Mean-value theorem for non-complete rational trigonometric sums

V. N. Chubarikov   ; H. M. Saliba

Čebyševskij sbornik, Tome 19 (2018) no. 4, pp. 252-258

Résumé

For $2k>0.5n(n+1)+1$ $0\leq l\leq 0,5k-w-1, w=[\ln n/\ln p,]$ the asymptotic formulas was proved for the number of solutions of the system of congruences $$ \left\{ \begin{array}{l} x_1+\dots+x_k\equiv y_1+\dots +y_k\pmod{p^m}\ \dots\qquad\dots\qquad\dots\qquad\dots\qquad \ x_1^n+\dots+x_k^n\equiv y_1^n+\dots +y_k^n\pmod{p^m}, \end{array} \right. $$ where unknowns $x_1,\dots ,x_k,y_1,\dots,y_k$ run values up $1$ to $p^{m-l}$ from the complete system residues by modulo $p^{m}.$ The finding formula for $2k\leq 0.5n(n+1)+1$ has no the place. Let be $1\leq s$ Then as $2k>s+r+\dots +n$ for the number of the system of congruencies $$ \left\{ \begin{array}{l} x_1^s+\dots+x_k^s\equiv y_1^s+\dots +y_k^s\pmod{p^m} \ x_1^r+\dots+x_k^r\equiv y_1^r+\dots +y_k^r\pmod{p^m} \ \dots\qquad\dots\qquad\dots\qquad\dots\qquad\ x_1^n+\dots+x_k^n\equiv y_1^n+\dots +y_k^n\pmod{p^m}, \end{array} \right. $$ where unknowns $x_1,\dots ,x_k,y_1,\dots,y_k$ run values up $1$ to $p^{m-l}$ from the complete system residues by modulo $p^m,$ was found the asymptotic formula. This formula has no place as $2k\leq s+r+\dots +n.$

Détail

Keywords: non-complete rational trigonometric sums, Hua Loo-keng's method, the exponent of convergence of the average value of non-complete trigonometric sums.

Bibliographie

[1] I. M. Vinogradov, Metod trigonometricheskih summ v teorii chisel, Nauka, M., 1980 | MR

[2] L. K. Hua, Selected Papers, Springer Verlag, New York Inc, 1983, 888 pp. | MR | Zbl

[3] G. I. Arhipov, Izbrannye trudy, Izd-vo Orlovskogo gos. un-ta, Orel, 2013, 464 pp.

[4] G. I. Arhipov, A. A. Karacuba, V. N. Chubarikov, Teoriya kratnyh trigonometricheskih summ, Nauka, M., 1987 | MR

[5] G. I. Arkhipov, V. N. Chubarikov, A. A. Karatsuba, Trigonometric Sums in Number Theory and Analysis, de Gruyter Expositions in Mathematics, 39, Walter de Gruyter, Berlin–New York, 2004 | MR | Zbl

[6] V. N. Chubarikov, “Ob asimptoticheskih formulah dlya intngrala I. M. Vinogradova i ego obobshchenij”, Tr. MIAN, 157, 1981, 214–232 | Zbl

[7] V. N. Chubarikov, “Kratnye polnye racional'nye arifmeticheskie summy ot znachenij mnogochlena”, Dokl. RAN, 478:1 (2018), 22–24 | Zbl

[8] L. G. Arhipova, V. N. Chubarikov, “Pokazatel' skhodimosti osobogo ryada odnoj mnogomernoj problemy”, Vestn. Mosk. un-ta. Ser. I, Matematika, mekhanika, 2018, no. 5, 59–62

[9] H. M. Saliba, “On non-complete rational trigonometric sums”, Chebyshevskii sbornik, 19:3 (2018) | MR

[10] V. N. Chubarikov, “Ob odnoj teoreme o srednem”, Vestn. Mosk. un-ta. Ser. I, Matematika, mekhanika, 2019, no. 1, 59–62