An explicit van der Corput estimate for \(\zeta(1/2+it)\) (Q266162)

Let \(\zeta(s)\) denote the Riemann zeta-function. A fundamental problem is the estimation of \(\zeta(1/2+it)\), and the (yet unproved) Lindelöf hypothesis states that \(\zeta(1/2+it) \ll |t|^\varepsilon\) for any given postive \(\varepsilon\). The classical result of \textit{J. G. van der Corput} [Math. Ann. 89, 215--254 (1923; JFM 48.0181.04)] is \(\zeta(1/2+it) \ll |t|^{1/6}\log|t|\), and the exponent 1/6 was replaced, in the course of about hundred years of extensive research, by several slightly smaller numbers. These were attained by a sophisticated method for the estimation of exponential sums. However, it is of interest to obtain explicit estimates of van der Corput type, and recently \textit{D. J. Platt} and \textit{T. S. Trudgian} [J. Number Theory 147, 842--851 (2015; Zbl 1382.11059)] showed that NEWLINE\[NEWLINE |\zeta(1/2+it)| \leq 0.732t^{1/6}\log t\qquad(t\geq 2).\leqno(1) NEWLINE\]NEWLINE The present author improves (1) by showing that \(|\zeta(1/2+it)| \leq 1.461\) for \(0\leq t\leq 3\) and NEWLINE\[NEWLINE |\zeta(1/2+it)| \leq 0.63t^{1/6}\log t\qquad(t\geq 3). NEWLINE\]NEWLINE The proof is based on several ingredients: an explicit ``third derivative'' exponential sum estimate (explicit version of the AB-process in the theory of exponent pairs), a precise version of the Riemann-Siegel formula for \(|\zeta(1/2+it)|\) and several other explicit estimates. The basic idea is to divide the sum of length \(\ll \sqrt{t}\) in the Riemann-Siegel formula for \(|\zeta(1/2+it)|\) into subsums of length \(\ll t^{1/3}\). Each of these subsums is then estimated by the bound of Lemma 1.2.