Short cubic exponential sums with Möbius function

Z. Kh. Rakhmonov ; F. Z. Rahmonov

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 281-305.

Résumé

The work is dedicated to the conclusion of non-trivial estimates of short cubic exponential sums with Möbius function of the form $$ S_3(\alpha;x,y) = \sum_{x-y\le x} \mu(n) e(\alpha n^3), $$ over minor arcs $\mathfrak{m}(\mathscr L^{32(B+18)})$ for $y\ge x^\frac{4}{5}\mathscr L^{8B+944}$ and $\tau=y^5x^{-2}\mathscr L^{-32(B+18)}.$

Détail

Keywords: shorts double exponential sum, Möbius function, method for estimating exponential sums with prime numbers, nontrivial estimate, minor arcs.

Bibliographie

[1] Davenport H., “On some infinite series involving arithmetical functions (II)”, The Quarterly Journal of Mathematics, 8:1 (1937), 313–320 | DOI | MR

[2] Hua L. K., Additive theory of prime numbers, American Mathematical Soc., 1965, 190 pp. | MR | Zbl

[3] Baker R. C., Harman G., “Exponential sums formed with the Mobius function”, J. London Math. Soc., 2-43:2 (1991), 193–198 | DOI | MR | Zbl

[4] Liu J. Y., Zhan T., “Exponential sums involving the Mobius function”, Indag. Math. (N.S.), 7:2 (1996), 271–278 | DOI | MR | Zbl

[5] Zhan T., “Davenport's theorem in short intervals”, Chin. Ann. of Math., 12 B:4 (1991), 421–431 | MR | Zbl

[6] Zhan T., “On the representation of large odd integer as a sum of three almost equal primes”, Acta Math Sinica. New ser., 7:3, 135–170 | MR

[7] Liu J. Y., Zhan T., “Estimation of exponential sums over primes in short intervals I”, Monatshefte für Mathematik, 127:1 (1999), 27–41 | DOI | MR | Zbl

[8] Lu G. S., Lao H. X., “On exponential sums over primes in short intervals”, Monatshefte fur Mathematik, 151:2 (2007), 153–164 | DOI | MR | Zbl

[9] Kumchev A V., “On Weyl sums over primes in short intervals”, Arithmetic in Shangrila, Proceedings of the 6th China-Japan Seminar on Number Theory, Series on Number Theory and Its Applications, 9, World Scientific, Singapore, 2012, 116–131 | MR

[10] Rakhmonov Z. Kh., Rakhmonov F. Z., “Short Cubic Exponential Sums over Primes”, Proceedings of the Steklov Institute of Mathematics, 296 (2017), 211–233 | DOI | DOI | MR | Zbl

[11] Rakhmonov Z. Kh., Rakhmonov F. Z., Zamonov B.M., “Estimates of short cubic double exponential sums with a long continuous summation”, Chebyshevskii Sbornik, 17:1 (2016), 217–231 | MR | Zbl

[12] Rakhmonov Z. Kh., Zamonov B. M., “Short cubic double exponential sums, with a long continuous summation”, Izvestiya Akademii nauk Respubliki Tajikistan. Otdeleniye fiziko-matematicheskikh, khimicheskikh, geologicheskikh i tekhnicheskikh nauk, 2014, no. 4(157), 7–23 (in Russian)

[13] Rakhmonov Z. Kh., Rakhmonov F. Z., “Sum of short exponential sums over prime numbers”, Doklady Mathematics, 90:3 (2014), 699–700 | DOI | MR | Zbl

[14] Rakhmonov Z. Kh., Rakhmonov F. Z., “The sum of short double trigonometric sums”, Doklady Akademii nauk Respubliki Tajikistan, 56:11 (2013), 853–860 (in Russian)

[15] Rahmonov F. Z., “Estimate of quadratic trigonometric sums with prime numbers”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2011, no. 3, 56–60 | Zbl

[16] Rakhmonov Z. Kh., “Theorem on the mean value of $\psi(x,\chi)$ and its applications”, Russian Academy of Sciences. Izvestiya Mathematics, 43:1 (1994), 49–64 | DOI | MR | Zbl

[17] Vinogradov I. M., Special variants of the method of trigonometric sums, Izdat. “Nauka”, M., 1976, 119 pp. (Russian) | MR

[18] Mardjhanashvili K. K., “An estimate for an arithmetic sum”, Doklady Akad. Nauk SSSR, 22:7 (1939), 391–393