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A note on order estimates of the $q$-analogue of the Riemann zeta function

Hideki Murahara, Tomokazu Onozuka

TL;DR

The paper derives growth estimates on vertical lines for the $q$-analogue of the Riemann zeta function, $\zeta_q(s,t)$, and its specialization $\zeta_q(s)$, by decomposing a key representation into a finite part and a controllable tail. Using Stirling-type bounds on binomial coefficients and an optimal truncation parameter $N$ that scales with the imaginary height $|v|$, it establishes explicit bounds that depend on the sign of $\Re(t)$. The results show polynomial growth when $\Re(t)\ge0$ and exponential decay when $\Re(t)<0$, with a corollary giving analogous bounds for $\zeta_q(s)$ and a definition of $\mu_q(\sigma)$ capturing the vertical-growth rate. The work highlights a gap with the classical Lindelöf hypothesis due to fixed $q$ and anticipates convergence to the classical bounds as $q\to1$, contributing toward understanding $q$-deformations of zeta-type objects.

Abstract

At the first step of studying order estimates for the $q$-analogue of the Riemann zeta function, we estimate bounds for it on vertical lines for a fixed parameter $q$.

A note on order estimates of the $q$-analogue of the Riemann zeta function

TL;DR

The paper derives growth estimates on vertical lines for the -analogue of the Riemann zeta function, , and its specialization , by decomposing a key representation into a finite part and a controllable tail. Using Stirling-type bounds on binomial coefficients and an optimal truncation parameter that scales with the imaginary height , it establishes explicit bounds that depend on the sign of . The results show polynomial growth when and exponential decay when , with a corollary giving analogous bounds for and a definition of capturing the vertical-growth rate. The work highlights a gap with the classical Lindelöf hypothesis due to fixed and anticipates convergence to the classical bounds as , contributing toward understanding -deformations of zeta-type objects.

Abstract

At the first step of studying order estimates for the -analogue of the Riemann zeta function, we estimate bounds for it on vertical lines for a fixed parameter .

Paper Structure

This paper contains 2 sections, 5 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Let $\varepsilon>0$, $q\in (0,1)$, and let $s=\sigma+iv$ and $t$ be complex numbers. Fix $\varepsilon$, $q$, $\sigma$, and $\Re(t)$. Assume that $t$ satisfies $\inf_{r\in \mathbb{Z}_{\ge 0}} |1-q^{t+r}|>\varepsilon$. Then, for sufficiently large $|v|$, we have Here the implicit constants depend on $\epsilon$, $q$, $\sigma$, and $\Re(t)$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['main']}