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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 [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

$\text{Card}\left\{N\in \phantom{\rule{4pt}{0ex}}\right]x,2x\right],\phantom{\rule{4pt}{0ex}}\exists {p}_{1},{p}_{2}:N={p}_{1}+{p}_{2}\right\}\ge x/5·$

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Theorem 2 to Theorem 1 is done by means of a generalization of a theorem of Ostmann on sum sets (J. M. Deshouillers, unpublished).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

${r}_{2}\left(N\right)=\sum _{{p}_{1}+{p}_{2}=N,\phantom{\rule{4pt}{0ex}}{p}_{1}\ge \sqrt{x},\phantom{\rule{4pt}{0ex}}{p}_{2}\le x}log{p}_{2}\phantom{\rule{2.em}{0ex}}\left(x

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

${r}_{2}\left(N\right)\le {R}_{2}\left(N\right)=\sum _{y+{p}_{2}=N,\phantom{\rule{4pt}{0ex}}{p}_{2}\le x}\beta \left(y\right)log{p}_{2}·$

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

$\sum _{d\le {z}^{2}}{w}_{d}\underset{a\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}d}{{\sum }^{*}}T\left(a/d\right)\phantom{\rule{4pt}{0ex}}e\left(-Na/d\right),$

where

${w}_{d}=\sum _{{d}_{1},{d}_{2},d|\left[{d}_{1},{d}_{2}\right]}{\lambda }_{{d}_{1}}{\lambda }_{{d}_{2}}{\left(\left[{d}_{1},{d}_{2}\right]\right)}^{-1},\phantom{\rule{1.em}{0ex}}T\left(\alpha \right)=\sum _{p\le x}logp·e\left(p\alpha \right)·$

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

$R=\sum _{N\in \phantom{\rule{4pt}{0ex}}\right]x,2x\right]}{\rho }_{2}^{-1}\left(N\right)\phantom{\rule{4pt}{0ex}}{r}_{2}\left(N\right)\phantom{\rule{2.em}{0ex}}\left({\rho }_{2}\left(n\right)=\sum _{p\mid N,\phantom{\rule{4pt}{0ex}}p\ne 2}\frac{p-1}{p-2}·\prod _{p\ge 3}\left(1-\frac{1}{{\left(p-1\right)}^{2}}\right)\right)·$

By careful numerical consideration it is shown (Proposition 1): For $x\ge exp\left(67\right)$ 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

${R}^{*}=\sum _{d\le {z}^{2}}{w}_{d}\underset{a\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}d}{{\sum }^{*}}T\left(a/d\right)\phantom{\rule{4pt}{0ex}}\overline{U}\left(a/d\right),$

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