Continued Fractions Count the Zeros of an Epstein Zeta Derivative on Vertical Lines (M3)
Year: 2026
Type: preprint
Abstract: How many zeros does a derivative of the Epstein zeta function have on a vertical line? The continued fraction of the line decides. For τ = θ + iy let Z(τ,s) be the Epstein zeta function of the lattice ℤ + ℤτ and f(τ) = ∂y(y Z(τ,s)) with s = D/2, D ≥ 3. The number ND(θ) of zeros of y ↦ f(θ+iy) is an arithmetic function of θ: there is one zero above the Ford circles, and every reduced fraction p/q with 0 < q²|θ − p/q| ≤ κ̂D adds two more, inside a small positive "lobe" in its Ford circle. For 3 ≤ D ≤ 6 nothing else happens outside a band of width below 0.01 around the threshold. Consequences. ND(θ) = ∞ for Lebesgue-almost every θ and for θ = e − 2; ND(θ) = 1 for every θ whose partial quotients are at most 3 (golden ratio, √2 − 1, √3 − 1) when 3 ≤ D ≤ 6; N4(π − 3) ≥ 9. These results are proved with finitely many ball-arithmetic checks and were first obtained in the source paper cited below; the verification was re-run for this note. New in this note: the threshold. The leading-order lobe threshold κD has a closed form in terms of Z and f on rectangular lattices; κD < 1/√D for every D and κD√D → 1 (proved), with κD√D = 1 − (2 log D − log 2 + 3/2)/D + o(1/D) (leading-order derivation, confirmed numerically up to D = 104). So in dimension D a convergent produces zeros once the next partial quotient is of size about √D. Values of κD to 15 digits are given. The note is self-contained and written for mathematicians: a primer on Ford circles and continued fractions, the geometric mechanism (directional form, SL2(ℤ)-covariance, a universal lobe in cusp coordinates), three figures, key takeaways, implications for mathematicians and physicists, and a table separating proved, computer-assisted, numerical and open statements. Files. PDF, LaTeX source and figures (also as an Overleaf-ready zip); companion code (Python/mpmath, two reference runs at 20 and 30 digits, log of the re-run ball-arithmetic verification, SHA-256 checksums).
Author: Krause, Thomas
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225463981