Publications of (and about) Paul Erdös
Autor: Erdös, Pál
Title: Remarks on two problems. (In Hungarian. RU, English summary)
Source: Mat. Lapok 11, 26-32 (1960).
Review: Using the elementary estimate \pi(x) > cx / log x first the existence of an absolute constant C is proved such that for every sufficiently large n there is an m < n with the property that d(m) > prod d (m+i) d(m-1) (1 \leq i < C log n/ log log n)2). The author remarks that there is a sufficiently large constant C such that d(n) < prod d(n+i) (1 \leq i < C log n/ log log n log log log n). Given a sufficiently large n let k = c(log n) ½ log log n where c is a suitable absolute constant and let i1,...,ik be any permutation of 1,...,k. Then there is an m, not exceeding n such that d(m+i1) < ··· < d(m+ik). Finally let f(n) be the largest natural number such that V(n) < ··· < V(n+f(n)). By utilizing the prime number theorem it is proved that limsup f(n) log log n/ log n 1/2 .
Classif.: * 11N64 Characterization of arithmetic functions
Index Words: number theory
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