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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 \(x \geq \exp (67)\) we have \[ \text{Card} \bigl\{ N \in\;] x,2x],\;\exists p_1, p_2 : N = p_1 + p_2 \bigr\} \geq x/5. \] 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Theorem 2 to Theorem 1 is done by means of a generalization of a theorem of Ostmann on sum sets (J. M. Deshouillers, unpublished).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 \[ r_2 (N) = \sum_{p_1 + p_2 = N,\;p_1 \geq \sqrt x,\;p_2 \leq x} \log p_2\qquad (x < N \leq 2x). \] 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 \[ r_2 (N) \leq R_2 (N) = \sum_{y + p_2 = N,\;p_2 \leq x} \beta (y) \log p_2. \] \(R_2 (N)\) can be written as \[ \sum_{d \leq z^2} w_d {\mathop {{\sum}^*}_{a \bmod d}} T(a/d)\;e(- Na/d), \] where \[ w_d = \sum_{d_1, d_2, d |[d_1, d_2]} \lambda_{d_1} \lambda_{d_2} \bigl( [d_1, d_2] \bigr)^{-1}, \quad T (\alpha) = \sum_{p \leq x} \log p \cdot e(p \alpha). \] 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 \[ R = \sum_{N \in\;]x,2x]} \rho_2^{-1} (N)\;r_2 (N)\qquad \left( \rho_2 (n) = \sum_{p \mid N,\;p \neq 2} {p - 1 \over p - 2} \cdot \prod_{p \geq 3} \left( 1 - {1 \over (p - 1)^2} \right) \right). \] By careful numerical consideration it is shown (Proposition 1): For \(x \geq \exp (67)\) we have \(R \geq 0.478 x^2 \log^{-1} x\).
Note that the factor \(0.478\) is very near to the expected optimal value \(1/2\). \(R\) is estimated from above by \[ R^* = \sum_{d \leq z^2} w_d {\mathop {{\sum}^*}_{a \bmod d}} T(a/d)\;\overline U(a/d), \] where \(U(\alpha) = \sum_{N \in\;]x,2x], r_2 (N) \neq 0} \rho_2^{-1} (N) e(N \alpha)\). \(R^*\) 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.

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
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Online Encyclopedia of Integer Sequences:

Minimal number of primes needed to sum to n.

References:

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