TY - JOUR

T1 - ON A SELF-SIMILAR BEHAVIOR OF LOGARITHMIC SUMS

AU - Федотов, Александр Александрович

AU - Лукашева, Ирина

PY - 2024/8/1

Y1 - 2024/8/1

N2 - 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.

AB - 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.

UR - https://www.mendeley.com/catalogue/695fbf19-c65d-31f2-8bef-4e9ca816de67/

U2 - 10.1007/s10958-024-07301-y

DO - 10.1007/s10958-024-07301-y

M3 - Article

VL - 283

SP - 690

EP - 698

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 4

ER -