Fourier Series Generated by Additive Prime Factor Functions

TL;DR

The paper addresses representing additive prime-factor statistics in a spectral-geometric framework by exploiting the exact prime-indexed decomposition of $B(n)$. It defines a sparse Fourier series $F_n(t)=\sum_{p\le n} v_p(n!)\, e^{i p t}$, whose $L^2$ structure is governed by factorial valuations and exploits mutually orthogonal prime frequencies via the lack of cross terms. The work links additive number theory to circulant Hermitian polygon transformations, framing $F_n$ as a continuous analogue of Fourier polygons with prime frequencies. Through analytic identities and experimental planar-geometry observations, it provides an arithmetic foundation for prime-related spectral geometry and motivates further theoretical and experimental investigations into the multiscale patterns of the resulting planar 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 $B(x) \sim \frac{π^2 x^2}{12 \log x}$ 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.

Theorems & Definitions (6)

• Theorem 2.1: Vartziotis--Tzavellas VT
• Lemma 3.1
• proof
• Definition 4.1
• Proposition 4.2
• proof