The Collision Transform and the Critical Strip

The centered sum converges at $s = 1$. What about below?

The strip

The critical strip is the region $0 < \text{Re}(s) < 1$ where the zeros of $L$-functions live. The Riemann Hypothesis says they all sit at $\text{Re}(s) = 1/2$. The Generalized Riemann Hypothesis says the same for all Dirichlet $L$-functions.

The centered collision sum $F^\circ(s)$ converges at $s = 1$. Pushing it below $s = 1$ means entering the strip. The sum involves $P(s, \chi) = \sum_p \chi(p)/p^s$, which has singularities at the zeros of $L(s, \chi)$.

Conditional penetration

If every odd $L$-function modulo $b^2$ has no zeros for $\text{Re}(s) > \sigma_0$, then $F^\circ(s)$ converges for real $s > \sigma_0$.

Under GRH ($\sigma_0 = 1/2$): $F^\circ(s)$ converges all the way to $s = 1/2$. The collision invariant reaches the center of the strip.

Without GRH: the penetration depth is limited by the zero-free region. The classical zero-free region gives convergence only slightly below $s = 1$. The collision transform can see into the strip, but not deeply, without help from the zeros.

What this means

The collision invariant at $s = 1$ is unconditional. It converges because the centering removes the Mertens term.

The collision invariant below $s = 1$ is conditional. Its depth depends on where the $L$-function zeros are. If the zeros are well-behaved (on the critical line), the invariant sees deep. If not, it stops early.

The collision transform is a probe. The strip is the territory. The zeros are the obstacles.

Try it yourself

./nfield 7               # collision deviation data
./nfield 97              # feeds into the sum

Code: github.com/alexspetty/nfield

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Alexander S. Petty
June 2025
.:.