A Weil-positivity lock detects and counts genuine off-line zeros of an Epstein zeta function
Year: 2026
Type: preprint
Abstract: Weil's explicit formula turns the location of the zeros of a zeta function into the sign of a quadratic form, and the resolvent lock of the author's earlier note (doi:10.5281/zenodo.23459270) reads that sign, together with the exact count of negative directions, from the real zeros of a single polynomial. We build the truncated Weil form of a two-dimensional Epstein zeta function directly from its own data — the gamma factor, one constant and a finite set of generalised primes obtained from the logarithmic derivative of its Dirichlet series by an exact recursion — so that the construction uses the function's own data only. The Epstein family has no Euler product and, for suitable lattices, genuine zeros off the critical line. On the lattice D = 7, whose zeros all lie on the line, the lock is closed (κ = 0). On the lattice Δ = 0.800942… the lock opens with κ = 4, and the index law splits the count as 4 = 2 + 2 + 0. Subtracting the zero-side contributions of the three off-line quadruples inside the band closes the lock again (κ = 0); subtracting any single one leaves it open. Along the way the explicit-formula identity, evaluated from function values, located two off-line zeros beyond the published one, at heights 17.21 and 23.06 (the latter with real part 1.326). All numbers were reproduced on two machines. English and Turkish versions.
Author: Oltulu, Mehmet Akif
Institutions: Andover Eye Associates, Oldham Council
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222885344