The Collision Spectrum

Abstract

For a prime base $b$ and primitive odd Dirichlet character $\chi$ modulo $b^2$, the collision transform coefficient $\hat{S}^{\circ}(\chi)$ admits an exact factorization: \[ \hat{S}^{\circ}(\chi) = -\frac{B_{1,\overline{\chi}} \cdot \overline{S_G(\chi)}}{\phi(b^2)}, \] where $B_{1,\overline{\chi}}$ is the generalized first Bernoulli number and $S_G(\chi)$ is the diagonal character sum. By the standard Bernoulli--$L$-value formula, $|B_1| = (b/\pi)\, |L(1, \chi)|$, so the collision invariant's Fourier spectrum encodes $L$-function special values. A Parseval identity gives an exact formula for the weighted second moment $\sum |L(1, \chi)|^2 \cdot |S_G(\chi)|^2$ 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.