The zero strip of the completed Davenport-Heilbronn function

Section draft for WP2 (2026-08-16). Instrument: strip.py; decided values in strip_results.json. Vocabulary per MISSION.md: decided means an interval or ball whose exact endpoints settle a sign, stated with backend and precision; measured means one float route.

Frame (added 2026-08-16, GATE.md closure item (a)). sigma_0 is a point in the s plane and is frame-free. Delta = sigma_0 - 1/2 and Delta^2/2 are not: they are stated here in the normalization s = 1/2 + iz (Stopple, arXiv:1301.3158, and this hunt). In the normalization s = (1+iz)/2 used by de Bruijn as usually quoted, Newman, Rodgers-Tao, Polymath 15 and Dobner, Delta doubles to 1.7902723164702194421227177424... and Delta^2/2 quadruples to 1.6025374835598228. Conversion table and derivation: FRAME.md. sigma_0 itself is unchanged in either.

Superseded constant (added 2026-08-18). Nothing below is altered and nothing below is wrong. This document derives sigma_0 by coefficient domination, sum_{n>=2} |a_n| n^{-sigma} < 1, which discards all cancellation. Two facts a reader of this page alone would otherwise miss. First, the quantity sigma_0 bounds is not new: Bombieri and Ghosh, Around the Davenport-Heilbronn function, Russian Math. Surveys 66:2 (2011), 221-270, Theorem 7, determine the least upper bound of the real parts of the zeros of this very function exactly, and publish sigma(tau_+, 1) = 1.120362 (BOMBIERI-GHOSH.md). Second, that sharper abscissa is now derived and decided in-tree, on both backends, from an elementary phase obstruction in the Euler products of the two Dirichlet L-functions whose combination f is: STRIP2.md and strip2.py decide 1.12036249819, giving Delta^2/2 = 0.1924248145802688766381 narrow and 0.7696992583210755065522 wide, a factor 2.082030697360155 better than the number below. The upper bound of record is the one in STRIP2.md. This page is kept because the two derivations are independent and a reader should be able to see both, and because its sections 3(d), 3(e) and 5 are used unchanged there.

1. Statement

Let f(s) = sum_{n>=1} a_n n^{-s} with the period-5 Davenport-Heilbronn coefficients a_n given by the pattern (1, kappa, -kappa, -1, 0) on n = 1, 2, 3, 4, 0 (mod 5), where kappa is the functional-equation constant, decided by the Arb linear solve of instrument.kappa_ball (backend python-flint, 500 bits, ball width 3.1e-148) to lie in

kappa in [0.284079043840412296028291832393126169091088088, 0.284079043840412296028291832393126169091088089],

inside the pinned 40-digit reference KAPPA_REF +/- 1e-39 of zeta/epstein.py (exact rational comparison; the ball also decides 0 < kappa < 1, which several bounds below use). Let

gamma(s) = (pi/5)^{-(s+1)/2} Gamma((s+1)/2), F(s) = gamma(s) f(s),

the completed function, which is entire and satisfies F(s) = F(1-s) (structurally: F = c Lambda(s, chi) + conj(c) Lambda(s, chibar) for the odd primitive character chi mod 5, each completed L entire; in-tree: the defect zeta.epstein.dh_functional_equation_defect is measured ~1e-50 at dps 50 and test-pinned). Define

g(sigma) = sum_{n>=2} |a_n| n^{-sigma} - 1 (sigma > 1).

Claim (decided constant, exact argument). g has a unique root sigma_0 > 1, and every zero of F lies in the open strip

1 - sigma_0 < Re s < sigma_0.

Decided enclosures for sigma_0 (section 4):

backend python-flint (Arb), 192 bits, bisection width < 1e-25: sigma_0 in [1.3951361582351097210613588712, 1.3951361582351097210613589375] backend mpmath.iv, dps 40, bisection width < 1e-15: sigma_0 in [1.395136158235109178, 1.395136158235109747]

The intervals overlap, and both lie inside the scouted value 1.39513615823511 +/- 5e-15 (half an ulp of the scout's last digit; the scout is a 14-decimal rounding, so containment runs in that direction).

Consequently every zero of Xi_DH (equivalently H_0 in the flow_repair normalisation, where Xi_DH(z) = F(1/2 + iz); validate.py check 1 pins the identification at four points) satisfies

|Im z| < Delta = sigma_0 - 1/2,

with Delta in [0.8951361582351097210613588712, 0.8951361582351097210613589375] (flint endpoints, outward). This is the strip that de Bruijn's Theorem 13 converts into the upper bound Lambda_DH <= Delta^2/2 in WP2.

2. The identity behind g, verified before use

Splitting the absolutely convergent sum (Re s > 1) over residue classes mod 5, with |a_n| equal to 1 on n = 1, 4 (mod 5), kappa on n = 2, 3 (mod 5), and 0 on n = 0 (mod 5), and using 5^{-s} zeta(s, c/5) = sum_{j>=0} (5j + c)^{-s}:

sum_{n>=1} |a_n| n^{-s} = 5^{-s} [zeta(s, 1/5) + zeta(s, 4/5)]

• kappa 5^{-s} [zeta(s, 2/5) + zeta(s, 3/5)].

