Collision-uniform kernel asymptotics for Báez–Duarte zero packets
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Abstract
We determine the high-height kernel of the harmonic-step space underlying the Báez–Duarte approximation problem. For packets with unequal real parts in a fixed compact rectangle inside the critical strip, an exact arithmetic gauge gives an explicit limiting Mellin kernel. Convergence holds with every fixed number of mixed derivatives. Complex Newton–Hermite divided differences then give a relative Gram comparison uniform through arbitrary collisions, without a minimum-separation condition. The resulting native zero-jet projection bound retains the exact unequal powers and oscillatory phases of the target. We derive a two-point interference formula and a multiple-zero lower bound with a logarithmic multiplicity factor. The proof works directly with the arithmetic step kernel and pays both ends of its infinite series.
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- Collision_uniform_arithmetic_packets.pdf96.7 KiB · pdf
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Research details
- Subjects
- Mathematics · Number Theory · Complex Analysis · Functional Analysis
- Keywords
- Báez–Duarte approximation, Mellin transform, reproducing kernels, divided differences, multiple zeros
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
For each fixed packet size and compact real-part and height-offset range, the exactly gauged arithmetic step kernel has an explicit mixed-derivative limit and a collision-uniform relative Gram law. Native finite-zero-jet distance bounds retain the exact target phases and include multiplicity limits.
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