Email this article Waring's problem in natural numbers of special form
- Authors: Eminyan K.M.1
-
Affiliations:
- Financial University under the Government of the Russian Federation
- Issue: Vol 211, No 5 (2020)
- Pages: 126-142
- Section: Articles
- URL: https://journal-vniispk.ru/0368-8666/article/view/133332
- DOI: https://doi.org/10.4213/sm9289
- ID: 133332
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Abstract
Let $\mathbb N_0$ be the set of positive integers whose binary decompositions contain an even number of ones. We give a bound for the trigonometric sum of special form over numbers in $\mathbb N_0$; using this bound, we derive an asymptotic formula for the number of solutions to Waring's equation in positive integers in $\mathbb N_0$, and also a bound for the number of terms in the last equation, which is sufficient for the equation to be solvable in integers in $\mathbb N_0$. Bibliography: 9 titles.
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About the authors
Karapet Mkrtichevich Eminyan
Financial University under the Government of the Russian Federation
Email: eminyan@mail.ru
Candidate of physico-mathematical sciences, Associate professor
References
- A. O. Gelfond, “Sur les nombres qui ont des proprietes additives et multiplicatives donnees”, Acta Arith., 13 (1968), 259–265
- К. М. Эминян, “О проблеме делителей Дирихле в некоторых последовательностях натуральных чисел”, Изв. АН СССР. Сер. матем., 55:3 (1991), 680–686
- Р. Вон, Метод Харди–Литтлвуда, Мир, М., 1985, 184 с.
- D. Hilbert, “Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl $n^mathrm{ter}$ Potenzen (Waringsches Problem)”, Math. Ann., 67:3 (1909), 281–300
- И. М. Виноградов, Метод тригонометрических сумм в теории чисел, 2-е изд., Наука, М., 1980, 144 с.
- А. А. Карацуба, Основы аналитической теории чисел, 2-е изд., Наука, М., 1983, 240 с.
- J. M. Thuswaldner, R. F. Tichy, “Waring's problem with digital restrictions”, Israel J. Math., 149 (2005), 317–344
- O. Pfeiffer, J. M. Thuswaldner, “Waring's problem restricted by a system of sum of digits congruences”, Quaest. Math., 30:4 (2007), 513–523
- С. М. Воронин, А. А. Карацуба, Дзета-функция Римана, Физматлит, М., 1994, 376 с.
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