Erdős, Paul; Kiss, Péter; Pomerance, Carl On prime divisors of Mersenne numbers. (English) Zbl 0733.11003 Acta Arith. 57, No. 3, 267-281 (1991). Let f(n) be the sum of the reciprocals of the distinct prime divisors of the n-th Mersenne number \(f(n)=\sum p^{-1}(p/2^n-1).\) By elementary, but complicated arguments the authors show that for each \(k\geq 2\) and infinitely many n \[ \min (f(n),f(n+1),...,f(n+k-1))\geq \log_{k+2}n+c \log_{k+3}n \] (c is an absolute negative constant, \(\log_kn\) denotes the k-fold iterated logarithm). If the Extended Riemann Hypothesis for certain Dedekind zeta functions is assumed, then for all \(k\geq 2\) und n sufficiently large the above min is \(\leq 3 \log_{k+2}n+ck\). Finally, the average order of f in short intervals is studied. Reviewer: D.Wolke (Freiburg i.Br.) Cited in 1 ReviewCited in 4 Documents MSC: 11A41 Primes 11N37 Asymptotic results on arithmetic functions 11N25 Distribution of integers with specified multiplicative constraints Keywords:sum of reciprocals; distinct prime divisors; Mersenne number; average order; short intervals PDFBibTeX XMLCite \textit{P. Erdős} et al., Acta Arith. 57, No. 3, 267--281 (1991; Zbl 0733.11003) Full Text: DOI