## The number of powers of 2 in a representation of large even integers. II.(English)Zbl 0924.11086

It was shown by Yu. V. Linnik [Mat. Sb. Nov. Ser. 32(74), 3-60 (1953; Zbl 0051.03402)] that every sufficiently large even integer $$N$$ can be written as a sum of two primes and a bounded number of powers is 2 $N-p_1+ p_2+ 2^{\nu_1}+\cdots+ 2^{\nu_k}, \quad k\leq k_0.$ The method was simplified and improved by P. X. Gallagher [Invent. Math. 29, 125-142 (1975; Zbl 0305.10044)]. In the first part [Sci. China, Ser. A 41, 386-398 (1998)] the authors showed that on the generalized Riemann hypothesis 770 powers of 2 suffice. They now prove unconditionally that $$k_0= 54 000$$ is a possible numerical value for $$k_0$$. The proof follows Gallagher’s method, combined with explicit numerical bounds for zero free regions and density results for Dirichlet’s $$L$$-functions.

### MSC:

 11P32 Goldbach-type theorems; other additive questions involving primes 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11P55 Applications of the Hardy-Littlewood method 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$

### Citations:

Zbl 0990.46081; Zbl 0051.03402; Zbl 0305.10044
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### References:

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