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JOURNALS // Trudy Matematicheskogo Instituta imeni V.A. Steklova // Archive

Trudy Mat. Inst. Steklova, 2017 Volume 296, Pages 220–242 (Mi tm3774)

This article is cited in 4 papers

Short cubic exponential sums over primes

Z. Kh. Rakhmonov, F. Z. Rahmonov

Mathematics Institute and Computing Center, Academy of Sciences of the Republic of Tadzhikistan

Abstract: For $y\ge x^{4/5}\mathscr L^{8B+151}$ (where $\mathscr L=\log (xq)$ and $B$ is an absolute constant), a nontrivial estimate is obtained for short cubic exponential sums over primes of the form $S_3(\alpha ;x,y) = \sum _{x-y<n\le x}\Lambda (n) e(\alpha n^3)$, where $\alpha =a/q+\theta /q^2$, $(a,q)=1$, $\mathscr L^{32(B+20)}<q\le y^5x^{-2}\mathscr L^{-32(B+20)}$, $|\theta |\le 1$, $\Lambda $ is the von Mangoldt function, and $e(t)=e^{2\pi it}$.

UDC: 511.325

Received: May 6, 2016

DOI: 10.1134/S0371968517010174


 English version:
Proceedings of the Steklov Institute of Mathematics, 2017, 296, 211–233

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