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A Bombieri-type theorem for exponential sums

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Abstract

Let f k (n) be the characteristic function of n with Ω(n) = k, and

$$T_k \left( {x,\alpha } \right) = \sum\limits_{n \leqslant x} {f_k \left( n \right)e\left( {n\alpha } \right)} .$$

The main purpose of this paper is to establish a Bombieri-type mean-value theorem for T k (x, α), via using the modified Huxley-Hooley contour and the large-sieve type zero-density estimate for Dirichlet L-functions. The result plays an important role in handling the enlarge major arcs when we try to solve the Waring-Goldbach type problem by the circle method.

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Shanghai University, Shanghai, 200444, P. R. China

    Wei Li Yao

  2. School of Mathematics, Shandong University, Ji’nan, 250100, P. R. China

    Wei Li Yao

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  1. Wei Li Yao
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Correspondence to Wei Li Yao.

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Supported by National Natural Science Foundation of China (Grant No. 11271249) and the First-class Discipline of Universities in Shanghai

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Yao, W.L. A Bombieri-type theorem for exponential sums. Acta. Math. Sin.-English Ser. 29, 1997–2012 (2013). https://doi.org/10.1007/s10114-013-1519-3

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  • DOI: https://doi.org/10.1007/s10114-013-1519-3

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