This important article contains a dramatic progress concerning the problem of determining Schnirel’man’s constant, the least integer such that every integer can be written as a sum of at most 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 we have
For small numbers 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 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Theorem 2 to Theorem 1 is done by means of a generalization of a theorem of Ostmann on sum sets (J. M. Deshouillers, unpublished).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 is the well-known coefficient in Selberg’s sieve with a parameter (which will finally be chosen , and then
can be written as
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 we have .
Note that the factor is very near to the expected optimal value . is estimated from above by
where . requires a lot of effort, both in theoretical and numerical respect. The bounds for easily give Theorem 2.
The article is well organized and a pleasure to read.