The Lorentzian Weight

The zeros of L-functions control the distribution of primes, but individual zeros are hard to locate. What’s accessible is their collective behavior — how densely they cluster in different regions of the critical strip. Proving unconditional density bounds without the Riemann Hypothesis requires choosing the right test function.

Shiller uses Lorentzian spectral weights — peaked, bell-shaped functions that decay as 1/(1+t²) — to study the norm-form energy N = S_ζ² − d·S_L² for real quadratic fields. The Lorentzian weight has a specific technical advantage: it makes the low-lying zeros dominate the sum unconditionally. No assumption about zero locations needed. The weight naturally suppresses contributions from distant zeros, leaving the zeros nearest the critical line — the ones that matter most — in charge.

The through-claim: the weight function is the mathematical argument. Choosing the Lorentzian isn’t a convenience — it’s the mechanism by which unconditional results become possible. Different weights would require different (possibly stronger) hypotheses. The Lorentzian shape matches the structure of the problem.

The result: N remains negative for all squarefree d > 1, unconditionally. The density of exceptions where N > 0 has an effective bound via Jacobi-Anger resonance analysis. Under a finite-rank hypothesis (verified computationally for M ≤ 20), a sharp asymptotic formula holds.

The appendix certifies over 1,000 zeros of Dirichlet L-functions to 70 decimal places. Precision in service of proof — the zeros need to be known well enough for the truncation bounds to hold.