A conditional lower bound e^(-(4π+ε)n) for the bottom of the windowed Weil quadratic form
Year: 2026
Type: preprint
Abstract: We study κ(n), the infimum of the Weil quadratic form of the Riemann zeta function over smooth test functions supported in (−½ log n, ½ log n), normalised by the L² norm. The main result is conditional. Assume the Riemann Hypothesis and hypothesis (H3_θ): for some finite θ ≥ 0, |ζ′(ρ)|^(−1) ≤ C_ε |γ|^(θ+ε) at every nontrivial zero ρ = β + iγ. Then for every ε > 0, κ(n) ≥ e^(−(4π+ε)n) for all n ≥ n₀(ε, θ), where n₀ is ineffective. Without any hypothesis on the zeros we construct test functions showing κ(n) ≤ exp(−4πn + O(n/√log n)). Together, under RH and (H3_θ), −log κ(n)/n tends to 4π. For a deterministic model lattice in place of the zeros we prove, without assumptions, the lower bound c·n^(−C)·e^(−4πn). Under RH and (H3_θ) the lower bound excludes, for large n, the decay rate 2π²n predicted by Conjecture 12.1 of Zhu. Upper bounds of order e^(−4πn) were obtained earlier by Kastoris and by Mori; we claim no priority for that order. All statements involving RH or (H3_θ) are conditional; this paper neither asserts nor refutes the Riemann Hypothesis.
Authors: Wiktorowicz, Rafał, Wiktorowicz, Braun SI
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225315497