Estimates of short exponential sums with primes in major arcs
https://doi.org/10.22405/2226-8383-2021-22-4-200-224
Abstract
For a number of additive problems with almost equal summands, in addition to the estimates for short exponential sums with primes of the form
𝑆𝑘(𝛼; 𝑥, 𝑦) =
Σ︁
𝑥−𝑦<𝑛⩽𝑥
Λ(𝑛)𝑒(𝛼𝑛𝑘),
in minor arcs, we need to have an estimate of these sums in major arcs, except for a small neighborhood of their centers. We also need to have an asymptotic formula on a small
neighborhood of the centers of major arcs.
In this paper, using the second moment of Dirichlet 𝐿-functions on the critical line, we obtained a nontrivial estimate of the form
𝑆𝑘(𝛼; 𝑥, 𝑦) ≪ 𝑦L−𝐴,
for 𝑆𝑘(𝛼; 𝑥, 𝑦) in major arcs 𝑀(L𝑏), 𝜏 = 𝑦5𝑥−2L−𝑏1 , L = ln 𝑥𝑞, except for a small neighborhood of their centers |𝛼 − 𝑎
𝑞 | >
(︀
2𝜋𝑘2𝑥𝑘−2𝑦2
)︀−1, when 𝑦 ⩾ 𝑥1− 1
2𝑘−1+𝜂𝑘 L𝑐𝑘 , where
𝜂𝑘 =
2
4𝑘 − 5 + 2
√︀
(2𝑘 − 2)(2𝑘 − 3)
, 𝑐𝑘 =
2𝐴 + 22 +
(︁
2
√
√ 2𝑘−3
2𝑘−2 − 1
)︁
𝑏1
2
√︀
(2𝑘 − 2)(2𝑘 − 3) − (2𝑘 − 3)
,
and 𝐴, 𝑏1, 𝑏 are arbitrary fixed positive numbers. Furthermore, and we also proved an asymptotic formula on a small neighborhood of the centers of major arcs.
About the Author
Zarullo Khusenovich Rakhmonov
National Academy of Sciences of Tajikistan, A. Dzhuraev Institute of Mathematics
Tajikistan
Tajikistan
doctor of physical and mathematical sciences, professor,
Academician
References
1. Vinogradov, I. M., 1952, Izbrannye trudy. (Russian) [Selected works.], Izdat. Akad. Nauk SSSR, Moscow.
2. Vinogradov, I. M., & Karatsuba, A. A., 1986, “The method of trigonometric sums in number theory”, Proc. Steklov Inst. Math., vol. 168, pp. 3-30.
3. Haselgrove C. B., 1951, “Some theorems in the analitic theory of number”, J. London Math. Soc., vol. 26, pp. 273-277.
4. Statulevicius, V., 1955, "On the representation of odd numbers as the sum of three almost equal prime numbers Univ. Mokslo Darbai. Mat. Fiz. Chem. Mokslu Ser, vol. 3, pp. 5-23.
5. Pan Chengdong, Pan Chengbiao, 1990, “On estimations of trigonometric sums over primes in short intervals (III)”, Chinese Ann. of Math., vol. 2. pp. 138-147.
6. Zhan, T., 1991, “On the Representation of large odd integer as a sum three almost equal primes”, Acta Math Sinica. New ser., vol. 7, Is. 3. pp. 135 – 170.
7. Liu, J., & Zhan, T., 1999, “Estimation of exponential sums over primes in short intervals I”, Monatshefte fur Mathematik, vol. 127, Is. 1, pp. 27-41.
8. Liu, J., & Zhan, T., 2000, “Hua’s Theorem on Prime Squares in Short Intervals”, Acta Mathematica Sinica. English Series, vol. 16, Is. 4. pp. 669-690.
9. Liu J., & Lu, G., Zhan, T., 2006, “Exponential sums over primes in short intervals”, Science in China: Series A Mathematics, vol. 49, Is. 5, pp. 611-619. doi:10.1007/s11425-006-0611-x
10. Hua, L. K., 1938, “Some results in the additive prime number theory”, Quart. J. Math., vol. 9, Is. 1, pp. 68-80.
11. Liu J., & Zhan T., 1996, “Estimation of exponential sums over primes in short intervals II”, In Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam,
12. Birkhauser, pp. 571 – 606.
13. Jutila, M., 1991, “Mean value etstimates for exponential sums with applications to 𝐿-functions”, Acta Arithmetica, vol. 57, Is. 2. pp. 93-114.
