On the exceptional set of a system of linear equations with prime numbers

I. A. Allakov ; B. Kh. Abrayev

Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 15-37

Résumé

Let $X$ — be a sufficiently large real number, $b_{1},b_{2}$-integers with $1\le{{b}_{1}},{{b}_{2}}\le X, {{a}_{ij}}$,$(i=1,2; j=\overline{1,4})$ — positive integers, $ {{p}_{ 1}}, \ldots ,{{p}_{4}}- $prime numbers. Let $ B=\max\left\{ 3\left|{{a}_{ij}}\right| \right\},$ $({{i=1,2;j=\overline{1,4}}}),$ $\bar{b}=(b_{1},b_{2}),$ $K= 9\sqrt{2}B^{3}\left|\bar{b} \right|,$ $E_{2,4}(X)= \left\{{{b}_{i}} \bigm| 1\leq b_{i}\leq X, {{b}_{i}}\ne {{a}_{i1}}{{p}_{1}}+\cdots +{{a}_{i4}}{{p}_{4}}, i=1,2\right\}.$ The paper studies the solvability of a system of linear equations $ {{ b}_{i}}= {{a}_{i1}}{{p}_{1}}+\cdots +{{a}_{i4}}{{p}_{4}}, i=1,2,$ in primes $p_{1},\ldots,p_{4}$ and for the first time a power estimate for the exceptional set $E_{2,4}(X)$ and a lower estimate for $ R(\bar b)$ — the number of solutions of the system under consideration in prime numbers, are obtained, namely, that if $X$ is sufficiently large and $ \delta (0\delta1) $ is sufficiently small real numbers, then: there exists a sufficiently large number $ A, $ such that for $ X>{{B}^ {A}} $ estimate is fair ${{E}_{2,4}}(X) {{X}^{2-\delta }};$ and for $ R(\bar b) $ given $ \bar {b}=(b_{1},b_{2}),$ $1\le b_{1},b_{2} \le X $ fair estimate $R(\bar{b})\ge {K}^{2- {\delta }}{{\left( \ln K \right)^{-4}}}, $ for all $ \bar b=(b_{1},b_{2})$ except for at most $ {X}^{2-{\delta}}$ pairs of them.

Détail

Keywords: equation, system of linear equations, prime numbers, integer coefficients, natural numbers, determinant, solvability criteria, set, cardinality of a set, estimate, power estimate, Dirichlet series, Dirichlet character, exceptional zero.