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.
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. 
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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/

[1] I. M. Vinogradov, Special Variants of the Method of Trigonometric Sums, Nauka, M., 1976, 119 pp. | MR

[2] I. M. Vinogradov, “Distribution of quadratic residues and non-residues of the type $p+k$ with the respect to prime modulus”, Matem.sb., 3(45) (1938), 311–320 (abstract in English)

[3] I. M. Vinogradov, “An improvement of the method of estimating sums of primes”, Izv. Akad. Nauk SSSR Ser. Matem., 7 (1943), 17–34 (abstract in English)

[4] I. M. Vinogradov, “A new approach to estimating the values of $\chi(p+k)$”, Izv. Akad. Nauk SSSR Ser. Matem., 16 (1952), 197–210 (abstract in English) | Zbl

[5] I. M. Vinogradov, “An improved estimate of the sums of $\chi(p+k)$”, Izv. Akad. Nauk SSSR Ser. Matem., 17 (1953), 285–290 (abstract in English) | Zbl

[6] A. A. Karatsuba, “Character sums over primes”, Izv. Akad. Nauk SSSR Ser. Matem., 34 (1970), 299–321 (abstract in English)

[7] A. A. Karatsuba, “Character sums over primes belonging to the arithmetic progression”, Izv. Akad. Nauk SSSR Ser. Matem., 35 (1971), 469–484 (abstract in English) | Zbl

[8] I. M. Vinogradov, The method of trigonometric sums in number theory, 2-nd edition, revised and enlarged, Nauka, M., 1980, 144 pp. | MR

[9] I. M. Vinogradov, Elements of Number Theory, 10-th Edition, Publ. house ‘`Lan’", Sankt-Petersburg, 2004, 176 pp. | MR

[10] L. K. Hua, Selected Papers, Springer Verlag, New York, 1983, 888 | MR

[11] L. K. Hua, The method of trigonometric sums in number theory, Mir, M., 1964, 188

[12] G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, The theory of multiple trigonometric sums, Nauka, M., 1987, 368 pp. | MR

[13] G. I. Arkhipov, Selected Papers, Izd-vo Orlovskogo gosudarstvennogo universiteta, Orjol, 2013, 464 pp.

[14] G. I. Arkhipov, V. N. Chubarikov, A. A. Karatsuba, Trigonometric Sums in Number Theory and Analysis, de Gruyter Expositions in Mathematics, 39, Walter de Gruyter, Berlin–New York, 2004 | MR | Zbl