*(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 [*H. Riesel* and *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 $x\ge exp\left(67\right)$ we have

For small numbers $N$ the author uses numerical results of *A. Granville*, *J. van de Lune*, and *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Theorem 2 to Theorem 1 is done by means of a generalization of a theorem of Ostmann on sum sets (J. M. Deshouillers, unpublished).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

If ${\lambda}_{d}$ is the well-known coefficient in Selberg’s sieve with a parameter $z\in \phantom{\rule{4pt}{0ex}}]1,{x}^{1/2}]$ (which will finally be chosen $\approx {x}^{1/2}{(logx)}^{-1/4})$, and $\beta \left(y\right)={\left({\sum}_{d|y}{\lambda}_{d}\right)}^{2}$ then

${R}_{2}\left(N\right)$ can be written as

where

Following *H. N. Shapiro* and *J. Warga* [Commun. Pure Appl. Math. 3, 153-176 (1950; Zbl 0038.18602)] the author derives lower and upper bounds for the expression

By careful numerical consideration it is shown (Proposition 1): For $x\ge exp\left(67\right)$ we have $R\ge 0\xb7478{x}^{2}{log}^{-1}x$.

Note that the factor $0\xb7478$ is very near to the expected optimal value $1/2$. $R$ is estimated from above by

where $U\left(\alpha \right)={\sum}_{N\in \phantom{\rule{4pt}{0ex}}]x,2x],{r}_{2}\left(N\right)\ne 0}{\rho}_{2}^{-1}\left(N\right)e\left(N\alpha \right)$. ${R}^{*}$ 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.

##### MSC:

11P32 | Additive questions involving primes |