ISSN:
1573-8876
Keywords:
the number of prime divisors
;
Halácz estimate
;
Chen method
;
Selberg's sieve
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract 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 $$\sum\limits_{i = 0}^3 {\left| {\left\{ {n:n \leqslant x,g_2 \left( {n + a} \right) - g_1 \left( n \right) = m + i} \right\}} \right|} \geqslant c\left( {\mu ,a,T} \right)\frac{x}{{\sqrt {R\left( x \right)} }}.$$
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1026623624946
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