Effective Upper Bound Estimates for $|ζ'(1/2+it)|$ via Exponential Sums

TL;DR

This work addresses the problem of obtaining effective upper bounds for the derivative of the Riemann zeta function on the critical line, $|unction{zeta}'(1/2+it)|$. By employing exponential-sum techniques and a careful decomposition of $\zeta'(s)$ into main sums and a tractable remainder, the authors derive explicit bounds with controllable constants. The key contributions are two explicit bounds: (i) for $t \ge e^2$, a bound of the form $2 t^{1/2} \log t - 4 t^{1/2} + 8.047 \log t + 6.399$, and (ii) for $t \ge e^6$, a refined bound $|\zeta'(1/2+it)| \le Q_1 t^{1/6} (\log t)^2 + Q_2 t^{1/6} \log t + Q_3 t^{1/6} + Q_4 (\log t)^2 + Q_5 \log t + Q_6$, with constants determined by auxiliary parameters. The method involves detailed bounds on oscillatory integrals, Euler–Maclaurin remainders, and dyadic-range exponential sums, culminating in a structured path to tighter, computable bounds on $\zeta'(1/2+it)$. These results advance the concrete understanding of zeta-derivative behavior on the critical line and offer a framework for further tightening constants or extending to higher derivatives.

Abstract

In this paper, we use methods of exponential sums to derive a formula for estimating effective upper bounds of $|ζ'(1/2+it)|$. Different effective upper bounds can be obtained by choosing different parameters.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7