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**Zentralblatt MATH**

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

**Zbl.No: ** 154.29403

**Autor: ** Erdös, Pál

**Title: ** Remarks on number theory. I (In Hungarian)

**Source: ** Mat. Lapok 12, 10-16, 161-168 (1961).

**Review: ** I. Denote by n_{k}(p) the smallest positive k-th power non-residue (mod p). Mirsky asked the author to find an asymptotic formula for **sum**_{p \leq x} n_{k} (p). The author proves using the large sieve of Linnik if p_{1} < p_{2} < ··· is the sequence of consecutive primes that **sum**_{p \leq x} n_{2}(p) = (1+o(1))**sum**_{k = 1}^{oo} {p_{k} \over 2^{k}} {x \over log x}. It is very likely true that **sum**_{p < x} n_{k}(p) = (1+o(1)) {c_{k} x \over log x}.

II. Let \phi(n) = \phi_{1} (n) be Euler's \phi function and put \phi_{k}(n) = \phi(\phi_{k-1}(n)). The author proves that if we neglect a sequence of density 0 then for k \geq 2

**lim**_{n ––> oo}{\phi_{k}(n) log log log n \over \phi_{k-1}(n)} = c^{-c}, where C is Euler's constant. Several other problems and results are stated about the \phi function.

**Classif.: ** * 11N69 Distribution of integers in special residue classes

11A25 Arithmetic functions, etc.

**Index Words: ** number theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag