Hardy-Littlewood series and even continued fractions

Abstract

For any $s\in (1/2,1]$, the series$F_s(x)=\sum_{n=1}^{\infty} e^{i\pi n^2 x}/n^s$ converges almost everywhere on $[-1,1]$ by a result of Hardy-Littlewood, but not everywhere. However, there does not yet exist an intrinsic description of the set of convergence for $F_s$. In this paper, we define in terms of even or regular continued fractions certain subsets of points of $[-1,1]$ of full measure where the series converges. Our method is based on an approximate function equation for $F_s(x)$. As a by-product, we obtain the convergence of certain series defined in term of the convergents of the even continued fraction of an irrational number.