On a self-similar behavior of logarithmic sums

The logarithmic sums $S_N (\omega, \zeta) = \sum \limits_{n = 0}^{N-1} \ln \left(1 + e^{-2 \pi i (\omega n + \frac \omega{2} + \zeta)} \right)$, where $\omega$ and $\zeta$ are parameters, are related to trigonometric products from the theory of quasiperiodic operators, as well as to a special function kindred to the Malyuzhinets function from the diffraction theory, hyperbolic Ruijsenaars $G$-function arising in connection with the theory integrable systems, and the Faddeev quantum dilogarithm, which plays an important role in the knot theory, Teichmüller quantum theory and complex Chern-Simons theory. Assuming that $\omega \in (0,1)$ and $\zeta \in\mathbb C _-$, and using renormalization formulas similar to the ones well known in the theory of the Gauss exponential sums, we describe the behavior of the logarithmic sums for large $N$.