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.