Power series which diverges precisely at the roots of unity, converges elsewhere

Is there a complex power series $\sum a_nz^n$ with radius of convergence $1$ which diverges at the roots of unity (e.g., $z=e^{2\pi i\theta}$, $\theta \in \mathbb{Q}$) and converges elsewhere on the unit circle ($z=e^{2\pi i\theta}$, $\theta \in \mathbb{R} \setminus \mathbb{Q}$)?

I know $\sum \frac{z^n}{n}$ is a series with radius of convergence $1$ which converges everywhere on the unit circle except $1$. Perhaps we can play around with this to get the desired result.

I also know that $\sum \frac{z^{n!}}{n}$ diverges at the roots of unity, but I am not aware of a result that it converges at all other points on the unit circle.

Note similar questions have been asked here before, but they do not directly answer the question posed above.

Solution 1:

Because your question asks whether this is possible when the set of divergence is a certain countable set, the answer is YES by Theorem 1 of [2] below.

Results including and related to what you’ve asked are discussed in the following Stack Exchange questions:

Behaviour of power series on their circle of convergence

Examples of Taylor series with interesting convergence along the boundary of convergence?

Complex power series divergent and convergent on dense subsets of the circumference of convergence?

Power series with funny behavior at the boundary

What are the subsets of the unit circle that can be the points in which a power series is convergent?

Because the Duke Mathematical Journal papers by Herzog/Piranian mentioned in some of the above are not freely available, I’ve included some relevant excerpts from them. Incidentally, I have not bothered to include excerpts from [3] because it is freely available.

In these excerpts, additional notes by me are enclosed in double square brackets [[ ... ]].

[1] Fritz Herzog and George Piranian, Sets of convergence of Taylor series I, Duke Mathematical Journal 16 #3 (September 1949), 529-534. MR 11,91f; Zbl 34.04806

Introduction (p. 529): Let $\sum_n a_n z_n$ be a Taylor series of radius of convergence one, with $\sum_n |a_n| = \infty$ and $\lim_n a_n = 0.$ We consider the point set $M$ on the unit circle $C,$ on which the series converges. As Landau [2; 13-14] [[= Landau (1929, Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie, 2nd edition)]] points out, the cardinal number of the set of such Taylor series is $\mathfrak c$ [[at least $\mathfrak c$ --- add any constant to a specific such Taylor series; at most $\mathfrak c$ --- every such Taylor series represents a continuous function and there are $\mathfrak c$ many continuous functions]], while the cardinal number of the set of subsets of $C$ is $\mathfrak f$ [[= $2^{\mathfrak c}$]]; hence, there exists a set $M$ on $C$ such that no Taylor series converges on $M$ and diverges on $C-M.$ It follows that if a set $M$ on $C$ is such that some Taylor series converges on $M$ and diverges on $C-M,$ the set must have certain special properties. Lusin [3] [[= Luzin (1911, Über eine potenzreihe)]] (see also Landau [2; 69-71]) has constructed a Taylor series whose coefficients tend to zero and which diverges on the entire unit circle $C.$ Sierpiński (see Landau [2; 71]) has modified Lusin's example to obtain divergence at all points of $C$ except one. For every closed arc $A$ on $C,$ Neder [5] [[= Neder (1919, Zur Konvergenz der Trigonometrischen Reihen, einschliesslich der Potenzreihen auf dem Konvergenzkreise, Ph.D. Dissertation)]] constructed a Taylor series which converges on $C-A$ and whose partial sums are unbounded at every point of $A.$ Mazurkiewicz [4] [[= Mazurkiewicz (1922, Sur les séries de puissances)]] used Neder's example to prove the following proposition: If $M$ is a closed set on $C,$ there exists a Taylor series which converges on $M$ and diverges on $C-M,$ and a Taylor series which diverges on $M$ and converges on $C-M.$ The present paper is devoted to the extension of these results. Its method is inspired by Lusin's example. Theorem 1 (p. 531): If $M$ is a set of type $F_{\sigma}$ on the unit circle $C$ there exists a Taylor series which converges everywhere in $M$ and diverges everywhere in $C-M.$ Theorem 2 (p. 532): If $F$ is a closed set on the unit circle $C,$ there exists a Taylor series which converges uniformly in $F$ and diverges everywhere in $C-F.$ Theorem 3 (p. 533): If $M$ is a set on the unit circle $C,$ such that some Taylor series converges uniformly on $M$ and diverges on $C-M,$ then $M$ is a closed set. Theorem 4 (p. 533): If $M$ is a set on the unit circle $C,$ such that some Taylor series converges on $M$ and diverges on $C-M,$ then $M$ is of type $F_{\sigma \delta}.$ (last paragraph of the paper, at bottom of p. 533) [[References are on p. 534]] The "no-man's-land" between Theorems 1 and 4 is considerable; it consists of all sets of type $F_{\sigma \delta}$ on $C$ that are not of type $F_{\sigma}.$ It is not known whether every set of type $F_{\sigma \delta}$ is the set of convergence of some Taylor series. However, if $M$ is the complement on the unit circle of an arbitrary denumerable set, there exists a Taylor series for which $M$ is the set of convergence (the construction of such a series will be described in a later paper); this implies that not every set of convergence of a Taylor series is of type $F_{\sigma}.$ [[Note that if $D$ is a set that is dense in $C$ and we let $M = C-D,$ then $M$ is such a set. This follows from the Baire category theorem in the same way that one usually shows that the set of irrational numbers in $\mathbb R$ is not an $F_{\sigma}$ set.]]

