On the approximation of real numbers by the sums of two powers of primes
https://doi.org/10.22405/2226-8383-2025-26-4-329-343
Abstract
In the article we have for any fixed 𝑐 estimate of 𝜅(𝑐) such that 𝑁 > 𝑁0(𝜀) can be approached by the sum of powers of two primes 𝑝𝑐 1 + 𝑝𝑐 2 by a distance not exceeding 𝐻 = 𝑁𝜅(𝑐)+𝜀, where 𝜀
is an arbitrary positive number. These results were obtained using the density technique developed by Yu.V. Linnik in the 1940s. The density technique is based on applying explicit formulas expressing sums over prime numbers with sums over nontrivial zeros of the Riemann zeta function and using density
theorems that estimate the number of nontrivial zeros of the zeta function lying in the critical strip such that their real part is greater than some 𝜎, 1 > 𝜎 ≥ 1/2.
The results obtained in this paper are based on the application of modern density theorems obtained by A. Ivich. In addition, the proof used the theorem of Baker, Harman, and Pintz: one can approach a given real number 𝑁 > 𝑁0(𝜀) by a prime number by a distance no more than 𝐻 = 𝑁21/40+𝜀. Also, the following result obtained by M. Huxley: |𝜁(1/2 + 𝑖𝑡)| ≪ 𝑡32/205+𝜀.
About the Author
Anton Pavlovich Naumenko
Lomonosov Moscow State University; JSC “Infotex”
Russian Federation
Russian Federation
postgraduate student
References
1. Huxley, M. N. 1972, “On the difference between consecutive primes”, Inventiones Mathematicae, 15(1), pp. 164–170.
2. Gritsenko, S. A. 1994, “The refinement of a constant in the density theorem”, Mathematical Notes, 55(2), pp. 59–61.
3. Ivi´c, A. 1985, The Riemann Zeta-Function, New York: John Wiley and Sons.
4. Ivi´c, A. 1983, Topics in Recent Zeta-Function Theory, Orsay: Universit´e de Paris-Sud.
5. Ivi´c, A. 1979, “A note on the zero-density estimates for the zeta-function”, Archiv der Mathematik, 33, pp. 155–164.
6. Ivi´c, A. 1980, “Exponent pairs and the zeta-function of Riemann”, Studia Scientiarum Mathematicarum Hungarica, 15, pp. 157–181.
7. Linnik, Yu. V. 1945, “On the possibility of a unified method in certain questions of additive and multiplicative number theory”, Doklady of the Academy of Sciences of the USSR, 49(1), pp. 3–7.
8. Linnik, Yu. V. 1945, “On a theorem of the theory of primes”, Doklady of the Academy of Sciences of the USSR, 47(1), pp. 7–9.
9. Voronin, S. M. and Karatsuba, A. A. 1994, The Riemann Zeta Function, Moscow: Fizmatlit.
10. Baker, R. C., Harman, G., and Pintz, J. 2001, “The difference between consecutive primes, II”, Proceedings of the London Mathematical Society, 83(3), pp. 532–562.
11. Girko, V. V. and Gritsenko, S. A. 2006, “On a diophantine inequality with primes”, Chebyshevskii Sbornik, 7(4), pp. 26–30.
12. Gritsenko, S. A. and Nguyen Thi Tcha, 2012, “On diophantine inequalities with primes”, Nauchnyye Vedomosti BelGU. Series: Mathematics. Physics, 23(142), 29, pp. 186–190.
13. Naumenko, A. P. 2018, “On nonlinear diophantine inequalities with primes”, in Algebra, Number Theory and Discrete Geometry: Modern Problems and Applications: Proceedings of the XV International Conference, Tula, pp. 239–241.
14. Naumenko, A. P. 2019, “On some nonlinear diophantine inequalities with primes”, Mathematical Notes, 105(6), pp. 943–948.
15. Naumenko, A. P. 2019, “On the approximation of real numbers by the sums of two squares of primes”, Vestnik Moskovskogo Universiteta. Seriya 1: Matematika. Mekhanika, 5, pp. 3–8.
16. Karatsuba, A. A. 1993, Basic Analytic Number Theory, Berlin: Springer-Verlag.
17. Huxley, M. N. 2005, “Exponential sums and the Riemann zeta function V”, Proceedings of the London Mathematical Society, 90, pp. 1–41.
Review
For citations:
Naumenko A.P. On the approximation of real numbers by the sums of two powers of primes. Chebyshevskii Sbornik. 2025;26(4):329-343. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-4-329-343
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