Large deviations in Hooley's theorem (Q1054779)

Let \(\psi(x; a, k)\) denote the sum \(\displaystyle \sum_{m\le x} \Lambda(m)\), where \(\Lambda(m)\) is the von Mangoldt function. Let \((a,k)=1\), \(E(x;a,k) = \psi(x; a,k) - x/\varphi(k)\). Assuming that a certain conjecture holds \textit{C. Hooley} [J. Lond. Math. Soc., II. Ser. 16, 1--8 (1977; Zbl 0377.10023)] has proved that \[ \frac1{T-1} \mathrm{mes} \left\{t\in [1,T], E(e^t;a,k) < x(e^t \varphi(k)\ln k)^{\frac12}\right\} \] as \(T\to\infty\) converges to a continuous function \(F_k(x)\), such that \[ \lim_{k\to\infty} F_k(x) = \frac1{\sqrt{2\pi}} \int_{-\infty}^x e^{-u^2/2}\,du. \tag{1} \] An improvement of Hooley's theorem is presented in this paper. The rate of convergence in (1) is found and a theorem on large deviations for the function \(F_k(x)\) is proved, too.