The Collision Spectrum

Abstract

For a prime base $b$ and primitive odd Dirichlet character $\ddot{I}‡$ modulo $b^2$, the collision transform coefficient ${\hat{S}}^{\circ }(\ddot{I}‡)$ admits an exact factorization: \[ \hat{S}^{\circ}(χ) = -\frac{B_{1,\overlineχ} \cdot \overline{S_G(χ)}}{ϕ(b^2)}, \] where $B_{1,\overline{\ddot{I}}‡}$ is the generalized first Bernoulli number and $S_G(\ddot{I}‡)$ is the diagonal character sum. By the standard Bernoulli--$L$-value formula, $|B_1|=(b/\ddot{I}€)|L(1,\ddot{I}‡)|$, so the collision invariant's Fourier spectrum encodes $L$-function special values. A Parseval identity gives an exact formula for the weighted second moment $\sum |L(1,\ddot{I}‡){|}^2\cdot |S_G(\ddot{I}‡){|}^2$ in terms of the collision invariant's values on the finite group. The digit function computes this $L$-value moment exactly. Under a conditional zero-free hypothesis, the triangle inequality yields a separate bound connecting $L(1)$ to $L(s)$ for $s$ in the critical strip. At base~$5$, the factorization gives $|{\hat{S}}^{\circ }|\propto |L(1){|}^2$ exactly. For quadratic characters in the family, the decomposition specializes to class-number data.

6 pages