Abstract
The paper is devoted to methods of solution of binary additive problems with semiprime numbers, which form sufficiently “rare” subsequences of the natural series. Additional conditions are imposed on these numbers; the main condition is belonging to so-called Vinogradov intervals. We solve two problems that are analogs to the Titchmarsh divisor problem; namely, based on the Vinogradov method of trigonometric sums, we obtain asymptotic formulas for the number of solutions to Diophantine equations with semiprime numbers of a specific form.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Balog and K. J. Friedlander, “A hybrid of theorems of Vinogradov and Piatetski-Shapiro,” Pac. J. Math., 156, 45–62 (1992).
E. Bombieri, “On the large sieve,” Mathematica, 12, 201–225 (1965).
M. E. Changa, “Primes in special intervals and additive problems with such numbers,” Mat. Zametki, 73, No. 3, 423-436 (2003).
S. A. Gritsenko, “A problem of I. M. Vinogradov,” Mat. Zametki, 39, No. 5, 625–640 (1986).
S. A. Gritsenko, “The Goldbach ternary problem and the Goldbach–Waring problem with prime numbers lying in intervals of special form,” Usp. Mat. Nauk, 43, No. 4 (262), 203–204 (1988).
S. A. Gritsenko, “Three additive problems,” Izv. Akad. Nauk SSSR. Ser. Mat., 56, No. 6 (9262), 1198–1216 (1992).
S. A. Gritsenko and N. A. Zinchenko, “On an estimate for one trigonometric sum over prime numbers,” Nauch. Ved. Belgorod Univ. Ser. Mat. Fiz., 5 (148), No. 30, 48–52 (2013).
C. Hooley, Applications of Sieve Methods to the Theory of Numbers, Cambridge Univ. Press, Cambridge (1976).
A. A. Karatsuba, Foundations of Analytic Number Theory [in Russian], Nauka, Moscow (1983).
Yu. V. Linnik, Dispersion Method in Binary Additive Problems, Leningrad State Univ., Leningrad (1961).
E. C. Titchmarsh, “A divisor problem,” Rend. Circ. Mat. Palermo, 54, 414–429 (1930).
D. I. Tolev, “On a theorem of Bombieri–Vinogradov type for prime numbers from a thin set,” Acta Arithm., 81, No. 1, 57–68 (1997).
A. I. Vinogradov, “The density hypothesis for Dirichet L-series,” Izv. Akad. Nauk SSSR. Ser. Mat., 29, No. 4, 903–934 (1965).
I. M. Vinogradov, “A general property of prime numbers distribution,” Mat. Sb., 7 (49), No. 2, 365–372 (1940).
I. M. Vinogradov, Method of Trigonometric Sums in Number Theory [in Russian], Nauka, Moscow (1971).
I. M. Vinogradov, Foundations of Number Theory [in Russian], Nauka, Moscow (1972).
N. A. Zinchenko, “Binary additive problem with semi-prime numbers of a special form,” Chebyshev. Sb., 6, No. 2 (14), 145–162 (2005).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 166, Proceedings of the IV International Scientific Conference “Actual Problems of Applied Mathematics,” Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part II, 2019.
Rights and permissions
About this article
Cite this article
Zinchenko, N.A. On Additive Binary Problems with Semiprime Numbers of a Specific Form. J Math Sci 260, 175–193 (2022). https://doi.org/10.1007/s10958-022-05682-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-05682-6