On large oscillations of the remainder of the prime number theorems (Q5932679)

The author studies the deviations of the behaviour of the remainder in the prime number theorem. The most important result is an effective form of \textit{J. E. Littlewood}'s classical theorem [C. R. Acad. Sci. Paris 158, 1869-1872 (1914; JFM 45.0305.01)] that \(\psi(x)-x=\Omega_{\pm}(\sqrt{x}\log_3x)\) under the assumption of the Riemann Hypothesis, and of the corresponding bounds for the arithmetic progressions. More precisely, assume the Generalized Riemann Hypothesis for all \(L\) functions modulo \(q\); then, for all \(X>X_0(q,\varepsilon)\) (where \(X_0\) is effectively computable) there exists \(x_1\in(0,X)\) such that \[ \min_{a\not\equiv 1\bmod q} (\psi(x_1;q,1)-\psi(x_1;q,a)) > \left({1\over 2}-\varepsilon\right)\sqrt{x_1}\log_3 x_1, \] and there exists \(x_2\in(0,X)\) satisfying a similar inequality, with min replaced by max, \(>\) replaced by \(<\), and a minus sign in the right hand side. This improves on a result of \textit{J. Kaczorowski} [Q. J. Math., Oxf. II. Ser. 44, 451-458 (1993; Zbl 0799.11030)], who had an unspecified unbounded function in place of the \(\log_3\). The analysis leads to explicit expressions for moments of the weighted remainder term, as functions of linear combinations of the ordinates of the zeros of \(L\) functions. It is also interesting to notice that most of the paper is rather elementary.