14. Kumchev, A. V., 2012, “On Weyl sums over primes in short intervals”, “Arithmetic in Shangrila”—Proceedings of the 6th China-Japan Seminar on Number Theory. Series on Number
15. Theory and Its Applications, vol. 9, Singapore: World Scientific, pp. 116–131.
16. Yao, Y., 2014, “Sums of nine almost equal prime cubes”, Frontiers of Mathematics in China, vol. 9, Is. 5. pp. 1131-1140. doi:10.1007/s11464-014-0384-4.
17. Rakhmonov, Z. Kh., & Rakhmonov, F. Z., 2016, “Estimation of short cubic exponential sums with prime numbers in minor arcs”, Doklady Akademii nauk Respubliki Tajikistan, vol. 59,
18. no 7-8, pp. 273-277, (in Russian).
19. Rakhmonov, Z. Kh.,& Rakhmonov, F. Z., 2017, “Short Cubic Exponential Sums over Primes”, Proceedings of the Steklov Institute of Mathematics, vol. 296, pp. 211–233.
20. doi.org/10.1134/S0081543817010175
21. Rakhmonov, Z. Kh.,& Rakhmonov, F. Z., 2013, “The sum of short double trigonometric sums”, Doklady Akademii nauk Respubliki Tajikistan, vol. 56, no 11, pp. 853-860, (in Russian).
22. Rakhmonov, Z. Kh.,& Rakhmonov, F. Z., 2014, “Sum of short exponential sums over prime numbers”, Doklady Mathematics, vol. 90, No 3, pp. 699–700. doi.org/10.1134/S1064562414070138.
23. Rakhmonov, Z. Kh., & Zamonov, B. M., 2014, “Short cubic double exponential sums, with a long continuous summation”, Izvestiya Akademii nauk Respubliki Tajikistan. Otdeleniye fizikomatematicheskikh, khimicheskikh, geologicheskikh i tekhnicheskikh nauk, № 4(157), pp. 7-23, (in Russian).
24. Rakhmonov, Z. Kh., & Rakhmonov, F. Z., Zamonov, B.M., 2016, “Estimates of short cubic double exponential sums with a long continuous summation”, Chebyshevskii Sbornik, vol. 17,
25. Is. 1, pp. 217–231.
26. Rane, V. V., 1980, “On the mean square value of Dirichlet 𝐿-series”, J. London Math. Soc., vol. s2-21, Is. 2, pp. 203-215, doi-org.eres.qnl.qa/10.1112/jlms/s2-21.2.203.
27. Zhan T., 1992, “On the Mean Square of Dirichlet 𝐿-Functions”, Acta Mathematica Sinica. New Series, vol. 8, No 2, pp. 204-224.
28. Rakhmonov, Z. Kh., & Sobirov A. A., Fozilova P. M., 2020, “Behavior of short cubic exponential sums with primes in a small neighborhood of the center of major arcs”, Doklady Akademii nauk Respubliki Tajikistan, vol. 63, no 5-6, pp. 279-288, (in Russian).
29. Rakhmonov, Z. Kh., & Sobirov A. A., Fozilova P. M., 2020, “Estimate of short cubic exponential sums with primes in major arcs”, Doklady Akademii nauk Respubliki Tajikistan, vol. 63, no 7-8, pp.405-415, (in Russian).
30. Arkhipov G. I. & Chubarikov V. N. & Karatsuba A. A. 2004. Trigonometric sums in number theory and analysis, Berlin–New-York: Walter de Gruyter, 554 p.
31. Davenport H., 1967, Multiplicative Number Theory, Markham Publishing Company, Chigago.
32. Prachar K., 1957, Primzahlverteilung, Springer-Verlag.
33. Karatsuba A. A., 1993, Basic analytic number theory, Springer-Verlag, Berlin, xiv+222 pp.
34. Vinogradov I. M., 2003 Elements of Number Theory, Mineola, NY: Dover Publications, ISBN 0-486-49530-2.
Review
For citations:
Rakhmonov Z.Kh. Estimates of short exponential sums with primes in major arcs. Chebyshevskii Sbornik. 2021;22(4):200-224. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-4-200-224
Views:
247
Email this article 