The Collision Transform

TL;DR

This paper studies the collision transform's Fourier expansion over Dirichlet characters, analyzing its convergence, structure, and implications for prime sums and residue classes, revealing base-specific properties and connections to L-function zeros.

Contribution

It introduces the collision transform's Fourier analysis, explores its convergence behavior, and uncovers base-specific structural properties related to prime sums and residue classes.

Findings

Convergence of prime harmonic sums persists down to s=0.6 in base 10 and s=0.5 in base 3.

Only odd characters contribute to the collision transform's Fourier expansion due to reflection symmetry.

The collision invariant's structure is specific to each base, with negligible combined effects across bases.

Abstract

For a prime p and base b, the collision invariant $S_{\ell}(p)$, introduced in the companion paper, is a function of $p \bmod b^{\ell+1}$ and therefore lives on the finite group $(\mathbb{Z}/b^{\ell+1}\mathbb{Z})^{\times}$. Its Fourier expansion over Dirichlet characters modulo $b^{\ell+1}$ is the collision transform. The reflection identity forces all even-character coefficients of the centered invariant to vanish: only odd characters contribute. The centered prime harmonic sum $F^{\circ}(s) = \sum_p S^{\circ}_p / p^s$ is therefore a finite linear combination of non-trivial odd character sums $\sum_p \chi(p)/p^s$, with no principal-character term. At $s = 1$, each sum converges by Mertens' theorem for arithmetic progressions. Convergence below $s = 1$ is conditional on the absence of $L$-function zeros above a given depth. Computation indicates convergence persists to at least $s = 0.6$ in base 10 and to $s = 0.5$ in base 3. The real parts of the products $\hat{S}^{\circ}(\chi) \cdot P(s, \chi)$ have mixed signs, so convergence is a collective constraint on the joint zero distribution, not a test of each $L$-function individually. Aggregating the collision deviation across bases with a fixed convergent weighting produces the base sum, a function on primes that reveals mod-3 structure. For bases with $3 \nmid b$, the reflection $a \mapsto m - a$ fixes a unique residue class modulo 3, and the mean of $S$ over units in that class equals the grand mean $-1/2$ (the neutrality theorem). Removing the mod-3 component introduces a principal-character term that is absent from $F^{\circ}$. The base-summed harmonic sum is negligible: the collision invariant's structural content is base-specific.