A GEOMETRIC AND SPECTRAL DECOMPOSITION OF THE SUM-OF-DIVISORS FUNCTION VIA FOURIER ANALYSIS AND THE DIGAMMA FUNCTION

Abstract. We construct an exact analytical framework for the sum-of-divisors function σ1(x), starting from an elementary indicator function for divisibility. Extending this indicator to a periodic square wave and expanding it in a Fourier series, we obtain an infinite trigonometric representation of σ1(x). Using a classical identity for alternating harmonic series, we reduce this infinite series to an exact finite double sum whose coefficients are expressed through the digamma function ψ. We then give this finite sum a geometric reading: each inner sum is exactly k times the abscissa of the centroid of a polygon in the complex plane, whose k vertices lie at radii governed by ψ and atangles proportional to x. We prove that this centroid abscissa takes only two possible exact values, according to whether k divides x or not, and that this dichotomy induces a strict geometric separation : barycenters of non-divisors lie exactly on the vertical line Re(z) = −1/k, while barycenters of divisors lie exactly on the positive real axis. We further derive a parabolic local model, governed by an explicit moment-of-inertia constant, that predicts the height, width, and shape of each alignment peak in closed form.