A short note on the Lindelöf hypothesis (Q1881786)

For the Riemann zeta-function, the Lindelöf hypothesis asserts that \(\zeta({1\over 2}+it) \ll t^\varepsilon\) for any \(\varepsilon > 0\). \textit{R. Balasubramanian} [Hardy-Ramanujan J. 9, 1--10 (1986; Zbl 0662.10030)] proved that there are large values of \(\zeta({1\over 2}+it)\), \[ \max_{T\leq t \leq 2T} \left| \zeta\left({1\over 2}+it\right) \right| \geq \exp\left( {3\over 4}\; \sqrt{{{\log T}\over{\log\log T}}}\right), \] if \(T\) is large. The authors prove that this phenomenon does not occur too often: For any fixed \(\lambda > 0\) \[ \begin{multlined} \text{measure} \left\{ t \in [0,T]; \; \left| \zeta\left({1\over 2}+it\right) \right| \geq \lambda \, \exp\left( {3\over 4}\; \sqrt{{{\log T}\over{\log\log T}}}\; \right) \right\}\\ \ll T \left( {{\log T}\over \lambda} \right)^4 \exp \left( -3\, \sqrt{{{\log T}\over{\log\log T}}} \right).\end{multlined} \] In the opposite direction, it follows from a result of Hejhal (to appear) that for any positive monotone function \(f(T)\) \[ {1 \over T} \text{ measure } \left\{ t \in [0,T]; \;\left| \zeta\left({1\over 2}+it\right) \right| < f(T) \right\}= \Phi \left( {{\log f(T)} \over {\sqrt{{1\over 2} \log\log T}}} \right) + O\left( {{(\log\log\log T)^2} \over {\sqrt{\log\log T}}} \right), \] where \(\Phi(y) = {1\over{\sqrt{2\pi\,}}} \int_{-\infty}^y \exp( - {1\over2} u^2) \, du\).