# MathSciDoc: An Archive for Mathematician ∫

#### Number Theorymathscidoc:1701.24005

Acta Mathematica, 213, (2), 199-236, 2013.4
We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2\$s\$, with \$\${1\leqslant s\leqslant k^{2}-1}\$\$ . In this way, we prove the main conjecture for such even moments when \$\${1\leqslant s\leqslant \tfrac{1}{4}(k+1)^{2}}\$\$ , showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of\$s\$, these finding application to the asymptotic formula in Waring’s problem.
```@inproceedings{kevin2013on,
title={On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing},
author={Kevin Ford, and Trevor D. Wooley},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203403636815795},
booktitle={Acta Mathematica},
volume={213},
number={2},
pages={199-236},
year={2013},
}
```
Kevin Ford, and Trevor D. Wooley. On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing. 2013. Vol. 213. In Acta Mathematica. pp.199-236. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203403636815795.