A remark on the distribution of values of the divisor function (Q1925169)

The Erdös-Kac theorem implies that the logarithm of the divisor function has asymptotically normal distribution with mean \(\log 2\log\log n\) and standard deviation \(\log 2 \sqrt {\log\log n}\). In the paper under review, the author is interested in the finer distribution of the values of the divisor function \(\tau(n)\) and proves the following local limit theorem: Given positive integers \(n\) and \(l\), let \(\lambda=\sqrt {\log\log n}\) and \(y= (\log l-\lambda^2 \log 2)/(\lambda \log 2)\). Then, uniformly for \(|y|\leq\lambda^{1/3}\), one has \[ {1\over n} \sum_{{m \leq n\atop \tau(m)=l}} 1={e^{-y^2/2} \over\lambda \sqrt{2\pi}} R(l) \left(1+ O\left( {1+ |y|^3 \over\lambda} \right) \right). \] Here \(R(l)\) is a factor independent of \(n\), given by an explicit formula that depends on the multiplicative structure of \(l\). For example, in the case \(l\) is a power of 2, \(R(l)\) reduces to the infinite product \(\prod_p (1-{1\over p}) \sum_{k\geq 0} p^{1-2^k}\).