[2] Fritz Herzog and George Piranian, Sets of convergence of Taylor series. II, Duke Mathematical Journal 20 #1 (March 1953), 41-54. MR 14,738b; Zbl 50.07802

(first two sentences of the paper, on p. 41) In an earlier paper [7] the authors have shown that if $M$ is a set of type $F_{\sigma}$ on the unit circle $C,$ there exists a Taylor series which has $M$ as its set of convergence, i.e., which converges on $M$ and diverges on $C-M.$ The main purpose of the present paper is to exhibit Taylor series whose sets of convergence are not of type $F_{\sigma}.$ Theorem 1 (p. 45): If $M$ is a denumerable set on the unit circle $C,$ there exists a function $f(z)$ with the following properties: (i) $f(z)$ is schlicht in the region $|z| < 1;$ (ii) the Taylor series of $f(z)$ diverges on $M$ and converges on $C-M;$ (iii) the partial sums of the Taylor series of $f(z)$ are uniformly bounded on $C;$ (iv) the set of vertices of the Mittag-Leffler star of $f(z)$ consists of the set $\overline{M}.$ Theorem 2 (p. 48): If $M$ is a denumerable set on $C,$ and $N$ is a subset of $M,$ then there exists a function $f(z)$ with the following properties: (i) the Taylor series of $f(z)$ converges on $C-M,$ diverges on $M,$ and has uniformly bounded partial sums; (ii) $\lim_{r \rightarrow 1}f(re^{i \theta})$ exists when $e^{i \theta}$ lies in $C-N,$ does not exist when $e^{i \theta}$ lies in $N.$ (from p. 50) In [7] it was proved that every set of type $F_{\sigma}$ on $C$ is the set of convergence of some Taylor series. Together with the results established so far in this paper, this gives the result that every denumerable set on $C$ is both a set of convergence and a set of divergence. Since the construction in [7] can easily be modified to introduce arbitrarily large gaps, we also obtain the result that if $M$ is the union of a denumerable set and a set of type $G_{\delta}$ on $C,$ then $M$ is the set of divergence of some Taylor series. As far as the general theory of sets of convergence is concerned, the following [[= Theorem 3]] is the most important conclusion that can be drawn from Theorems 1 and 2: Theorem 3 (p. 50): Not every set of convergence of Taylor series is of type $F_{\sigma}.$ (first two sentences after Theorem 3 statement, on p. 50) This follows at once from the fact that if $M$ is a denumerable set, dense on $C,$ then $M$ is not of type $G_{\delta}$ (see [6; 138]). It should be remarked that Theorem 3 can be deduced from Fejér’s work and should therefore have been included in [7]. Theorem 4 (p. 50): There exists a function $f(z)$ which is holomorphic, bounded and schlicht in $|z| < 1$ and whose Taylor series diverges on a set which is locally non-denumerable on $C.$ [["Locally non-denumerable on $C$" means that the set has uncountable intersection with every arc of $C.$ Since the sets in question are Borel sets, and uncountable Borel sets have cardinality $\mathfrak c,$ it follows that "locally non-denumerable on $C$" can be replaced with "$\mathfrak c$-dense in $C$".]] Theorem 5 (p. 51): There exists a function which is holomorphic in $|z| < 1$ and continuous in $|z| \leq 1$ and whose Taylor series diverges on a set which is locally non-denumerable on $C.$ Theorem 6 (p. 51): There exists a function $f(z),$ bounded and schlicht in $|z| < 1,$ whose Taylor series converges everywhere on $C,$ but not uniformly on any arc of $C.$ Theorem 7 (p. 51): There exists a function $f(z)$ which is continuous in $|z| \leq 1$ and whose Taylor series converges everywhere on $C,$ but not uniformly on any arc of $C.$ Definition (p. 52): If $f(z)$ is defined everywhere on $C,$ a point $z_0$ on $C$ lies in the set of boundedness of $f(z)$ provided $f(z)$ is bounded on some open arc of $C$ which contains the point $z_{0}.$ The set of unboundedness of $f(z)$ is the complement [[relative to $C$]] of the set of boundedness. Theorem 8 (p. 52): A necessary and sufficient condition for a set $M$ on $C$ to be the set of unboundedness of some Taylor series converging everywhere on $C$ is that $M$ be closed and nowhere dense on $C.$ [[I suspect that requiring the behavior to not just be unbounded at a point, but to actually approach infinity at the point (i.e. being locally unbounded and finite subsequent limits are not allowed) will be characterized by being a scattered set, as is the case at Is there a function whose all limits at rational points approach infinity? (see my comments there).]]

[3] Fritz Herzog and George Piranian, Some point sets associated with Taylor series, Michigan Mathematical Journal 3 #1 (1955-1956), 69-75. MR 17,834a; Zbl 70.29501