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

Published online by Cambridge University Press:  17 February 2016

Bingrong Huang*
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China email [email protected]
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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].

Type
Research Article
Copyright
Copyright © University College London 2016 

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