Geodesic
On a self-similar behavior of logarithmic sums
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 279-292 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $N$. 
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A. A. Fedotov; I. I. Lukashova. On a self-similar behavior of logarithmic sums. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 279-292. http://geodesic.mathdoc.fr/item/ZNSL_2021_506_a17/

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