Extreme values of the zeta function at critical points (Q2833461)

The authors prove several results on extreme values of the zeta function at ``critical points''. More precisely, let \(\rho_1 = \beta_1+i\gamma_1\) denote complex zeros of the Riemann zeta-function \(\zeta(s)\) such that \(\zeta'(\rho_1) = 0\) and \(\beta_1\geqslant 1\). Assuming the Riemann hypothesis (RH; all complex zeros of \(\zeta(s)\) satisfy \(\mathrm{Re} (s) = 1/2\)) the authors prove that NEWLINE\[NEWLINE \limsup_{\gamma_1\to\infty} \frac{|\zeta(\rho_1)|}{\log\log\gamma_1} \leqslant \frac12 {\roman e}^{C_0}, \quad \liminf_{\gamma_1\to\infty}|\zeta(\rho_1)|\log\log\gamma_1 \geqslant \frac{\pi^2}{3}{\roman e}^{-C_0}. \leqno(1) NEWLINE\]NEWLINE Here \(C_0 = - \Gamma'(1)\) denotes Euler's constant. Unconditionally, the authors show that NEWLINE\[NEWLINE \limsup_{\gamma_1\to\infty} \frac{|\zeta(\rho_1)|}{\log\log\gamma_1} \geqslant \frac14 {\roman e}^{C_0}, \quad \liminf_{\gamma_1\to\infty}|\zeta(\rho_1)|\log\log\gamma_1 \leqslant \frac{2\pi^2}{3}{\roman e}^{-C_0}. \leqno(2) NEWLINE\]NEWLINE They remark that under the RH it is classically known that [\textit{E. C. Titchmarsh}, The theory of the Riemann zeta-function. 2nd ed., Oxford: Clarendon Press (1986; Zbl 0601.10026)] NEWLINE\[NEWLINE \limsup_{t\to\infty}\frac{|\zeta(1+it)|}{\log\log t} \leqslant 2{\roman e}^{C_0},\quad \liminf_{t\to\infty} |\zeta(1+it)|\log\log t \geqslant \frac{\pi^2}{12}{\roman e}^{-C_0}, NEWLINE\]NEWLINE and unconditionally that NEWLINE\[NEWLINE \limsup_{t\to\infty}\frac{|\zeta(1+it)|}{\log\log t} \geqslant {\roman e}^{C_0},\quad \liminf_{t\to\infty} |\zeta(1+it)|\log\log t \leqslant \frac{\pi^2}{6}{\roman e}^{-C_0}. NEWLINE\]NEWLINE These bounds are to be compared to (1) and (2). The proofs of (1) and (2) are based on altogether 13 lemmata, some of which are of independent interest.