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Mat. Zametki, 2004, Volume 75, Issue 6, Pages 877–894 (Mi mz78)  

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

The Concentration Function of Additive Functions with Nonmultiplicative Weight

N. M. Timofeev, M. B. Khripunova

Vladimir State Pedagogical University

Abstract: Suppose that $g(n)$ is a real-valued additive function and $\tau(n)$ is the number of divisors of $n$. In this paper, we prove that there exists a constant $C$ such that
$$ \sup_a\sum_{\substack n<N
g(n)\in[a,a+1)} \tau(N-n) \le C\frac{N \log N}{\sqrt{W(N)}}, $$
where
$$ W(N) =4+\min_\lambda(\lambda^2 +\sum_{p<N} \frac1p\min(1,(g(p)-\lambda\log p)^2)). $$
In particular, it follows from this result that
$$ \sup_a|\{m,n:mn<N,\;g(N-mn)=a\}| \ll N \log N  (\sum_{p<N, g(p)\ne0}(1/p))^{-1/2}. $$
The implicit constant is absolute.

DOI: https://doi.org/10.4213/mzm78

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English version:
Mathematical Notes, 2004, 75:6, 819–835

Bibliographic databases:

UDC: 511
Received: 16.08.2000
Revised: 10.11.2002

Citation: N. M. Timofeev, M. B. Khripunova, “The Concentration Function of Additive Functions with Nonmultiplicative Weight”, Mat. Zametki, 75:6 (2004), 877–894; Math. Notes, 75:6 (2004), 819–835

Citation in format AMSBIB
\Bibitem{TimKhr04}
\by N.~M.~Timofeev, M.~B.~Khripunova
\paper The Concentration Function of Additive Functions with Nonmultiplicative Weight
\jour Mat. Zametki
\yr 2004
\vol 75
\issue 6
\pages 877--894
\mathnet{http://mi.mathnet.ru/mz78}
\crossref{https://doi.org/10.4213/mzm78}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2085814}
\zmath{https://zbmath.org/?q=an:1067.11059}
\elib{http://elibrary.ru/item.asp?id=6618291}
\transl
\jour Math. Notes
\yr 2004
\vol 75
\issue 6
\pages 819--835
\crossref{https://doi.org/10.1023/B:MATN.0000030991.37899.cc}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000222492400025}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. M. Timofeev, M. B. Khripunova, “The Concentration Function of Additive Functions with Special Weight”, Math. Notes, 76:2 (2004), 244–263  mathnet  crossref  crossref  mathscinet  zmath  isi
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