Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  733.11003
Autor:  Erdös, Paul; Kiss, Péter; Pomerance, Carl
Title:  On prime divisors of Mersenne numbers. (In English)
Source:  Acta Arith. 57, No.3, 267-281 (1991).
Review:  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/2n-1). By elementary, but complicated arguments the authors show that for each k \geq 2 and infinitely many n

max (f(n),f(n+1),...,f(n+k-1)) \geq logk+2n+c logk+3n

(c is an absolute negative constant, logkn 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 logk+2n+ck. Finally, the average order of f in short intervals is studied.
Reviewer:  D.Wolke (Freiburg i.Br.)
Classif.:  * 11A41 Elemementary prime number theory
                   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

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