##
**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 2^{k}+p = n, k being a non-negative integer and p a prime number. The author proves that:

(I) there is a positive constant c_{0} such that f(n) > c_{0} 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 **sum**_{n = 1}^{x} f^{r}(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 2^{k}+p have positive density. The author generalizes Romanoff's theorem by proving, that if a_{1} < a_{2} < ··· is an infinite sequence of positive integers such that a_{k} | a_{k+1} for each k, then the positive integers expressible in the form p+a_{k} have positive density if and only if there exist positive numbers c1 and c_{2} such that {log a_{k} \over k} < c_{1} and **sum**_{d/ak} ^{1}/_{d} < c_{2} 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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag