On the tails of the limiting distribution function of the error term in the Dirichlet divisor problem (Q2773295)

If \(\Delta(x)\) is the error term in the Dirichlet divisor problem, it was shown by the reviewer [\textit{D. R. Heath-Brown}, Acta Arith. 60, 389-415 (1992; Zbl 0739.11036)] that there is a distribution function NEWLINE\[NEWLINED(u)=\lim_{X\to\infty} \text{meas} \bigl\{x\in [0,X]:x^{-1/4} \Delta(x)\leq u\bigr\}.NEWLINE\]NEWLINE The reviewer showed further that \(D(u)=\int^u_{-\infty} f(\alpha)\) \(d\alpha\) for an infinitely differentiable function \(f(\alpha)\), and [Advances in Number Theory 1991, Clarendon Press, Oxford, 31-35 (1993; Zbl 0791.11048)] that \(f(\alpha)\ll \exp(-|\alpha|^{4 -\varepsilon}))\) for any fixced \(\varepsilon>0\). NEWLINENEWLINENEWLINEIf one defines NEWLINE\[NEWLINE\text{tail} (u)=\left \{\begin{matrix} D(u), & u<0,\\ 1-D(u), & u\geq 0,\end{matrix} \right.NEWLINE\]NEWLINE to describe large deviations in the distribution of \(\Delta(x)\), then it is shown that there exist constants \(c_1\), \(c_2>0\) such that NEWLINE\[NEWLINE\exp\left(- c_1{|u|^4 \over\bigl( \log|u|\bigr)^\beta} \right)\ll \text{tail} (u)\ll\exp \left(-c_2{|u|^4 \over\bigl(\log|u|\bigr)^\beta}\right).NEWLINE\]NEWLINE where \(\beta=3(2^{4/3}-1) =4.559.\dots.\)NEWLINENEWLINENEWLINEAs the paper notes, similar results have been obtained by Montgomery, in unpublished work.