The Collision Transform

Everything before this was about one prime at a time. This paper looks across all of them.

The transform

The collision invariant $S(a)$ is a function on the finite group $(\mathbb{Z}/b^2\mathbb{Z})*$. Each prime $p$ contributes a value at its residue class $a = p \bmod b^2$. The Fourier transform over this group decomposes $S$ into components indexed by Dirichlet characters modulo $b^2$.

The transform is:

$$\hat{S}(\chi) = \frac{1}{\phi(m)} \sum_a S(a) \bar\chi(a)$$
Odd characters carry the signal. Even characters vanish (antisymmetry). The centered transform removes the principal character's contribution. What remains is the collision spectrum.

Convergence

The centered sum $F^\circ(s) = \sum_p S^\circ(p) / p^s$ converges at $s = 1$. The Mertens-rate divergence of the raw sum cancels exactly after centering. No principal-character term survives.

This is the collision analogue of the prime number theorem's error term. The main term (the class mean) is removed. The residual (the centered deviation) converges. The convergence is not conditional. It holds for every base and every lag.

Finite determination

The collision invariant at lag $\ell$ depends only on $p \bmod b^{\ell+1}$. The prime $p$ contributes to the transform through its residue class, not through $p$ itself. The infinite population of primes is compressed into a function on a finite group.

This is why the transform works: the characters are defined on the finite group, and every prime maps to a point in that group. The sum over primes becomes a weighted sum over residue classes.

What the transform sees

The odd characters carry the collision spectrum. The spectrum encodes how the digit function's collision structure distributes across the multiplicative group. The next paper will show that this distribution factors into L-function special values.

Try it yourself

./nfield 7               # S at p=7 in base 10
./nfield 97              # S at p=97
./nfield align 12        # alignment (related to S)

Code: github.com/alexspetty/nfield

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Alexander S. Petty
February 2025
.:.