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On Šnirel’man’s constant. (English) Zbl 0851.11057
This important article contains a dramatic progress concerning the problem of determining Schnirel’man’s constant, the least integer $n$ such that every integer $> 1$ can be written as a sum of at most $n$ primes. The value 7 gained here is much nearer to the conjectured value 3 than to 19, the best result known hitherto [{\it H. Riesel} and {\it R. C. Vaughan}, Ark. Mat. 21, 45-74 (1983; Zbl 0516.10044)]. More precisely, the author shows: Theorem 1. Every even integer is a sum of at most 6 primes. This is derived from Theorem 2. For $x \ge \exp (67)$ we have $$\text{Card} \bigl\{ N \in\ ] x,2x],\ \exists p_1, p_2 : N = p_1 + p_2 \bigr\} \ge x/5.$$ For small numbers $N$ the author uses numerical results of {\it A. Granville}, {\it J. van de Lune}, and {\it H. J. J. te Riele} [in Number theory and applications, Proc. NATO ASI, Banff/Can. 1988, NATO ASI Ser., Ser. C 265, 423-433 (1989; Zbl 0679.10002)]. For large $N$ the step from Theorem 2 to Theorem 1 is done by means of a generalization of a theorem of Ostmann on sum sets (J. M. Deshouillers, unpublished). The proof of Theorem 2 in principle follows the classical approach. One has to find an upper bound for $$r_2 (N) = \sum_{p_1 + p_2 = N,\ p_1 \ge \sqrt x,\ p_2 \le x} \log p_2\qquad (x < N \le 2x).$$ If $\lambda_d$ is the well-known coefficient in Selberg’s sieve with a parameter $z \in\ ]1,x^{1/2}]$ (which will finally be chosen $\approx x^{1/2} (\log x)^{- 1/4})$, and $\beta (y) = (\sum_{d |y} \lambda_d)^2$ then $$r_2 (N) \le R_2 (N) = \sum_{y + p_2 = N,\ p_2 \le x} \beta (y) \log p_2.$$ $R_2 (N)$ can be written as $$\sum_{d \le z^2} w_d {\mathop {{\sum}^*}_{a \bmod d}} T(a/d)\ e(- Na/d),$$ where $$w_d = \sum_{d_1, d_2, d |[d_1, d_2]} \lambda_{d_1} \lambda_{d_2} \bigl( [d_1, d_2] \bigr)^{-1}, \quad T (\alpha) = \sum_{p \le x} \log p \cdot e(p \alpha).$$ Following {\it H. N. Shapiro} and {\it J. Warga} [Commun. Pure Appl. Math. 3, 153-176 (1950; Zbl 0038.18602)] the author derives lower and upper bounds for the expression $$R = \sum_{N \in\ ]x,2x]} \rho_2^{-1} (N)\ r_2 (N)\qquad \left( \rho_2 (n) = \sum_{p \mid N,\ p \ne 2} {p - 1 \over p - 2} \cdot \prod_{p \ge 3} \left( 1 - {1 \over (p - 1)^2} \right) \right).$$ By careful numerical consideration it is shown (Proposition 1): For $x \ge \exp (67)$ we have $R \ge 0.478 x^2 \log^{-1} x$. Note that the factor $0.478$ is very near to the expected optimal value $1/2$. $R$ is estimated from above by $$R^* = \sum_{d \le z^2} w_d {\mathop {{\sum}^*}_{a \bmod d}} T(a/d)\ \overline U(a/d),$$ where $U(\alpha) = \sum_{N \in\ ]x,2x], r_2 (N) \ne 0} \rho_2^{-1} (N) e(N \alpha)$. $R^*$ requires a lot of effort, both in theoretical and numerical respect. The bounds for $R$ easily give Theorem 2. The article is well organized and a pleasure to read.