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On prime divisors of Mersenne numbers. (English) Zbl 0733.11003

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.

MSC:

11A41 Primes
11N37 Asymptotic results on arithmetic functions
11N25 Distribution of integers with specified multiplicative constraints
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