By A. V. Arhangel’skii (auth.), A. V. Arhangel’skii (eds.)

This ebook with its 3 contributions by way of Arhangel'skii and Choban treats vital subject matters ordinarily topology and their position in useful research and axiomatic set thought. It discusses, for example, the continuum speculation, Martin's axiom; the theorems of Gel'fand-Kolmogorov, Banach-Stone, Hewitt and Nagata; the foundations of comparability of the Luzin and Novikov indices. The publication is written for graduate scholars and researchers operating in topology, useful research, set concept and chance idea. it's going to function a reference and likewise as a consultant to contemporary learn results.

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Elem. Math. 14 (1959), 60–61. [6] Y. Wang, On the representation of large integer as a sum of prime and an almost prime. Sci. Sinica 11 (1962), 1033–1054. Originally published in Colloquium Mathematicum LXVIII (1995), 55–58 Andrzej Schinzel Selecta On integers not of the form n − ϕ(n) with J. Browkin (Warszawa) W. Sierpi´nski asked in 1959 (see [4], pp. 200–201, cf. [2]) whether there exist infinitely many positive integers not of the form n − ϕ(n), where ϕ is the Euler function. We answer this question in the affirmative by proving Theorem.

Erd˝os’ paper [1]. Using ideas and results from that paper we can prove the following theorem. Theorem 3. Let f (n) be an additive function satisfying condition 1 of Theorem 1 and let (1/p) be divergent, f (p) /p convergent, then the distribution function of f (p)=0 h-tuples f (m + 1), f (m + 2), . . , f (m + h) exists, and it is a continuous function. Proof. We denote by N (f ; c1 , c2 , . . , ch ) the number of positive integers m not exceeding n for which f (m + i) ci , i = 1, 2, . . , h, where ci are given constants.

P|m 888 G. Arithmetic functions Let us also consider the function fk (m) = f (p). We are going to show that p|m, p k the sequence N (fk ; c1 , c2 , . . , ch )/n is convergent. Since fk (m + A) = fk (m), where A= p, we can see that the integers m for which p k c fk (m + i) c ci (i = 1, 2, . . , h) are distributed periodically with the period A. Hence N (fk ; c1 , c2 , . . , ch )/n has a limit. To prove the existence of a limit of N (f ; c1 , c2 , . . , ch )/n it is sufficient to show that for arbitrary ε > 0 there exists k0 such that for every k > k0 and n > n(ε) N (f ; c1 , c2 , .