Appendix: Riemann's Function

It is interesting to point out that a certain lacunary (or lacunar) trigonometric series proposed by Riemann, as a possible example of a continuous function that is not differentiable anywhere, can be analyzed in the context of the relationship presented in this paper. Riemann proposed the sine series

$$S_{\rm s} = \sum_{k=1}^{\infty} \frac{1}{k^{2}}\, \sin\!\left(k^{2}\theta\right),$$

from which we can construct the FC cosine series $S_{\rm c}=\bar{S}_{\rm s}$,

$$S_{\rm c} = \sum_{k=1}^{\infty} \frac{1}{k^{2}}\, \cos\!\left(k^{2}\theta\right).$$

Both series pass the Weierstrass $M$-test and are therefore uniformly convergent to continuous functions over the whole periodic interval. Their term-by-term derivatives, however, are not convergent at all,

$$\begin{eqnarray*} \frac{dS_{\rm c}}{d\theta} & = & - \sum_{k=1}^{\infty} \s... ...ta} & = & \sum_{k=1}^{\infty} \cos\!\left(k^{2}\theta\right). \end{eqnarray*}$$

All these series can be understood as trigonometric series in which the coefficients $a_{k}$ corresponding to values of $k$ which are not perfect squares are all zero. Therefore, there are lots of missing terms, hence the name ``lacunary'' series. Since the two trigonometric series $S_{\rm c}$ and $S_{\rm s}$ are the FC series of each other, they can be used to construct the complex power series

$$S_{z} = \sum_{k=1}^{\infty} \frac{1}{k^{2}}\, z^{k^{2}}.$$

This is a lacunary power series which presumably has a countable infinity of soft singularities densely distributed over the unit circle. The ratio test tells us that the maximum disk of convergence of this series is indeed the open unit disk. We have therefore an inner analytic function defined on that disk,

$$\begin{eqnarray*} w(z) & = & \sum_{k=1}^{\infty} \frac{1}{k^{2}}\, z^{k^{2}... ...ho,\theta) + \mbox{\boldmath$\imath$} f_{\rm s}(\rho,\theta), \end{eqnarray*}$$

where

$$\begin{eqnarray*} f_{\rm c}(1,\theta) & = & S_{\rm c}, \\ f_{\rm s}(1,\theta) & = & S_{\rm s}. \end{eqnarray*}$$

Since the trigonometric series are convergent on the whole unit circle, due to Abel's theorem [6] the $\rho\to 1$ limits of this function from within the unit disk exists everywhere on that circle. The derivative $w'(z)$ of this function is also analytic on the open unit disk, and is given by

$$\begin{eqnarray*} w'(z) & = & \frac{dw(z)}{dz} \\ & = & \sum_{k=1}^{\infty} z^{k^{2}-1}. \end{eqnarray*}$$

The non-differentiability of the function $w(z)$ on the unit circle must therefore be related to the non-existence of the $\rho\to 1$ limits of the function $w'(z)$ from within the unit disk. Note that in terms of derivatives with respect to $\theta$ we may write

$$\begin{eqnarray*} \frac{dw(z)}{d\theta} & = & \mbox{\boldmath$\imath$} z\, ... ...& = & \mbox{\boldmath$\imath$} \sum_{k=1}^{\infty} z^{k^{2}}. \end{eqnarray*}$$

This relation holds true everywhere within the open unit disk. One might therefore consider analyzing the differentiability of $f_{\rm c}(\rho,\theta)$ and $f_{\rm s}(\rho,\theta)$ on the unit circle by examining the behavior of the $\rho\to 1$ limits of the function $\mbox{\boldmath$\imath$} zw'(z)$, which is proportional to the logarithmic derivative of $w(z)$. The expectation is that the function $w(z)$ has a densely distributed countable infinity of borderline soft singularities over the unit circle. It then follows that its logarithmic derivative has a densely distributed countable infinity of borderline hard singularities over the unit circle, and therefore fails to be defined at all points. This can be understood as a consequence of the extended version of Abel's theorem [6], so long as it can be shown that the series diverges to infinity everywhere over the unit circle.

The Riemann sine series has been widely studied [7], but its FC series less so, it would seem. There are many papers and results about lacunary functions related to lacunary power series [8], which could be of use here. In particular the Fabry and the Ostrowski-Hadamard gap theorems [9] seem to be directly relevant. Such a lacunary function can be characterized as one that cannot be analytically continued beyond the boundary of the maximum disk of convergence of its Taylor series. This would be the case if such a function had a densely distributed set of singularities over the unit circle. Note that using the correspondence established in this paper we may construct from any lacunary trigonometric series a corresponding lacunary power series, and vice-versa, so that these two subjects are completely identified with each other.