In [this question](https://mathoverflow.net/questions/273331/behaviour-at-natural-boundary) some experiments were used to conjecture that the zeros of partial sums of a series converging to a function with natural boundary on the unit circle were (weakly) converging *to* the unit circle. Well, nothing is new under the Sun, and this is a Theorem of Jentsch, from his recent PhD thesis:



```
R. Jentsch, Untersuchungen zur Theorie der Folgen analytischer 
Funktionen, Inauguraldissertation
Berlin, 1914,39 s.

```

Actually, Jentsch's theorem does not need the circle to be the natural boundary, but simply the circle of convergence, so the only thing it is lacking is the stronger statement that the empirical measure of the zeros converges to the Lebesgue measure on the unit circle.


**EXACT STATEMENT**
Let $$f(z) = \sum\_{i=0}^\infty a\_n z^n,$$ with radius of convergence $1.$ Then every point of the unit circle is an accumulation point of zeros of partial sums of $f.$


**END EXACT STATEMENT**


The question is: does anyone here know of a more accessible place where the proof of this result could be found (or maybe there is a 10 line argument someone could just post)?



zeros on the circle of convergence
**Disclaimer:** I learned the trickery below from N.K.Nikolskii, who was giving us a special topics course in complex analysis when I was a fourth year undegraduate student. I have no idea whether it can also be traced back 100 years but, certainly, it has been used by many people on many occasions and is worth teaching to our graduate students (IMHO).


**Main Lemma:** Let $r<1<R$. Assume that $P(z)=\sum\_{k=0}^n a\_kz^k$ is any polynomial of degree $n$ such that $|a\_k|\le R^n$ for all $k=0,\dots,n$ and $|a\_0|,|a\_n|\ge r^n$. Then $\frac 1n\log|P(z)|-\log\_+|z|$ is small in $L^1(\frac{dA(z)}{\max(1,|z|^4)})$ provided that $r$ and $R$ are close to $1$ and $n$ is large.


**Proof:** In the closed unit disk $\{|z|\le 1\}$, we have $|P(z)|\le(n+1)R^n$ whence $\frac 1n\log\_+|P(z)|\le \frac{\log(n+1)}n+\log R$. Also, by Jensen, the average value of $\log|P(z)|$ over the unit disk is at least $n\log r$. Hence, we can bound the average value of $\frac1n|\log|P||$ over the disk by $\frac{2\log(n+1)}n+\log (R^2/r)$.


To treat the part outside the unit disk, just notice that the measure $\frac{dA(z)}{\max(1,|z|^4)}$ is invariant under the mapping $z\mapsto 1/z$ and apply the argument above to $z^nP(1/z)$ instead of $P$.


**Corollary:** If we have a sequence of polynomials like above with the parameters $n\to\infty$, $r=r\_n\to 1$, and $R=R\_n\to 1$, then the normalized counting measures of their zeroes tend weakly to the Haar measure on the unit circle.


**Proof:** Just take the Laplacians of both sides and observe that for probability measures the convergence in the sense of generalized functions is equivalent to weak convergence. 


The applications to partial sums should be obvious now.


