Chebyshevskii Sbornik
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Chebyshevskii Sb.: Year: Volume: Issue: Page: Find

 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]
$$\alpha_k \leqslant 1-\frac{1}{k+1}, \quad k\geqslant 2.$$

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
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}