Fourier Series Generated by Additive Prime Factor Functions

TL;DR

This paper constructs a prime-indexed Fourier series based on additive prime factor functions, exploring its analytic and geometric properties, and establishing connections to circulant Hermitian transformations and planar geometry.

Contribution

It introduces a novel prime-indexed Fourier series linked to additive prime factor functions and investigates its properties from both analytic and geometric viewpoints.

Findings

Established norm identities for the Fourier series

Connected the series to circulant Hermitian polygon transformations

Explored planar geometry of sampled curves

Abstract

We introduce a rigorous arithmetic--spectral construction associating planar geometric objects with additive prime factor statistics. Let $\mathrm{sopfr}(n)$ denote the sum of prime factors of $n$, counted with multiplicity, and define the summatory function $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$. It is known that $B(x) \sim \frac{\pi^2 x^2}{12 \log x}$ as $x \to \infty$. We show that $B(n)$ admits an exact prime-indexed decomposition $B(n) = \sum_{p \le n} p\, v_p(n!)$, where $v_p(n!)$ denotes the $p$-adic valuation of $n!$. This identity motivates the definition of a sparse prime-indexed Fourier series $F_n(t) = \sum_{p \le n} v_p(n!) e^{i p t}$, which we investigate from analytic and geometric perspectives. We establish precise norm identities, relate the construction to circulant Hermitian polygon transformations whose eigenpolygons are discrete Fourier modes, and examine the planar geometry arising from sampled curves. All geometric observations are explicitly experimental. The results provide a rigorous arithmetic foundation for prime-related Fourier geometry and motivate further theoretical and experimental investigations.