Hurwitz zeta function

The Hurwitz zeta function generalises the Riemann zeta function by shifting the summation index:

ζ (s, a) = \sum_{n = 0}^{\infty} \frac{1}{(n + a)^{s}}, Re (s) > 1, a \in C ∖ {0, - 1, - 2, \dots}

Key relation with powers

Since shifting $a$ by 1 simply drops the first term:

ζ (s, x) - ζ (s, x + 1) = x^{- s} = \frac{1}{x^{s}} .

Equivalently: $ζ (s, x + 1) = ζ (s, x) - x^{- s}$ .

This makes $ζ (s, \cdot)$ a discrete antiderivative of $x^{- s}$ : its forward difference equals the power $x^{- s}$ , with no extra terms.

Special case: $ζ (s, 1) = ζ (s)$ recovers the Riemann zeta function.