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.

Bibliographie

[1] Diriclet L., “Über die Bestimmung der mittleren Werte in der Zahlentheorie”, Abh. Akad Berlin, Werke, v. 2, 49–66; Math. Abh., 1849, 69–83

[2] Voronoi G., “Sur un probléme du calcul des fonctions asymptitiques”, J. Fur die reine und angewandte, Math., 126 (1903), 241–282 | MR | Zbl

[3] Landau E., “Über die Anzahl der Gitterpunkte in gewissen Bereichen, Nacher”, K. Gas. Wiss. Göttingen, Math.-Phys. Klassen, 6 (1912), 687–771

[4] Hardy G., Littlewood J., “The approximate functional equation in the theory of the zeta-function, with applications to the divisor problems of Dirichlet and Piltz”, Proc. London Maht. Soc. 2, 21 (1922), 39–74 | MR

[5] D. R. Heath-Brown, “Recent Progress in Analytic Number Theory”, Symposium Durham, v. 1, Academic Press, London, 1979, 1981 pp. | MR

[6] Karatsuba A. A., “Uniform approximation of the remainder term in the Dirichlet divisor problem”, Izv. Akad. Nauk SSSR Ser. Mat., 36:3 (1972), 475–483

[7] Gelfond A. O., “Sur les nombres qui ont des propriétés additives et multiplicatives données”, Acta Arith., XII (1968), 259–265 | MR

[8] Eminyan K. M., “On the Dirichlet divisor problem in some sequences of natural numbers”, Izv. Akad. Nauk SSSR Ser. Mat., 55:3 (1991), 680–686 | Zbl

[9] Eminyan K. M., “The Goldbach problem with primes having binary expansions of a special form”, Izv. RAN. Ser. Mat., 78:1 (2014), 215–224 | DOI | MR | Zbl

[10] Eminyan K. M., “On the Mean Values of the Function $\tau_k(n)$ in Sequences of Natural Numbers”, Mat. Zametki, 90:3 (2011), 454–463 | DOI

[11] Mauduit C., Rivat J., “Sur un problème de Gelfond: la somme des chiffres des nombres premiers”, Annals of Mathematics. Second Series, 171:3 (2010), 1591–1646 | DOI | MR | Zbl

[12] Green B., “Three topics in additive prime number theory”, Current Developments in Mathematics, 2007 (2009), 1–41 | DOI | MR | Zbl

[13] Karatsuba A. A., Fundamentals of the analytical number theory, Second edition, Nauka, M., 1983, 240 pp. (Russian) | MR

[14] Voronin S. M., Karatsuba A. A., The Riemann zeta function, Fiziko-Matematicheskaya Literatura, M., 1994, 376 pp. (Russian) | MR

[15] Montgomery H. L., Topics in multiplicative nunmber theory, Springer, 1971 | MR | MR