Exponential sums over primes in short intervals and an application to the Waring--Goldbach problem

Abstract

Let $\Lambda (n)$ be the von Mangoldt function, $x$ real and $2\le y\le x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;α) = \sum_{x< n \leq x+y} Λ(n) e\left( n^k α\right) \] when $k\ge 3$ for $\alpha$ in the minor arcs. And then combined with the Hardy--Littlewood circle method, this enables us to investigate the Waring--Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers, which improves the results in Wei and Wooley [12].

15 pages. Comments are welcome! arXiv admin note: text overlap with arXiv:1409.3450 by other authors

References in corpus (1)

• Strong orthogonality between the Mobius function and nonlinear exponential functions in short intervals

Cited by in corpus (6)

• Discorrelation between primes in short intervals and polynomial phases
• Strong orthogonality between the Mobius function and nonlinear exponential functions in short intervals
• Möbius Disjointness for Nilsequences Along Short Intervals
• Möbius disjointness for a class of exponential functions
• Waring-Goldbach problem in short intervals
• On sums of powers of almost equal primes