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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
$$
W(N)
=4+\min_\lambda\biggl(\lambda^2
+\sum_{p<N}
\frac1p\min\bigl(1,(g(p)-\lambda\log p)^2\bigr)\biggr).
$$
In particular, it follows from this result that
$$
\sup_a\bigl|\bigl\{m,n:mn<N,\;g(N-mn)=a\bigr\}\bigr|
\ll N\,\log N\,
\biggl(\sum_{p<N,\,g(p)\ne0}(1/p)\biggr)^{-1/2}.
$$
The implicit constant is absolute.
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
Linking options:
https://www.mathnet.ru/eng/mzm78https://doi.org/10.4213/mzm78 https://www.mathnet.ru/eng/mzm/v75/i6/p877
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