Distribution of digits of primes in \(q\)-ary canonical form (Q1082367)

The author finds asymptotic formulas for the number of primes up to \(x\), with \(r\) fixed digits in the \(q\)-ary expansion. For very small values of \(r\) such a formula follows from the prime number theorem for progressions; in the paper, however, \(r\) can be as large as \(O(\sqrt{\log x})\), assuming that the fixed digits are not among the largest \(C \log\log x\) with a suitable \(C\) (Theorem 1), or this assumption can be dropped at the price of reducing the permitted range of \(r\) to \(O(\log\log x)\). These results are applied to obtain results on the distribution of \(q\)-additive functions on the set of primes.