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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 100.27201

**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 i_{1},...,i_{k} be any permutation of 1,...,k. Then there is an m, not exceeding n such that d(m+i_{1}) < ··· < d(m+i_{k}). 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} .

**Reviewer: ** I.S.Gál

**Classif.: ** * 11N64 Characterization of arithmetic functions

**Index Words: ** number theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag