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EXPONENTIAL SUMS OVER PRIMES IN SHORT INTERVALS AND AN APPLICATION TO THE WARING–GOLDBACH PROBLEM

Part of: Exponential sums and character sums Additive number theory; partitions

Published online by Cambridge University Press:  17 February 2016

Bingrong Huang
Bingrong Huang*
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China email brhuang@mail.sdu.edu.cn
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Abstract

Let ${\rm\Lambda}(n)$ be the von Mangoldt function, $x$ be real and $2\leqslant y\leqslant x$. This paper improves the estimate for the exponential sum over primes in short intervals

$$\begin{eqnarray}S_{k}(x,y;{\it\alpha})=\mathop{\sum }_{x<n\leqslant x+y}{\rm\Lambda}(n)e(n^{k}{\it\alpha})\end{eqnarray}$$
when $k\geqslant 3$ for ${\it\alpha}$ in the minor arcs. When combined with the Hardy–Littlewood circle method, this enables us to investigate the Waring–Goldbach problem concerning the representation of a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers, and improve the results of Wei and Wooley [On sums of powers of almost equal primes. Proc. Lond. Math. Soc. (3) 111(5) (2015), 1130–1162].

MSC classification

Primary: 11L20: Sums over primes
Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes
Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 508 - 523
Copyright
Copyright © University College London 2016 

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