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On a method of Davenport and Heilbronn I.

Abstract : Let $\lambda_1, \lambda_2, \lambda_3$ be nonzero reals with $\lambda_1/\lambda_3$ negative irrational. Let $\varphi_j(u)\,(1\leq j\leq3)$ be smooth functions with derivatives $<\!\!\!< u^{-1}(\log u)^C\,(u\geq3)$. We prove in this paper that the inequality $\vert\sum_{j=1}^3\lambda_j(p_j+\varphi_j(p))\vert < \exp(-(\log(p_1p_2p_3))^{1/2})$ holds for infinitely many triplets of primes $p_j$.
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K Ramachandra. On a method of Davenport and Heilbronn I.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1998, Volume 21 - 1998, pp.12-26. ⟨10.46298/hrj.1998.136⟩. ⟨hal-01109319⟩

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