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A new minor-arcs estimate for number fields. (English) Zbl 0927.11049

Ahlgren, Scott D. (ed.) et al., Topics in number theory. In honor of B. Gordon and S. Chowla. Proceedings of the conference, Pennsylvania State University, University Park, PA, USA, July 31–August 3, 1997. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 467, 151-161 (1999).
It is well-known that in 1937 I. M. Vinogradov succeeded in obtaining the striking Goldbach three primes theorem by his treatment on exponential sums over minor arcs defined as in the Hardy-Littlewood circle method. Later O. Körner [Math. Ann. 144, 224-238 (1961; Zbl 0163.04102)] obtained a number field version of Vinodradov’s bound on minor arcs. In a previous work, the author considered the number field \(\mathbb{K}\) having class number one and extended the main result of T. D. Wooley [Ann. Math. (2) 135, 131-164 (1992; Zbl 0754.11026)] with a blemish on the coefficient in the second main term, namely, the Waring-Siegel function \(G_{\mathbb{K}}(k)\) satisfies \[ G_{\mathbb{K}}(k)\leq k\log k+3k\log\log k+ O(k). \tag{1} \] In the present paper the author re-examines Körner’s results and obtains an explicit estimate in the number field version of Vinogradov’s bound on generalized minor arcs which is too lengthy to clearly mention here. The author claims that by those results in the paper for the proof of this new estimate, the coefficient 3 in (1) can be replaced by 1. So now there is the same upper bound for \(G_{\mathbb{K}}(k)\) as Wooley’s bound for the Waring function \(G(k)\) [ibid., Corollary 1.2.1].
For the entire collection see [Zbl 0913.00029].

MSC:

11P05 Waring’s problem and variants
11L07 Estimates on exponential sums
11R99 Algebraic number theory: global fields
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