On lower bounds for the Riemann zeta-function. (Q1432444)

The author announces a lower bound for the maxima of the Riemann zeta-function \(\zeta(s)\) in short intervals on the critical line. The main result is the existence of an absolute constant \(A>0\) such that \[ \max_{T\leq t \leq T+\Delta}| \zeta(\tfrac12+it)| \geq \exp(A\log\Delta \log T) \] for all sufficiently large \(T\) and \(0<\Delta \leq (\log T)^{-1}\). It is also noted that for \(\Delta = (\log T)^{-1}\) and \(\Delta = T^{-1}\) this lower bound is significanly stronger than unconditional and conditional (depending on the Riemann Hypothesis) lower bounds which can be deduced from the results given in the treatise by \textit{E. C. Titchmarsh} [The theory of the Riemann zeta-function, 2nd ed., Oxford Univ. Press, (1986; Zbl 0601.10026)]. Based upon this result the author formulates the conjecture that there exists an absolute constant \(A>0\) and a function \(\Delta = \Delta(T)\to 0\), as \(T\to\infty\), such that \[ \max_{T\leq t \leq T+\Delta}| \zeta(\tfrac12+it)| \geq T^{-A} \] for all sufficiently large \(T\), and surmises further that this conjecture is valid for \(\Delta = (\log\log T)^{-1}\) and more strongly for \(\Delta = (\log T)^{-1}\). Former results depending on the Riemann Hypothesis imply the truth of this conjecture with \(\Delta = {\log\log\log T\over \log\log T}\).