Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 617.10038
Autor: Erdös, Paul; Maier, H.; Sárközy, A.
Title: On the distribution of the number of prime factors of sums a+b. (In English)
Source: Trans. Am. Math. Soc. 302, 269-280 (1987).
Review: Let A and B be two subsets of positive integers not exceeding x. Let their cardinalities be denoted by |A| and |B|. For any positive integer n put \nu(n) = sump|n1 and \Omega(n) = sump\alpha|| n \alpha. For real v, put \Phi(v) = (2\pi)-intv-ooe-u2/2 du. The authors continue a series of investigations by A. Balog, P. Erdös and A. Sárközy. The result of this paper is a surprising link of \nu(a+b) and \Omega(a+b) (a in A, b in B) with the normal distribution function \Phi(v) as in the famous Erdös-Kac theorem.
The authors prove the following Theorem: There exist absolute constants x0, C1 such that if x > x0 and \ell is any arbitrary positive integer then we have ||{(a,b); a in A, b in B, \nu(a+b) \leq \ell}| -\Phi(\frac{\ell- log log x}{(log log x) ½}) |A| |B||
< C1 x(|A| |B|) ½(log log x)-1/4. The same is true with \Omega(a+b) in place of \nu(a+b).
The theorem is proved by using the Hardy-Littlewood method. For \ell = 0,1,2,... put
S(x,\ell,\alpha) = sumn \leq x,\nu(n) \leq \ell e(n\alpha) where e(\alpha) = \exp(2\pi i\alpha). Also put
E(x,\ell) = \Phi(\frac{(\ell- log log x)} {(log log x) ½}). The main lemma of the paper is the following: There exist absolute constants x1, C2 such that for x > x1, \ell = 0,1,2,.... and any real number \alpha, we have
|S(x,\ell,\alpha)- E(x,\ell)sum[x]n = 1e(n,\alpha)| \leq C2 x(log log x)-1/4. The method of proof of this lemma bears some resemblance with that of Vinogradov applied in the proof of his three primes theorem.
Reviewer: K.Ramachandra
Classif.: * 11K65 Arithmetic functions (probabilistic number theory)
11P55 Appl. of the Hardy-Littlewood method
11P32 Additive questions involving primes
Keywords: arithmetic properties; dense sequences; sum sequences; Vinogradov method; normal distribution function; Erdös-Kac theorem; Hardy-Littlewood method
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