GENERALIZED PROBLEM OF DIVISORS WITH NATURAL NUMBERS WHOSE BINARY EXPANSIONS HAVE SPECIAL TYPE
https://doi.org/10.22405/2226-8383-2016-17-1-270-283
Abstract
Let τk(n) be the number of solutions of the equation x1x2 · · · xk = n in natural numbers x1, x2, . . . , xk. Let Dk(x) = X n6x τk(n). The problem of obtaining of asymptotic formula for Dk(x) is called Dirichlet divisors problem when k = 2, and generalyzed Dirichlet divisors problem when k > 3. This asymptotic formula has the form Dk(x) = xPk−1(log x) + O(xαk+ε), where Pk−1(x) — is the polynomial of the degree k − 1, 0 < αk < 1, ε > 0 — is arbitrary small number. Generalyzed Dirichlet divisor problem has a rich history. In 1849, L. Dirichlet [1] proved , that αk 6 1 − 1
k , k > 2. In 1903, G. Voronoi [2] αk 6 1 − 1 k + 1 , k > 2. (see also [3]) In 1922, G. Hardy and J. Littlewood [4] proved that αk 6 1 − 3 k + 2 , k > 4. In 1979, D. R. Heath-Brown [5] proved that
αk 6 1 − 3 k , k > 8. In 1972, A. A. Karatsuba got a remarkable result [6]. His uniform estimate of the remainder term has the form O(x1− c k2/3 (c1 log x)k), where c > 0, c1 > 0 — are bsolute constants. Let N0 — 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 X n6X n∈N0 τ (n) = 1 2 X n6X τ (n) + O(Xω ln2 X), where τ (n) — the number of divisors n, ω = 1 2
About the Author
K. M. Eminyan
Bauman Moscow State Technical University
Financial University under the Government of the Russian Federation
Russian Federation
Russian Federation
Candidate of Physico-Mathematical Sciences, Associate
Professor.
References
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Review
For citations:
Eminyan K.M. GENERALIZED PROBLEM OF DIVISORS WITH NATURAL NUMBERS WHOSE BINARY EXPANSIONS HAVE SPECIAL TYPE. Chebyshevskii Sbornik. 2016;17(1):270-283. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-1-270-283
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