The Euler-product defect of the Davenport-Heilbronn function and its zeros: an exact identity, tested
Year: 2026
Type: preprint
Abstract: The Davenport-Heilbronn function f = c L(s, chi) + conj(c) L(s, conj(chi)) (chi the odd quartic character mod 5) has a functional equation and no Euler product. Its von Mangoldt coefficients are split as Lambda_f = Lambda Re chi + D, where Lambda Re chi is the von Mangoldt function of the Euler product sqrt(L(s, chi) L(s, conj(chi))), and the defect is shown to have the closed form sum D(n) n^(-s) = P'(s) tan(P(s) - arctan kappa), P = Im log L(s, chi); hence D lives on the integers all of whose prime factors are congruent to 2 or 3 mod 5, and at two prime factors it is exactly the multiplicativity defect a(pq) - a(p)a(q) weighted by log pq. Subtracting the explicit formula of L(s, chi) from that of f gives an exact identity with no archimedean term: the zeros of f minus the zeros of L(s, chi), tested against h, equal -2 sum D(n) n^(-1/2) g(log n). The identity is tested with Gaussian h at 18 heights, in Arb, using the certified census of the 1,268 off-line pairs of f below height 20,000 (doi:10.5281/zenodo.23676415), the 25,078 on-line zeros, and 55,382 zeros of L(s, chi), L(s, conj(chi)) computed here; the two sides agree within their enclosures at all 36 (centre, width) combinations (differences <= 2e-9). At pair heights the defect sum is positive in 95-99.8% of cases against about 50% at control heights, but the off-line displacement itself explains only 14-20% of it; the rest is the local excess of zeros of f over zeros of L(s, chi) L(s, conj(chi)) at the pair. Nothing assumes RH or GRH. This version (4 November 2026) updates the citations to published records and corrects one label: 1.3951 is a zero-free half-plane from the Weil lemma, not the supremum of the real parts of the zeros (which is Bombieri-Ghosh's 1.120362). Written jointly with Claude (Anthropic); the human author is responsible for the content.
Author: Berzai, Nathan
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225224429