caa a22 4500 475740904 CHVBK 20180406123459.0 cr unu---uuuuu 170329e20001101xx s 000 0 eng 10.1023/A:1026623624946 doi (NATIONALLICENCE)springer-10.1023/A:1026623624946 On the Difference between the Number of Prime Divisors from Subsets for Consecutive Integers [Elektronische Daten] [N. Timofeev, M. Khripunova] Suppose that E 1,E 2 are arbitrary subsets of the set of primes and $$g_{\text{1}} \left( n \right),g_{\text{2}} \left( n \right)$$ are additive functions taking integer values such that $$g_i \left( p \right) = 1{\text{ if }}p \in E<Subscript>i ,{\text{ and }}g_i \left( p \right) = 0$$ otherwise, i=1,2. Set $$E_i (x) = \sum \limits_{p \leqslant x, p \in E_i} \frac{1}{p}, i = 1, 2.$$ It is proved in this paper that if $$R\left( x \right) = \max \left( {E_1 \left( x \right),E_2 \left( x \right)} \right){\text{ }}a \ne 0$$ is an integer, then $$\mathop {\sup }\limits_m \left| {\left\{ {n:n \leqslant x,{\text{ }}g_2 \left( {n + a} \right) - g_1 \left( n \right) = m} \right\}} \right| \ll \frac{x}{{\sqrt {R\left( x \right)} }}.$$ If, moreover, $$E<Subscript>i \left( x \right) \geqslant T{\text{ for }}x \geqslant x_0$$ , where T is a sufficiently large constant and $$\left| {m - \left( {E<Subscript>2 \left( x \right) - E<Subscript>1 \left( x \right)} \right)} \right| \leqslant \mu \sqrt {R\left( x \right)} ,$$ then there exists a constant $$c\left( {\mu ,a,T} \right) >0$$ such that for $$x \geqslant x_0$$ we have $$\sum\limits_{i = 0}^3 {\left| {\left\{ {n:n \leqslant x,g_2 \left( {n + a} \right) - g_1 \left( n \right) = m + i} \right\}} \right|} \geqslant c\left( {\mu ,a,T} \right)\frac{x}{{\sqrt {R\left( x \right)} }}.$$ Plenum Publishing Corporation, 2000 the number of prime divisors nationallicence Halácz estimate nationallicence Chen method nationallicence Selberg's sieve nationallicence Timofeev N. Vladimir State Pedagogical University, Russia aut Khripunova M. Vladimir State Pedagogical University, Russia aut Mathematical Notes Kluwer Academic Publishers-Plenum Publishers 68/5-6(2000-11-01), 614-626 0001-4346 68:5-6<614 2000 68 11006 https://doi.org/10.1023/A:1026623624946 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1026623624946 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Timofeev N. Vladimir State Pedagogical University, Russia aut NATIONALLICENCE 700 1- Khripunova M. Vladimir State Pedagogical University, Russia aut NATIONALLICENCE 773 0- Mathematical Notes Kluwer Academic Publishers-Plenum Publishers 68/5-6(2000-11-01), 614-626 0001-4346 68:5-6<614 2000 68 11006 Metadata rights reserved Springer special CC-BY-NC licence nationallicence SWISSBIB 474864782 BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer