Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 133.29905
Autor: Erdös, Pál
Title: On the distribution of divisors of integers in the residue classes (mod d) (In English)
Source: Bull. Soc. Math. Grece, N. Ser. 6, No.1, 27-36 (1965).
Review: Let k and l denote integers satisfying 0 < l < k,(l,k) = 1. Denote by f(x; k,l) the number of positive integers less than x which have a divisor congrent to l (mod k), by F(x; k) the number of positive integers less than x which have a divisor congruent to l (mod k) for every l, by Q(x) the number of positive integers less than x which have no divisor of the form p(kp+1), by d(n; k,l) the number of divisors of n which are congruent to l (mod k). The author proves or outlines the proof of the following theorems: 1. Let \epsilon > 0 be fixed but arbitrary, k < 2(1-\epsilon) log log x. Then uniformly in k F(x; k) = x+o(x). 2. Let \epsilon > 0 be fixed but arbitrary, k > 2(1+\epsilon) log log x. Then uniformly in k and l f(x; k,l) = x/l+o(x). 3. Q(x) = (1+o(1))e-c x/ log 2 log log x where c is Euler's constant. 4. Let \epsilon > 0 be fixed but arbitrary k < 2[(1-\epsilon) log log x]/2. Then for every \eta > 0 we have for every l1 and l2, for all but o(x) integers less than x 1-\eta < d(n; k,l1)/d(n; k,l2) < 1+\eta. The proofs of Theorems 1 and 4 are based on recent group theoretic results of the author and A. Rényi [J. Analyse math. 14, 127-138(1965)], the theorem of Siegel and Walfisz on the distribution of primes in an arithmetic progression, and a theory of Hardy, Ramanujan, and Turan on the number of prime factors of n. The proofs of Theorems 2 and 3 are outlined only.
Reviewer: R.C.Entringer
Classif.: * 11N69 Distribution of integers in special residue classes
Index Words: number theory
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