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Chebyshevskii Sb., 2016, Volume 17, Issue 1, Pages 270–283 (Mi cheb469)  

Generalized problem of divisors with natural numbers whose binary expansions have special type

K. M. Eminyanab

a Bauman Moscow State Technical University
b Financial University under the Government of the Russian Federation, Moscow

Abstract: Let $\tau_k(n)$ be the number of solutions of the equation $x_{1}x_{2}\cdots x_{k}=n$ in natural numbers $x_{1}$, $x_{2}$, $\ldots,$ $ x_{k}$. Let
$$ D_k(x)=\sum_{n\leqslant x}\tau_k(n). $$
The problem of obtaining of asymptotic formula for $D_k(x)$ is called Dirichlet divisors problem when $k=2$, and generalyzed Dirichlet divisors problem when $k\geqslant 3$.
This asymptotic formula has the form
$$ D_k (x)=x P_{k-1}(\log x)+O(x^{\alpha_k +\varepsilon}), $$
where $ P_{k-1}(x)$ — is the polynomial of the degree $k-1$, $0<\alpha_k<1$, $\varepsilon >0$ — is arbitrary small number.
Generalyzed Dirichlet divisor problem has a rich history.
In 1849, L. Dirichlet [1] proved , that
$$ \alpha_k \leqslant 1-\frac{1}{k}, \quad k\geqslant 2. $$
In 1903, G. Voronoi [2]
$$ \alpha_k \leqslant 1-\frac{1}{k+1}, \quad k\geqslant 2. $$

(see also [3])
In 1922, G. Hardy and J. Littlewood [4] proved that
$$ \alpha_k \leqslant 1-\frac{3}{k+2}, \quad k\geqslant 4. $$
In 1979, D. R. Heath-Brown [5] proved that
$$ \alpha_k \leqslant 1-\frac{3}{k}, \quad k\geqslant 8. $$
In 1972, A. A. Karatsuba got a remarkable result [6].
His uniform estimate of the remainder term has the form
$$ O(x^{1-\frac{c}{k^{2/3}}}(c_{1}\log x)^{k}), $$
where $c>0$, $c_1>0$ — are absolute constants.
Let $\mathbb{N}_{0}$ — be a set of natural numbers whose binary expansions have even number of ones.
In 1991, the autor [8] solved Dirichlet divisors problem and got the formula
$$ \sum_{\substack{n\leqslant Xn\in \mathbb{N}_{0}}}\tau(n)=\frac{1}{2}\sum_{n\leqslant X}\tau(n)+O(X^{\omega }\ln^{2}X), $$
where $\tau(n)$ — the number of divisors $n$, $\omega=\frac{1}{2}(1+\log_{2}\sqrt{2+\sqrt{2}})=0.9428\ldots$.
In this paper, we solve the generalyzed Dirichlet divisors problem in numbers from $\mathbb{N}_{0}$.
Bibliography: 15 titles.

Keywords: generalized problem of divisors, binary expansions, asymptotic formula, uniform estimate of the remainder term.

Full text: PDF file (747 kB)
References: PDF file   HTML file
UDC: 511
Received: 18.12.2015
Accepted:11.03.2016

Citation: K. M. Eminyan, “Generalized problem of divisors with natural numbers whose binary expansions have special type”, Chebyshevskii Sb., 17:1 (2016), 270–283

Citation in format AMSBIB
\Bibitem{Emi16}
\by K.~M.~Eminyan
\paper Generalized problem of divisors with natural numbers whose binary expansions have special type
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 1
\pages 270--283
\mathnet{http://mi.mathnet.ru/cheb469}
\elib{https://elibrary.ru/item.asp?id=25795090}


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