Asymptotic Behavior of Non-Analytic Function of the Eigenvalues    Hello,
Let $A_n = (a_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda_{n,i};\;i = 0,1,\ldots,n-1)$. 
If $A_n$ were a sequence of Hermitian Toeplitz matrices, and if $\sum_k|a_k|<\infty$, then Szego theorem states that for any continues function $F(\cdot)$ on $[\alpha,\beta]$ we have
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}F(\lambda_{n,k}) = \frac{1}{2\pi}\int_0^{2\pi}F(f(\xi))d\xi
$$
where
$$
f(\xi) = \sum_{k = -\infty}^{\infty}a_ke^{ik\xi}
$$
and $\alpha = \text{ess}\inf f$ and $\beta = \text{ess} \sup f$. 
If however $A_n$ are not-Hermitian, then the above hold only for polynomial functions, namely
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}p(\lambda_{n,k}) = \frac{1}{2\pi}\int_0^{2\pi}p(f(\xi))d\xi
$$
where $p(\cdot)$ is some polynomial function. 
My question: is there any result regard the asymptotic behavior of the modulus of the eigenvalues for the non-Hermitian case, namely
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}|\lambda_{n,k}| = ?.
$$

EDITION: due to the thankful Alexandersson comment, I add the factor $1/n$... 
 A: To create a partial answer, 
eigenvalues of banded Toeplitz matrices accumulate on some real algebraic curves in $\mathbb{C}$, (this has been proved by Schmidt and Spitzer around 1970.).
Now, if we also attach a point-mass at each eigenvalue, (for fixed matrix size) with equal mass at each point, and with total mass one, we get a probability measure.
These measures converge in a certain sense to some limit measure,
see for example the book by Bender and Böttcher .
Now, the sum $c_n = \frac{1}{n} \sum_{k=1}^n |\lambda_{n,k}|$ can then be interpreted 
as the center of mass mean distance to 0 of all the eigenvalues from matrix of size $n \times n.$
Say that the associated point measure is called $\mu_n$ (mass $1/n$ at each eigenvalue),
then we know that $\mu_n \to \mu$ for some $\mu$ in some sense.
Note that $c_n = \int_{\mathbb{C}} |z| d\mu_n(z)$ and then what you are looking for is $c = \int_{\mathbb{C}} |z| d\mu(z)$.
As an explicit example, taking your matrix to be tridiagonal, will give rise to characteristic polynomials which are also a family of orthogonal polynomials, w.r.t some measure (the limit of point measures, actually), see e.g. this paper i just googled.
