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

K. M. Eminyan

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 270-283

Résumé

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 $k-1$, $0\alpha_k1$, $\varepsilon >0$ — 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 $$ \sum_{\substack{n\leqslant X\\ n\in \mathbb{N}_{0}}}\tau(n)=\frac{1}{2}\sum_{n\leqslant X}\tau(n)+O(X^{\omega }\ln^{2}X), $$ where $\tau(n)$ — the number of divisors $n$, $\omega=\frac{1}{2}\big(1+\log_{2}\sqrt{2+\sqrt{2}}\big)=0.9428\ldots$. In this paper, we solve the generalyzed Dirichlet divisors problem in numbers from $\mathbb{N}_{0}$. Bibliography: 15 titles.

Détail

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