What is the asymptotic order of
$$
\int\_0^1 \left| \sum\_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx
$$
as $N \to \infty$. This should be known, but I cannot find it in the literature.



$L^1$ norm of exponential sum of $n^2 x$
[Jurkat and van Horne](https://projecteuclid.org/download/pdf_1/euclid.dmj/1077314936) showed that the $L^1$ norm is asymptotic to a constant times $\sqrt{N}$ (see Theorems 4 and 5 in their paper which compute all moments). For other related work see [Jurkat and van Horne](http://projecteuclid.org/download/pdf_1/euclid.dmj/1077303327) and [Marklof](http://projecteuclid.org/download/pdf_1/euclid.dmj/1077228505). Finally [Vaughan and Wooley](http://blms.oxfordjournals.org/content/30/2/113.abstract) considered Weyl sums for powers larger than $2$, and formulated some conjectures -- the case for squares behaves differently from higher powers. 


