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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
$$
\left(\frac{p+k_s}{q_s}\right)=\vartheta_s, s=1,\dots ,r,
$$
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.
Received: 19.05.2019 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
Linking options:
https://www.mathnet.ru/eng/cheb774 https://www.mathnet.ru/eng/cheb/v20/i2/p336
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Abstract page: | 207 | Full-text PDF : | 67 | References: | 19 |
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