A Diophantine approximation problem with two primes and one $k$-th power of a prime

Alessandro Gambini; Alessandro Languasco; Alessandro Zaccagnini

A Diophantine approximation problem with two primes and one $k$-th power of a prime

Number Theory 2018-02-14 v3

Authors: Alessandro Gambini , Alessandro Languasco , Alessandro Zaccagnini

Abstract

We refine a result of the last two Authors of [8] on a Diophantine approximation problem with two primes and a $k$ -th power of a prime which was only proved to hold for $1<k<4/3$ . We improve the $k$ -range to $1<k\le 3$ by combining Harman's technique on the minor arc with a suitable estimate for the $L^4$ -norm of the relevant exponential sum over primes $S_k$ . In the common range we also give a stronger bound for the approximation.

Keywords

Diophantine approximation primes exponential sums minor arcs Harman's technique L^4-norm k-th power

Cite

@article{arxiv.1706.00343,
  title  = {A Diophantine approximation problem with two primes and one $k$-th power of a prime},
  author = {Alessandro Gambini and Alessandro Languasco and Alessandro Zaccagnini},
  journal= {arXiv preprint arXiv:1706.00343},
  year   = {2018}
}

Comments

arXiv admin note: text overlap with arXiv:1703.02381

A Diophantine approximation problem with two primes and one $k$-th power of a prime

Abstract

Keywords

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