## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  012.01201
Autor:  Erdös, Pál; Turán, Pál
Title:  Ein zahlentheoretischer Satz. (A number-theoretical theorem.) (In German)
Source:  Mitteil. Forsch.-Inst. Math. Mech. Univ. Tomsk 1, 101-103 (1935).
Review:  Let a be a fixed integer, and let l(k) be defined (for any k prime to a) as the least positive integer for which al(k)\equiv 1 (mod k). Generalising a result of N.P.Romanoff (Zbl 009.00801), the authors prove here that sumk {1 \over kl(k)\epsilon} converges for every \epsilon > 0. It suffices to prove that sum 1/k extended over those k for which l(k) < (log k)2 \over \epsilon converges. For this it suffices that the number of divisors \leq n of (a-1) (a2-1)...(aN-1) should be O(n/ log 2 n), where N = [(log n)2 \over \epsilon ]. This is proved by estimating the number of prime factors, and considering separately those divisors with more than \sqrt{log n} different prime factors and those with less.
Reviewer:  Davenport (Cambridge)
Classif.:  * 11B25 Arithmetic progressions
11N13 Primes in progressions
Index Words:  Number theory

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