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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 \(L(s, \chi)\)
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