Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  041.36807
Autor:  Erdös, Pál
Title:  Some asymptotic formulas in number theory. (In English)
Source:  J. Indian Math. Soc., n. Ser. 12, 75-78 (1948).
Review:  If n is a positive integer, let f(n) denote the number of solutions of the equation 2k+p = n, k being a non-negative integer and p a prime number. The author proves that:
(I) there is a positive constant c0 such that f(n) > c0 log log n for infinitely many positive integers n,
(II) if r is a fixed positive integer, there exists a positive number c(r) such that sumn = 1x fr(n) < c(r)x,
(III) if n\equiv 7629217 (mod 11184810); f(n) = 0.
(II) contains a theorem of N.P.Romanoff (Zbl 009.00801) to the effect that the positive integers expressible in the form 2k+p have positive density. The author generalizes Romanoff's theorem by proving, that if a1 < a2 < ··· is an infinite sequence of positive integers such that ak | ak+1 for each k, then the positive integers expressible in the form p+ak have positive density if and only if there exist positive numbers c1 and c2 such that {log ak \over k} < c1 and sumd/ak 1/d < c2 for every k.
Reviewer's remark: The proof of (I) uses a result of A.Page (Zbl 011.14905) on the number of prime numbers in an arithmetic progression with relatively large difference. In applying this result, the author forgets to take account of a possible exceptional real primitive residue character which occurs in Page's work. This difficulty can be easily overcome in much the same manner as an analogous difficulty was handled in a joint paper of the author, the reviewer, and S.Chowla (Zbl 036.30702, see p. 170 of the work).
Reviewer:  P.T.Bateman
Classif.:  * 11N56 Rate of growth of arithmetic functions
Index Words:  Number theory

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