q-Zeta Function

📐 Definition

One common q-analogue (among several in the literature) is the exponentially weighted series for $0<q<1$ :

\zeta_q(s) = \sum_{n=1}^{\infty} \frac{q^{n}}{[n]_q^{\,s}}, \qquad [n]_q = \frac{1 - q^{n}}{1 - q}

Domain and Codomain

For fixed $0<q<1$ , the factor $q^n$ ensures absolute convergence for all $s\in\mathbb{C}$ . Since $[n]_q>0$ in this regime, each term $[n]_q^{-s}=\exp(-s\log [n]_q)$ is an entire function of $s$ , so $\zeta_q$ is analytic in $s$ term-by-term.

⚙️ Key Properties

It obeys the classical limit (for $\Re(s)>1$ ):

\lim_{q \to 1^{-}} \zeta_q(s) = \zeta(s)

🎯 Special Cases and Limits

$s=2$ yields a q-sum of inverse squares of q-numbers with exponential weighting.
As $q\to 0^{+}$ , $[n]_q \to 1$ for each fixed $n$ , so $\zeta_q(s)\to \sum_{n\ge 1} q^n = \frac{q}{1-q}$ .
As $q\to 1^{-}$ , $\zeta_q(s)$ approaches the classical zeta in the regime where both are defined.

Riemann zeta is recovered at $q \to 1$ (in the appropriate scaling/limit). q-numbers and q-exponentials share the same q-calculus primitives used in building many basic hypergeometric functions.

Usage in Oakfield

Oakfield provides a q-zeta (Hurwitz-style) implementation primarily for scripting and special-function support:

Special function core implements sim_q_zeta and sim_q_zeta_safe (with diagnostic reports and domain checks).
Lua API exposure: available as sim.qzeta(s, a, q) for experimentation; currently not used by built-in simulation operators by default.

Historical Foundations

📜 q-Analogs in Number Theory

q-analogs replace integer indexing with geometric progressions, producing deformations of classical Dirichlet-type series that connect naturally to q-series and basic hypergeometric function theory.

🌍 Modern Perspective

q-zeta variants are used as deformation tools in analytic and computational contexts where geometric spacing is more natural than uniform spacing.

📚 References

Gasper & Rahman, Basic Hypergeometric Series
Andrews, Askey, Roy, Special Functions