Geodesic
The mean value of products of Legendre symbol over primes
Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 336-347

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In the paper the asymptotical formula as $N\to\infty$ for the number of primes $p\leq N,$ satisfying to the system of equations $$ \left(\frac{p+k_s}{q_s}\right)=\vartheta_s, s=1,\dots ,r, $$ where $q_1,\dots ,q_r$ — different primes, $\vartheta_s$ may be take only two values $+1$ or $-1,$ but natural numbers $k_s$ take noncongruent values on modulus $q_s, s=1,\dots ,r,$ i.e. $k_s\not\equiv k_t\pmod{q_s}, t=1,\dots ,r,$ is found. The finding asymptotics is nontrivial as $q=q_1\dots q_r\gg N^{1+\varepsilon},$ moreover the number of $r$ may grow up as $o(\ln N).$ Here $\varepsilon>0$ is an arbitrary constant.
Mots-clés : the Legendre symbol
Keywords: the Vinogradov method of estimating on sums over primes, the Dirichlet's character, the Vinogradov's combinatorial sieve, the method of double sums. 
V. N. Chubarikov. The mean value of products of Legendre symbol over primes. Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 336-347. http://geodesic.mathdoc.fr/item/CHEB_2019_20_2_a25/
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