# $L^1$ norm of exponential sum of $n^2 x$

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.

Jurkat and van Horne 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 and Marklof. Finally Vaughan and Wooley considered Weyl sums for powers larger than $2$, and formulated some conjectures -- the case for squares behaves differently from higher powers.
• The answer is conjectured to be $\frac{1}{2}\sqrt{\pi} \sqrt{N}$, and the numerical evidence seems to support this conjecture (see the cited paper of Vaughan and Wooley). The situation is not different for squares versus higher powers for the first moment. The differences appear for fourth moments, since the major arc contribution takes over for squares at the fourth moment, and this has a logarithmic factor. – tdw Jun 4 '15 at 10:37