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 Chebyshevskii Sb., 2019, Volume 20, Issue 2, Pages 336–347 (Mi cheb774)

The mean value of products of Legendre symbol over primes

V. N. Chubarikov

Mechanics and Mathematics Faculty, Moscow State University named after M. V. Lomonosov (Moscow)

Abstract: In the paper the asymptotical formula as $N\to\infty$ for the number of primes $p\leq N,$ satisfying to the system of equations
$$(\frac{p+k_s}{q_s})=\vartheta_s, s=1,…,r,$$
where $q_1,…,q_r$ — different primes, $\vartheta_s$ may be take only two values $+1$ or $-1,$ but natural numbers $k_s$ take noncongruent values on modulus $q_s, s=1,…,r,$ i.e. $k_s\not\equiv k_t\pmod{q_s}, t=1,…,r,$ is found.
The finding asymptotics is nontrivial as $q=q_1…q_r\gg N^{1+\varepsilon},$ moreover the number of $r$ may grow up as $o(\ln N).$ Here $\varepsilon>0$ is an arbitrary constant.

Keywords: the Legendre symbol, the Vinogradov method of estimating on sums over primes, the Dirichlet's character, the Vinogradov's combinatorial sieve, the method of double sums.

DOI: https://doi.org/10.22405/2226-8383-2018-20-2-336-347

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UDC: 511.3
Accepted:12.07.2019

Citation: V. N. Chubarikov, “The mean value of products of Legendre symbol over primes”, Chebyshevskii Sb., 20:2 (2019), 336–347

Citation in format AMSBIB
\Bibitem{Chu19} \by V.~N.~Chubarikov \paper The mean value of products of Legendre symbol over primes \jour Chebyshevskii Sb. \yr 2019 \vol 20 \issue 2 \pages 336--347 \mathnet{http://mi.mathnet.ru/cheb774} \crossref{https://doi.org/10.22405/2226-8383-2018-20-2-336-347}