Exponential sums over primes and the prime twin problem

Author: Buttkewitz Yvonne  

Publisher: Springer Publishing Company

ISSN: 0236-5294

Source: Acta Mathematica Hungarica, Vol.131, Iss.1-2, 2011-04, pp. : 46-58

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

The purpose of this paper is to obtain an effective estimate of the exponential sum $sum_{nle x}Lambda(n)eleft(left(frac{a}{q}+betaright)nright)$ (where e(α)=e 2π i α , α,β∈ℜ, (a,q)=1 and Λ is the von Mangoldt function) in the range ${(log x)}^{1/2+varepsilon}le qle frac{x}{{(logloglog x)}^{1+varepsilon}}$ and $|beta| . It improves Daboussi’s estimate [2, Theorem 1] in the range q≤(log x) D and x(log x)D q, D>0 and is valid in a wider range for β.