Geodesic
On the exponents of the convergence of singular integrals and singular series of a multivariate problem
Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 46-57 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper we continue studies on the theory of multivariate trigonometric sums, in the base of which lies of the I.M.Vinogradov's method. Here we obtain for $n=r=2$ lower estimates of the convergence exponent of the singular series and the singular integral of the asymptotic formulas for $P\to\infty$ for the number of solutions of the following system of Diophantine equations $$ \sum_{j=1}^{2k}(-1)^jx_{1,j}^{t_1}\dots x_{r,j}^{t_r}=0, 0\leq t_1,\dots, t_r\leq n, $$ where $n\geq 2,r\geq 1, k$ are natural numbers, moreover an each variable $x_{i,j}$ can take all integer values from $1$ to $P\geq 1.$
Keywords: exponent of the convergence, singular integrals, singular series. 
@article{CHEB_2019_20_4_a3,
     author = {L. G. Arkhipova and V. N. Chubarikov},
     title = {On the exponents of the convergence of singular integrals and singular series of a multivariate problem},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {46--57},
     year = {2019},
     volume = {20},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a3/}
}
TY  - JOUR
AU  - L. G. Arkhipova
AU  - V. N. Chubarikov
TI  - On the exponents of the convergence of singular integrals and singular series of a multivariate problem
JO  - Čebyševskij sbornik
PY  - 2019
SP  - 46
EP  - 57
VL  - 20
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a3/
LA  - ru
ID  - CHEB_2019_20_4_a3
ER  - 
%0 Journal Article
%A L. G. Arkhipova
%A V. N. Chubarikov
%T On the exponents of the convergence of singular integrals and singular series of a multivariate problem
%J Čebyševskij sbornik
%D 2019
%P 46-57
%V 20
%N 4
%U http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a3/
%G ru
%F CHEB_2019_20_4_a3
L. G. Arkhipova; V. N. Chubarikov. On the exponents of the convergence of singular integrals and singular series of a multivariate problem. Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 46-57. http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a3/

[1] Vinogradov I. M, The method of trigonometric sums in the number theory, Nauka, M., 1980, 144 pp. | MR

[2] Vinogradov I. M., Elements of the number theory, Ed. 10th, Publ. House «Lan'», Sankt-Petersburg, 2004, 176 pp. | MR

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

[4] Hua L.-K., The method of trigonometric sums and its applications in the number theory, Mir, M., 1964, 188 pp.

[5] Arkhipov G. I., Selected Works, Publ. House of Orjol University, Orjol, 2013

[6] Arkhipov G. I., Chubarikov V. N., Karatsuba A. A., The theory of multiple trigonometric sums, Nauka, M., 1987 | MR

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

[8] Saliba H. M., Chubarikov V. N., “On the multivariate Arkhipov–Karatsuba system of congruences”, Dokl. RAS, 472:6 (2017), 631–633 | Zbl

[9] Saliba H. M., Chubarikov V. N., “The theorem on a mean value for non complete rational trigonometric sums”, Chebyshev Sbornik, 20:1(69) (2019), 31–37

[10] Chubarikov V. N., “On a multiple integral”, Dokl. AS USSR, 227:6 (1976), 1308–1310 | MR | Zbl

[11] Chubarikov V. N., “On multiple rational trigonometric sums and multiple integrals”, Math. notes, 20:1 (1976), 61–68 | MR | Zbl

[12] Chubarikov V. N., “On the convergence exponent of the singular integral of a multivariate additive problem”, Dokl. RAS, 46:5 (2015), 530–532

[13] Chubarikov V. N., “On the convergence exponent of the mean value of complete rational arithmetical sums”, Chebyshev Sbornik, 16:4(56) (2015), 303–318 | MR | Zbl

[14] Arkhipova L. G., Chubarikov V. N., “On the convergence exponent of the singular series of a multivariate problem”, Vestn. Moscow University. Ser. 1, Mathematics, Mechanics, 2018, no. 5, 68–71 | Zbl

[15] Demidovich B. P., The collection of problems on the mathematical analysis, Ed. 19th, correct., Publ. House «Lan'», Sankt-Petersburg, 2017, 624 pp.

[16] Arkhipov G. I., Sadovnichii V. A., Chubarikov V. N., Lectures on the mathematical analysis, Textbook, Ed. 4th., correct., DROFA, M., 2004, 640 pp.