On some additive problems of Goldbach’s type
https://doi.org/10.22405/2226-8383-2021-22-3-179-195
Abstract
In this paper, we find an asymptotic formula with power-saving remainder term for the number of solutions of one non-linear ternary problem with primes. The proof is based on the "precise formula"for Chebyshev’s function involving the zeros of Riemann zeta function. In fact, a ternary problem "at each zero"is solved. I. M. Vinogradov’s solution of the ternary Goldbach problem (1937, see [1], [2]) opened the way of solving a wide class of problems of the above type. In 1938, he found a power-saving estimate (with respect to the length of the summation interval) for the mean value of the modulus of the exponential sum with primes (see [2], theorem 3, p.82; theorems 6 and 7, p.86). Starting at 1996, G.I.Arkhipov, K.Buriev and the author have obtained several results concerning the exceptional sets in some binary problems of Goldbach’s type. These results used both the tools of the theory of Diophantine approximations and the "precise formulas"from Riemann’s zeta function theory. At the same time, the method of estimating of linear sums with primes based on the measure theory was derived in the papers of G. L. Watson, D. Bruedern, R. D. Cook and A. Perelli.
About the Authors
Holem Mansour Saliba
Notre Dame University-Louaize
Lebanon
Lebanon
Ph.D. in physics and mathematics
Vladimir Nikolaevich Chubarikov
Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
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Review
For citations:
Saliba H., Chubarikov V.N. On some additive problems of Goldbach’s type. Chebyshevskii Sbornik. 2021;22(3):179-195. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-3-179-195
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