Fourier Series Generated by Additive Prime Factor Functions

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 $$B(x) \sim \frac{π² x^2}{12 \log x}$$ 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.

• The paper proves the exact identity B(n)=∑_{p≤n} p v_p(n!), expressing the summatory sopfr function through prime valuations of n! and complementing its asymptotic growth of (π²/12)x²/log x.
• The paper constructs the sparse series F_n(t)=∑_{p≤n}v_p(n!)e^{ipt} and derives the norm formula ‖F_n‖²₂=2π∑_{p≤n}v_p(n!)², with small primes dominating because valuations are largest there.
• The paper connects these series conceptually to circulant Hermitian polygon geometry and reports experimental multiscale curves, while making no rigorous claims about fractal dimensions or quantitative spectral convergence.

Arithmetic foundation

The paper builds on the completely additive function $$\mathrm{sopfr}(n) = \sum_{p^\alpha \parallel n} \alpha p$$, which sums the prime factors of $n$ with multiplicity. Its summatory function $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$ satisfies the asymptotic

$B(x) \sim \frac{\pi^2}{12}\frac{x^2}{\log x},$

a result due to Vartziotis and Tzavellas (Vartziotis et al., 2016). The constant $\pi^2/12$ reflects the average contribution of each prime $p$ weighted by its density in the integers, and the asymptotic is the analytic anchor for everything that follows.

Exact prime decomposition via factorial valuations

The central elementary result is an exact identity: for every $n$,

$B(n) = \sum_{p \le n} p\, v_p(n!),$

where $v_p(n!) = \sum_{k\ge 1} \lfloor n/p^k \rfloor$ is Legendre's formula. The proof is a direct consequence of complete additivity: summing $\mathrm{sopfr}$ over all $n$0 interchanges the order of summation over primes and integers, and the inner sum collects exactly the exponent of $n$1 in $n$2. This identity is exact rather than asymptotic, which is what licenses the subsequent spectral construction without approximation error.

A sparse prime-indexed Fourier series

Motivated by the decomposition, the paper defines

$n$3

a Fourier series supported on prime frequencies with coefficients given by factorial valuations. Because distinct primes index distinct frequencies, orthogonality of the exponentials on $n$4 yields immediately that $n$5 with norm identity

$n$6

The coefficients grow roughly like $n$7 (from Legendre's formula), so the norm is dominated by small primes; the series is sparse but not uniformly bounded in frequency content as $n$8 grows. The construction places the object squarely in the tradition of exponential sums over primes studied by Hardy–Littlewood and Montgomery, though here the emphasis is on a deterministic coefficient sequence derived from valuations rather than on cancellation estimates.

Connection to circulant Hermitian polygon geometry

The paper connects $n$9 to the framework of circulant Hermitian polygon transformations [VW2], in which planar polygons are vectors in $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$0 and such matrices are diagonalized by the discrete Fourier transform, with eigenvectors corresponding to discrete Fourier polygons. The series $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$1 is presented as a continuous analogue of a linear combination of Fourier polygons with prime-indexed coefficients, providing a bridge between additive number theory and spectral polygon geometry. This link is conceptual rather than developed into new theorems; no quantitative statement ties properties of $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$2 to eigenvalue distributions of specific circulant operators.

Experimental geometric observations

Sampling $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$3 on dense grids produces planar curves exhibiting visually rich structure with apparent repetition across scales. Box-counting procedures are used to quantify geometric complexity, drawing on standard fractal-geometry methodology [Falconer; Tricot]. Crucially, the paper makes no claim regarding Hausdorff or Minkowski dimensions; all geometric findings are explicitly labeled experimental. This restraint is appropriate: visual multiscale structure in parametrized complex curves does not by itself establish any fractal dimension, and the box-counting results reported are descriptive rather than theorem-backed.

Limitations and open questions

The rigorous content consists of the valuation identity and the $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$4 norm computation; both are elementary consequences of additivity and orthogonality. Several questions remain open:

• Do the sampled curves possess well-defined Hausdorff or Minkowski dimensions, and can these be computed or bounded analytically?
• Is there a precise sense in which $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$5 approximates or relates to classical prime exponential sums, where cancellation phenomena are central?
• Can the connection to circulant Hermitian polygon transformations be made quantitative, e.g., through convergence of discrete eigenpolygons to features of $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$6?

The paper concedes these gaps explicitly, positioning the geometric material as motivation rather than established theory.

Conclusion

The paper contributes a clean exact identity linking the summatory function of $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$7 to factorial valuations, and uses it to define a sparse prime-frequency Fourier series with immediate $B(x) = \sum_{n \le x} \mathrm{sopfr}(n)$8 control. Its value lies in providing a rigorous arithmetic substrate for prime-related spectral constructions, while the fractal-flavored geometric observations remain experimental and unproven. The natural next step is an analytic treatment of the dimension and regularity of the associated planar curves.