Sums of the Form 1/x_1^k + ... + 1/x_n^k Modulo a Prime

Ernie Croot

Sums of the Form 1/x_1^k + ... + 1/x_n^k Modulo a Prime

Number Theory 2007-05-23 v2 Combinatorics

Authors: Ernie Croot

Abstract

We show that for every $0 < \epsilon \leq 1$ and integer $k\geq 1$ , there exists an integer $n = n(\epsilon,k)$ so that for all primes $p$ , and integers $0 \leq a \leq p-1$ , there exist integers $1 \leq x_1 < ... < x_n \leq p^\epsilon$ such that $a \equiv x_1^{-1} + ... + x_n^{-1} \pmod{p}$ . This extends a result of I. Shparlinski.

Keywords

sums of reciprocals modulo prime exponential sums shparlinski result prime fields

Cite

@article{arxiv.math/0403360,
  title  = {Sums of the Form 1/x_1^k + ... + 1/x_n^k Modulo a Prime},
  author = {Ernie Croot},
  journal= {arXiv preprint arXiv:math/0403360},
  year   = {2007}
}

Comments

Light Corrections. The parameter h in the definition of T had to be a lot larger

Sums of the Form 1/x_1^k + ... + 1/x_n^k Modulo a Prime

Abstract

Keywords

Cite

Comments

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