## 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 nk(p) the smallest positive k-th power non-residue (mod p). Mirsky asked the author to find an asymptotic formula for sump \leq x nk (p). The author proves using the large sieve of Linnik if p1 < p2 < ··· is the sequence of consecutive primes that

sump \leq x n2(p) = (1+o(1))sumk = 1oo {pk \over 2k} {x \over log x}.

It is very likely true that sump < x nk(p) = (1+o(1)) {ck x \over log x}.
II. Let \phi(n) = \phi1 (n) be Euler's \phi function and put \phik(n) = \phi(\phik-1(n)). The author proves that if we neglect a sequence of density 0 then for k \geq 2

limn ––> oo{\phik(n) log log log n \over \phik-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

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