About one additive problem Hua Loo Keng’s
https://doi.org/10.22405/2226-8383-2019-20-4-32-45
Abstract
Let X be enough big real number and $k\geq2$ be a natural number, M be a set of natural numbers n not exceeding X, which cannot be written as a sum of prime and fixed degree a prime, $E_k (X)=card M.$ In present paper is proved theorem.
Theorem. For it is enough greater $X-$equitable estimation $ E_k (X)\ll X^{\gamma},$ where
$$ \gamma<\left\{
\begin{array}{lll}
1-(17612,983k^2 (\ln k+6,5452))^{-1}, & \text{при} & 2\leq k\leq 205,\\[1mm]
1-(68k^3 (2\ln k+\ln\ln k+2,8))^{-1}, & \text{при} & k>205,\\[1mm]
1-(137k^3 \ln k)^{-1}, & \text{при} & k>e^{628}.
\end{array}\right.
$$}
In particular from this theorems follows that estimation $$\gamma<1-(137k^3 \ln k)^{-1},$$ got by V. A. Plaksin for it is enough greater k,
remains to be equitable under $\ln k>628$.
About the Authors
Ismail Allakov
Termez state University, faculty of physics and mathematics
Uzbekistan
doctor of physical and mathematical Sciences, Professor
Uzbekistan
doctor of physical and mathematical Sciences, Professor
Abduvohit Safarov
Termez state University
Uzbekistan
graduate student
Uzbekistan
graduate student
References
1. Allakov I. 2000, "On the representation of numbers by the sum of two primes from an arithmetic progression", Izvestiya VUZov. "MatematikaKazan’, vol. 8(459), pp. 3–15.
2. Allakov I. 2012, "Some of the following are some analytic problems of the theory with analytical methods", Monografija. T.: "Ta’lim200 c.
3. Allakov I. 1991, "On a binary additive problem with primes from an arithmetic progression", Dokl. AN RUz., vol. 7„ pp. 9.
4. Allakov I. 2002, "On numbers representable as the sum of a prime and a fixed power of a prime", Algebra i yeyo prilozheniya: Tezisy dokladov mezhdunar. konf. 5-9 avgusta 2002. Krasnoyarsk, pp. 3–4.
5. Arkhipov G. I., Chubarikov V. N. 2003, "On a ternary problem with prime numbers", Algebra i teoriya chisel: Sovremennyye problemy i prilozheniya: Tez. dokl. V mezhdunar. konf. 19–20 maya 2003, Tula, pp. 18–19.
6. Vinogradov A. I. 1985, "On the Hardy-Littlewood binary problem", Acta arithm, Varshava, vol. 46, pp. 33–56.
7. Zhukova A. A. 2000, "The Hardy-Littlewood Problem", Izv. Vuzov, Matematika, vol. 2(453), pp. 41–49.
8. Plaksin V. A. 1984, "The exceptional set of the sum of a prime and a fixed power of a prime", Petrozavodsk, 33 pp. Dep. v VINITI, 23.10.84. vol. 7010-84.
9. Plaksin V. A. 1990, "On one question of Hua-Lo-Ken", Mat. zametki, vol. 3(47), pp. 78–90.
10. Chubarikov V. N. 2011, "Multidimensional problems of the theory of prime numbers" Chebyshevskiy sbornik, vol. 12, iss. 4, pp. 176–256.
11. Davenport. H. 2000, "Multiplicative Number Theory", Third edition, Graduate Texts in Math. 71,New York: Sringer-Verlag, 178 p.
12. Jorg Brudern. 2008, "Representations of natural numbers as the sum of a prime and a k-th power", Tsukuba Journal of Mathematics, vol. 32, no. 2, pp. 349–360.
13. Montgomery H.L., 1975, Vaughan R.C. "The exceptional set in Goldbach’ s problem", Acta arithm., vol. 27, pp. 353–370.
14. Montgomery H.L. and Vaughan R.C. 2006, "Multiplicative number theory. I. Classical theory", Published in the United States of America by Cambridge University Press, New York, 552 p.
15. Vaughan R.C. 1997, "The Hardi-Littlewood method", Cambridge University Press, Nev York, 233 p.
16. Vinogradov A.I. 1980, "The metod of trigonometrical Sums in the Theory of Numbers", Мoskov: Nauka, 144 p.
Review
For citations:
Allakov I., Safarov A. About one additive problem Hua Loo Keng’s. Chebyshevskii Sbornik. 2019;20(4):32-45. (In Russ.) https://doi.org/10.22405/2226-8383-2019-20-4-32-45
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