Applying the resonance method to \(\mathrm{Re} \big(e^{-i \theta} \log \zeta (\sigma+it)\big)\) (Q6612917)

Let \(\zeta(s)\) denote the Riemann zeta-function. The purpose of this paper is to prove a Montgomery-type result for \(\mathrm{Re} \left(e^{-i \theta} \log \zeta (\sigma+it)\right)\) in the strip \(1/2 < \sigma < 1\) for \(t \in [T^{\beta}, T]\), \(\beta \in (0, 1)\) and real \(\theta\). Actually, it is shown that there exists a function \( v:(1/2, 1)\to \mathbb{R}_+\) which satisfies \N\[v(\sigma)>c \min\left \lbrace 1, (e^2-e) \frac{|\log (2\sigma-1)|}{|\log (2\sigma-1)|+1} \right\rbrace, \]\Nwhere \(c\) is an independent positive constant and \N\[\max_{t \in [T^\beta, T]}\mathrm{Re} \left(e^{-i \theta} \log \zeta (\sigma+it)\right)\geq v(\sigma) \frac{\kappa^{1-\sigma}}{\sqrt{|\log (2\sigma-1)|}} \frac{(\log T) ^{1-\sigma}}{(\log_2 T)^\sigma}, \]\Nwhere \(0 < \kappa < \min(\sigma - 1/2, 1 - \beta)\).\N\NAs consequence, by taking \(\theta = \pi\), we obtain the following inequality \[\max_{t \in [T^\beta, T]} -\log |\zeta(\sigma+it)|\geq v(\sigma) \frac{\kappa^{1-\sigma}}{\sqrt{|\log (2\sigma-1)|}} \frac{(\log T) ^{1-\sigma}}{(\log_2 T)^\sigma}.\] The above result can then be converted into estimate of the upper bound of the minimum of \(|\zeta(\sigma+it)|\) for \(t\in [T^\beta, T]\). The proof of the main theorem relies on the resonance method introduced by \textit{K. Soundararajan} [Math. Ann. 342, No. 2, 467--486 (2008; Zbl 1186.11049)].