The Collision Spectrum and the L-Function Landscape
This is the paper where everything connects. The collision spectrum factors into L-function special values. The digit function computes analytic number theory from integer arithmetic.
The decomposition theorem
For a primitive odd Dirichlet character $\chi$ modulo $b^2$:
$$\hat{S}^\circ(\chi) = -\frac{B_{1,\bar\chi} \cdot \overline{S_G(\chi)}}{\phi(b^2)}$$
The collision Fourier coefficient is the product of a generalized Bernoulli number and a diagonal character sum. The Bernoulli number encodes $|L(1, \chi)|$ through the class number formula. The diagonal sum encodes the bin geometry.
The collision spectrum is built from L-values. Not correlated with them. Built from them.
The moment identity
Parseval's identity, applied through the decomposition theorem, gives:
$$\sum_{\chi \text{ prim. odd}} |L(1, \chi)|^2 \cdot |S_G(\chi)|^2 = \frac{\pi^2 \phi(m)}{b^2} \sum_a |S\circ(a)|2$$
The left side is a weighted second moment of L-function special values. The right side is computable from the collision invariant's values on the finite group. Integer arithmetic computes an L-value moment exactly.
The Apostol mechanism
The short partial sum $P = \sum_{k=1}^{b-1} \bar\chi(k)$ decomposes via the classical identity into $L(1, \bar\chi) + \Delta(\chi)$, where $\Delta$ is a packet of twisted L-values. Computation shows the packet has constant magnitude relative to $L(1)$ (mean $\approx 1.08$) and uniform phase. The variance of $|\Delta|/|L|$ decays as $c/\log b$.
This variance decay is why the collision spectrum correlates with L-values: at large bases, the packet becomes nearly constant, so $|P| \approx c \cdot |L(1)|$, and the collision coefficient inherits the L-value's magnitude.
The critical strip
Under a conditional zero-free hypothesis, the triangle inequality yields a cross-moment bound connecting $L(1, \chi)$ (at the edge) to $P(s, \chi)$ (in the strip). The collision invariant, through the decomposition theorem, provides data about L-values at two points simultaneously.
Try it yourself
./nfield spectral 29 # spectral data feeding the decomposition
./nfield decompose 12 # coherence decomposition
Code: github.com/alexspetty/nfield
Alexander S. Petty
November 2025
.:.