On the fourth derivative test for exponential sums

Olivier Robert (1)

1 CTN - Combinatoire, théorie des nombres

We give an upper bound for the exponential sum $\sum_{m=1}^Mexp(2i\pi f(m))$ where $f$ is a real-valued function whose fourth derivative has the order of magnitude $\lambda>0$ small. Van der Corput's classical bound, in terms of $M$ and $\lambda$ only, involves the exponent $1/14$. We show how this exponent may be replaced by any $\theta<1/12$ without further hypotheses. The proof uses a recent result by Wooley on the cubic Vinogradov system.

Keywords: Exponential sums; Vinogradov system

Domains: Number Theory [math.NT]

Olivier Robert. On the fourth derivative test for exponential sums. Forum Mathematicum, 2016, 28 (2), pp.403-404. ⟨10.1515/forum-2014-0216⟩. ⟨hal-01464788⟩