Hardy-Littlewood series and even continued fractions

By a result of Hardy-Littlewood concerning the growth of the sums \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\nolimits_{n = 1}^N {{e^{i\pi {n^2}x}}} $$\end{document}, for each s ∈ (1/2, 1], the series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_s}(x) = \sum\nolimits_{n = 1}^\infty {{e^{i\pi {n^2}x/{n^s}}}} $$\end{document} converges almost everywhere but not everywhere on [−1, 1]. However, there does not yet exist an intrinsic description of the set of convergence for Fs. In this paper, we define in terms of even continued fractions a subset of [−1, 1] of full measure where the series converges. As an intermediate step, we prove that for s > 0, the sequence of functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{n = 1}^N {\frac{{{e^{i\pi {n^2}x}}}}{{{n^S}}} - {e^{{\text{sign}}(x)i\frac{\pi }{4}}}|x{|^{S - \frac{1}{2}}}\sum\limits_{n = 1}^{\left\lfloor {N|x|} \right\rfloor } {\frac{{{e^{ - i\pi {n^2}/x}}}}{{{n^S}}}} } $$\end{document} converges as N → ∞ to a function Ωs continuous on [−1, 1] \ {0} with (at most) a singularity at x = 0 of type x(s−1)/2 (s ≠ 1) or a logarithmic singularity (s = 1). We provide an explicit expression for Ωs and the error term. Finally, we study thoroughly the convergence properties of certain series defined in terms of the convergents of the even continued fraction of an irrational number.