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Exponential sums over primes in short intervals

Authors:

LIU Jianya L Guangshi ZHAN Tao

Institution:

Department of Mathematics, Shandong University, Jinan 250100, China

Abstract:

In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua’s result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as

$N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + p_5^2 , with |p_j - \sqrt {N/5} | \leqslant U = N^{\frac{1}{2} - \frac{1}{{20}} + \varepsilon }$

, where p _j are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.

Keywords:

exponential sums over primes circle method Waring-Goldbach problems

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