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

This article is cited in 1 scientific paper (total in 2 paper)

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

N. M. Timofeev, M. B. Khripunova

Vladimir State Pedagogical University

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
Received: 20.07.1999

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
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\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}


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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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Luca, F, “On the values of the divisor function”, Monatshefte fur Mathematik, 154:1 (2008), 59  crossref  mathscinet  zmath  isi  scopus  scopus
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