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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 xexp(67) we have

CardN ]x,2x], p 1 , p 2 : N = p 1 + p 2 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Theorem 2 to Theorem 1 is done by means of a generalization of a theorem of Ostmann on sum sets (J. M. Deshouillers, unpublished).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 (N)= p 1 +p 2 =N,p 1 x,p 2 x logp 2 (x<N2x)·

If λ d is the well-known coefficient in Selberg’s sieve with a parameter z]1,x 1/2 ] (which will finally be chosen x 1/2 (logx) -1/4 ), and β(y)=( d|y λ d ) 2 then

r 2 (N)R 2 (N)= y+p 2 =N,p 2 x β(y)logp 2 ·

R 2 (N) can be written as

dz 2 w d * amodd T(a/d)e(-Na/d),

where

w d = d 1 ,d 2 ,d|[d 1 ,d 2 ] λ d 1 λ d 2 [ d 1 , d 2 ] -1 ,T(α)= px logp·e(pα)·

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= N]x,2x] ρ 2 -1 (N)r 2 (N)ρ 2 (n)= pN,p2 p-1 p-2· p3 1-1 (p-1) 2 ·

By careful numerical consideration it is shown (Proposition 1): For xexp(67) we have R0·478x 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 * = dz 2 w d * amodd T(a/d)U ¯(a/d),

where U(α)= N]x,2x],r 2 (N)0 ρ 2 -1 (N)e(Nα). 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:
11P32Additive questions involving primes