Hardy-Littlewood series and even continued fractions

Abstract

By a result of Hardy-Littlewood concerning the growth of the sums $\sum\nolimits_{n = 1}^N {{e^{i\pi {n^2}x}}}$, for each s ∈ (1/2, 1], the series ${F_s}(x) = \sum\nolimits_{n = 1}^\infty {{e^{i\pi {n^2}x/{n^s}}}}$ converges almost everywhere but not everywhere on [−1, 1]. 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 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

$$\sum\limits_{n = 1}^N {\frac{{{e^{i\pi {n^2}x}}}}{{{n^S}}} - {e^{{\text{sign}}(x)i\frac{\pi }{4}}}|x{|^{S - \frac{1}{2}}}\sum\limits_{n = 1}^{\left\lfloor {N|x|} \right\rfloor } {\frac{{{e^{ - i\pi {n^2}/x}}}}{{{n^S}}}} } $$

converges as N → ∞ to a function Ω _s 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 Ω _s 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.