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 Mat. Zametki, 2000, Volume 68, Issue 5, Pages 725–738 (Mi mz993)

On the Difference between the Number of Prime Divisors from Subsets for Consecutive Integers

N. M. Timofeev, M. B. Khripunova

Abstract: Suppose that $E_1$, $E_2$ are arbitrary subsets of the set of primes and $g_1(n)$, $g_2(n)$ are additive functions taking integer values such that $g_i(p)=1$, if $p\in E_i$ and $g_i(p)=0$ otherwise, $i=1,2$. Set
$$E_i(x)=\sum_{\substack{p\le x, p\in E_i}}\frac 1p,\quad i=1,2.$$
It is proved in this paper that if $R(x)=\max(E_1(x),E_2(x))$, $a\ne0$ is an integer, then
$$\sup_m|\{n:n\le x, g_2(n+a)-g_1(n)=m\}| \ll\frac x{\sqrt{R(x)}}.$$
If, moreover, $E_i(x)\ge T$ for $x\ge x_0$, where $T$ is a sufficiently large constant and
$$|m-(E_2(x)-E_1(x))|\le\mu\sqrt{R(x)},$$
then there exists a constant $c(\mu,a,T)>0$ such that for $x\ge x_0$ we have
$$\sum_{i=0}^3|\{n:n\le x,g_2(n+a)-g_1(n)=m+i\}|\ge c(\mu,a,T)\frac x{\sqrt{R(x)}}.$$

DOI: https://doi.org/10.4213/mzm993

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English version:
Mathematical Notes, 2000, 68:5, 614–626

Bibliographic databases:

UDC: 511

Citation: N. M. Timofeev, M. B. Khripunova, “On the Difference between the Number of Prime Divisors from Subsets for Consecutive Integers”, Mat. Zametki, 68:5 (2000), 725–738; Math. Notes, 68:5 (2000), 614–626

Citation in format AMSBIB
\Bibitem{TimKhr00} \by N.~M.~Timofeev, M.~B.~Khripunova \paper On the Difference between the Number of Prime Divisors from Subsets for Consecutive Integers \jour Mat. Zametki \yr 2000 \vol 68 \issue 5 \pages 725--738 \mathnet{http://mi.mathnet.ru/mz993} \crossref{https://doi.org/10.4213/mzm993} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1835454} \zmath{https://zbmath.org/?q=an:1022.11044} \transl \jour Math. Notes \yr 2000 \vol 68 \issue 5 \pages 614--626 \crossref{https://doi.org/10.1023/A:1026623624946} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000166684000009} 

• http://mi.mathnet.ru/eng/mz993
• https://doi.org/10.4213/mzm993
• http://mi.mathnet.ru/eng/mz/v68/i5/p725

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This publication is cited in the following articles:
1. G. I. Arkhipov, V. G. Zhuravlev, V. A. Iskovskikh, A. A. Karatsuba, M. B. Levina-Khripunova, V. N. Chubarikov, A. A. Yudin, “Nikolai Mikhailovich Timofeev (obituary)”, Russian Math. Surveys, 58:4 (2003), 773–776
2. Luca, F, “On the values of the divisor function”, Monatshefte fur Mathematik, 154:1 (2008), 59
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