1)

Arguments

s

Numeric vector of exponents. The series converges for s > 1; below 1 the value is the analytic continuation (so hurwzeta(0.5, 0.5) is \(-0.6049\)), and s = 1 is a pole (NA).

q

Numeric scalar, the lower limit; default 1 (the Riemann zeta). Any real number but zero or a negative integer (poles, NA). For a negative non-integer q the value is real only for a whole s; a fractional s gives NA.

Value

A numeric vector the length of s.

Details

For \(s > 1\) and \(q > 0\) the series is evaluated in compiled code by Euler-Maclaurin summation: direct terms, the integral of the tail, and Bernoulli-number corrections. For \(s < 1\) the value is the analytic continuation: exact Bernoulli polynomials for whole \(s > -5\) (\(\zeta(-n, q) = -B_{n+1}(q)/(n+1)\)), Riemann's functional equation at \(q = 1\), Hurwitz's Fourier series for \(s \le -5\), Hermite's integral (by stats::integrate()) for \(-5 < s < 1\), and the asymptotic series for large \(q\), with the shift \(\zeta(s, q) = q^{-s} + \zeta(s, q + 1)\) to move \(q\) into range. A negative non-integer q uses the reflection of its negative terms.

Values that do not exist are NA, not errors: s = 1 (a pole), q zero or a negative integer (poles), a fractional s with a negative q (a complex value), and a non-finite s. Errors when s or q is logical or not numeric, and when q is not a single finite number.

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(hurwzeta(2), pi^2 / 6)
#> [1] 1.644934 1.644934
c(hurwzeta(4), pi^4 / 90)
#> [1] 1.082323 1.082323

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

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

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

# several exponents at once, including the analytic continuation
hurwzeta(c(2, 0.5, -1))  # -1/12 at s = -1
#> [1]  1.64493407 -1.46035451 -0.08333333

# one lower limit per call
try(hurwzeta(2, q = c(1, 2)))
#> Error : `q` must be a single finite number