Authors: Min Min Oo
This paper presents a block-structured decomposition of the Dirichlet eta functionη(s)=∑_(n=1)^00 (-1)^(n-1) n^(-s), where s=σ+it, based on an exponential partitioning of the summation index into intervals of the form N_k∼2e^(2kπ/t). The series is reorganized into finite segments, which are analyzed as dynamically scaling components with approxipmately geometric decay behavior.A ratio structure between successive block contributions is derived, leading to an approximate exponential scaling law of the form R_k (s)∼e^(-2πσ/t). This allows the eta function to be expressed as a coupled system of real and imaginary components, each defined over partitioned summation blocks. Using this decomposition, a real—imaginary interaction structure is introduced, where intersection-type conditions between the real and imaginary parts are studied through a determinant-based formulation. The resulting system suggests a structured relationship between block interactions and phase behavior in the complex plane. The framework provides a new perspective on the analytic structure of η(s)through interval scaling, phase coupling, and approximate geometric recursion. In particular, the model highlights a special role of the parameter σ=1/2within the interaction structure, emerging from symmetry considerations in the decomposed representation. This work is exploratory in nature and aims to develop a structured analytical model for studying oscillatory behavior in alternating Dirichlet series via block decomposition techniques.
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