k-free数对与连续square-full数
数论
2014-03-20 v2
摘要
我们考虑直到x的k-free整数对个数的渐近公式的误差项。我们的误差项改进了Heath-Brown、Brandes和Dietmann/Marmon的结果。然后我们将结果推广到k-free数的r元组，改进了Tsang之前的结果。此外，我们建立了连续square-full整数的误差项。最后，我们将证明对于所有θ<3和几乎所有D，与Pell方程x^2-Dy^2=1相关的基本解ε_D满足ε_D > D^θ。这改进/恢复了Fouvry和Jouve之前的结果。我们工作的主要工具是近似行列式方法。
引用
@article{arxiv.1212.3150,
title = {Pairs of k-free Numbers, consecutive square-full Numbers},
author = {T. Reuss},
journal= {arXiv preprint arXiv:1212.3150},
year = {2014}
}
备注
28 pages. The proof of the theorem about consecutive k-free numbers has been reworked and errors and typos have been corrected. The error exponent is now much stronger for k>3. A further application of the method about the size of the fundamental solution of a Pell equation has been added. References to work of Dietmann/Marmon and Fouvry/Jouve have been added. The paper has been restructured