# Volume 21

### 1. On a method of Davenport and Heilbronn I.

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 $p_j$.

### 2. On the Barban-Davenport-Halberstam theorem : X

We return to the topics of the third and ninth papers of the series, and the main aim is to remove some conditions on the previous results by means of a large-sieve estimate due to H. L. Montgomery.