The sums SNω,ζ=∑n-0N-11n1+e-2πiωn+ω2+ζ, where ω and ζ are parameters, are related to trigonometric products from the theory of quasi-periodic operators as well as to a special function kindred to the Malyuzhinets function from the diffraction theory, the hyperbolic Ruijsenaars G-function, which arose in connection with the theory of integrable systems, and the Faddeev quantum dilogarithm, which plays an important role in the knot theory, Teichmuller quantum theory and the complex Chern–Simons theory. Assuming that ω ∈ (0, 1) and ζ ∈ ℂ−, we describe the behavior of logarithmic sums for large N using renormalization formulas similar to those well-known in the theory of Gaussian exponential sums.