Spectral Geometry of Fourier Curves with Prime Frequencies: A Comparative Experimental Study
Dimitris Vartziotis
TL;DR
This work investigates whether the multiscale geometry of planar curves generated by a prime-frequency Fourier series reflects intrinsic arithmetic structure or is a generic outcome of sparse representations. It defines $F_n(t)=\sum_{p\le n} v_p(n!)\, e^{i p t}$, samples to obtain the planar curve $\Gamma_n$, and normalizes it for comparison. By contrasting against three randomized controls—random frequency, Cramér-type, and shuffled-prime models—and using box counting to compute the effective exponent $\widehat{d}(\varepsilon)=\frac{\log N(\varepsilon)}{\log(1/\varepsilon)}$ across dyadic scales, the study finds that the prime-frequency curves exhibit stable, intermediate scaling with $\widehat{d}(\varepsilon)$ between 1 and 2 that is not reproduced by the controls. This supports the view that the observed geometric complexity encodes a structured arithmetic coupling between the prime frequency set and factorial valuations, motivating further theoretical analysis of arithmetic contributions to geometric scaling and potential extensions to other arithmetic functions.
Abstract
We present a comparative experimental study of planar curves arising from a Fourier series whose frequencies are the prime numbers, together with several randomized control models. Starting from the series $F_n(t)=\sum_{p\le n} v_p(n!)\, e^{i p t},~t\in[-π,π]$, introduced and motivated in a companion work, we investigate the geometric complexity of the associated planar curves obtained by sampling in the complex plane. To test whether the observed multiscale behavior reflects arithmetic structure or can be reproduced as a generic consequence of sparsity or density, we compare the prime frequency model with randomized alternatives, including random frequency sets, a Cramér type random model, and a shuffled coefficient model. Using consistent box counting protocols and Monte Carlo ensembles, we observe stable scale dependent behavior for the prime frequency curves that is not reproduced by the randomized models. All results are experimental and are presented as evidence motivating further theoretical investigation.