The Collision Spectrum

For a prime base $b$ and primitive odd Dirichlet character $\chi$ modulo $b^2$, the collision transform coefficient ${\widehat{S}}^{\circ }(\chi )$ admits an exact factorization: $${\widehat{S}}^{\circ }(\chi )=-\frac{B_{1,\bar{\chi }}\cdot \overline{S_G(\chi )}}{\phi (b^2)},$$ where $B_{1,\bar{\chi }}$ is the generalized first Bernoulli number and $S_G(\chi )$ is the diagonal character sum. By the standard Bernoulli–$L$-value formula, $\mathrm{\mid }{B}_1\mathrm{\mid }=(b\mathrm{/}\pi )\mathrm{\mid }L(1,\chi )\mathrm{\mid }$, so the collision invariant’s Fourier spectrum encodes $L$-function special values. A Parseval identity gives an exact formula for the weighted second moment $\sum \mathrm{\mid }L(1,\chi ){\mathrm{\mid }}^2\cdot \mathrm{\mid }{S}_G(\chi ){\mathrm{\mid }}^2$ 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 $\mathrm{\mid }{\widehat{S}}^{\circ }\mathrm{\mid }\propto \mathrm{\mid }L(1){\mathrm{\mid }}^2$ exactly. For quadratic characters in the family, the decomposition specializes to class-number data.