Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 328.10004
Autor: Erdös, Paul
Title: Bemerkungen zu einer Aufgabe in den Elementen. (Remarks on a problem in the elements.) (In German)
Source: Arch. der Math. 27, 159-163 (1976).
Review: Let p denote an odd prime and \ell (p) the order of 2 mod p. Write E(r) for the number of odd primes p with \ell (p) = r and A(x,\delta) for the number of odd primes p with p \leq x and \ell (p) > p\delta, 0 < \delta < 1. Jaeschke and Bundschuh (Aufgabe 618 in ``Elemente der Mathematik'', 1971, hence the uniformative title of the present paper) proved that \hbox{(i)} E(r) \leq {r log 2 \over log r}, \hbox{(ii)} A(x, \delta) = (1+o(1)){x \over log x}, 0 < \delta < 1/2 ,
\hbox{(iii)} A(x, 1/2) \geq (1- log 2+o(1)){x \over log x}. In a few lines, using the Chinese remainder theorem and a weak form of Stirling's formula, (i) is sharpened to
E(r) \leq (1/2 +o(1)){r log 2 \over log r}. The main result of the paper is that (ii) also holds for \delta = 1/2 . The proof of this is rather complicated and uses among other things an estimate from the sieve method of Brun. There are some conjectures on E(r) and A(x, \delta) which seem very difficult to prove. To mention one of them: E(r) = o(r\epsilon), \epsilon > 0.
Reviewer: H.Jager
Classif.: * 11A05 Multiplicative structure of the integers
11N37 Asymptotic results on arithmetic functions
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