The n = 1 term (j = 0 of the c = 1 class) contributes 1, so g(sigma) = (right side) - 2. The identity was not trusted on derivation alone: check_identity compares the Hurwitz-form full sum (flint midpoint, 192 bits) against a direct float64 sum over n <= 10^6 at sigma = 1.5, 2, 3. Measured differences 1.03e-3, 5.14e-7, 2.57e-13, each positive and below the integral-test tail bound N^{1-sigma}/(sigma-1) (2e-3, 1e-6, 5e-13): exactly the discarded tail, as the identity requires. A wrong residue split or a wrong n = 1 accounting would miss at order one.

3. The argument

(a) Uniqueness of the root. Each term |a_n| n^{-sigma} with a_n != 0 is strictly decreasing in sigma (kappa > 0 is decided by the ball), so g is strictly decreasing and continuous on (1, inf). As sigma -> 1+ the n = 1 (mod 5) subsum alone diverges, so g -> +inf; as sigma -> inf, g -> -1. Hence exactly one root sigma_0, and any decided sign bracket encloses it.

(b) No zeros of f in the open half-plane Re s > sigma_0. For Re s = sigma > sigma_0, strict monotonicity gives sum_{n>=2} |a_n| n^{-sigma} < sum_{n>=2} |a_n| n^{-sigma_0} = 1, so

|f(s)| >= 1 - sum_{n>=2} |a_n| n^{-sigma} > 0.

(c) No zeros of f on the line Re s = sigma_0 (exact phase argument). Suppose f(sigma_0 + it) = 0. Then W := sum_{n>=2} a_n n^{-s} = -1, and since |W| = 1 = sum_{n>=2} |a_n| n^{-sigma_0}, the triangle inequality is an equality: every nonzero term a_n n^{-s} is a nonnegative multiple of a common unit vector, and the sum being -1 makes each term strictly negative real, i.e. n^{-it} = -1 when a_n > 0 and n^{-it} = +1 when a_n < 0. Take n = 3 (a_3 = -kappa < 0): 3^{-it} = 1. Take n = 4 (a_4 = -1 < 0): 4^{-it} = 1. Take n = 12 (12 = 2 mod 5, a_12 = kappa > 0): 12^{-it} = -1. But 12^{-it} = 3^{-it} 4^{-it} = 1. Contradiction; the case t = 0 fails the same constraints (2^{0} = 1 != -1). So f has no zero with Re s >= sigma_0.

(d) The gamma factor and the reflection. (pi/5)^{-(s+1)/2} is entire and never zero; Gamma((s+1)/2) is never zero and is analytic except for simple poles at s = -1, -3, -5, .... All those poles have Re s = -1 or less, so on Re s >= sigma_0 > 1 the factor gamma is analytic and nonvanishing, and F = gamma f has no zero with Re s >= sigma_0 by (b), (c). Since F is entire with F(s) = F(1-s), F also has no zero with Re s <= 1 - sigma_0. Every zero of F therefore lies in the open strip 1 - sigma_0 < Re s < sigma_0.

(e) Trivial zeros: why the claim is about F, not about f. F is entire while gamma has a simple pole at each s_m = -(2m+1), m >= 0. Writing f = F / gamma = F * (1/gamma), the function 1/gamma has a simple zero at s_m, so f vanishes there to order 1 + ord_F(s_m) >= 1: f has "trivial" zeros at s = -1, -3, -5, ..., forced by the functional equation exactly as zeta's trivial zeros are, shifted to the odd negative integers because chi is odd (the in-tree statement is in the zeta.epstein.completed_dh docstring). These lie outside the strip, so "all zeros of f lie in the strip" would be false. The decided statement is: all zeros of F lie in the strip. Away from the points s_m the zeros of F and of f coincide with multiplicity (gamma is finite and nonzero there); at s_m, ord_F = ord_f - 1. So the zeros of F are exactly the nontrivial zeros of f, and it is those that the map z -> s = 1/2 + iz carries to the zeros of Xi_DH, giving |Im z| < Delta in section 1.

(f) What is decided versus what is exact. Steps (a), (b), (c) are exact elementary analysis given the coefficient pattern and 0 < kappa < 1 (decided); step (d) uses the classical nonvanishing of Gamma and the structural facts that F is entire with F(s) = F(1-s), measured in-tree to defect ~1e-50 and test-pinned, with the classical derivation cited above. The numerics enter only to locate sigma_0. A fully decided rational form, with no reference to sigma_0 at all, is: g(sigma*) < 0 is decided at the rational sigma* = 1.3951361582351097210613589375 (the flint upper endpoint), hence every zero of F has 1 - sigma* < Re s < sigma*.

4. Decided numbers

All intervals are outward-rounded decimal strings; the printed interval contains the exact rational endpoints (which are dyadic-over-100 bisection points, exact by construction). Full detail in strip_results.json.

Decided inequalities, both backends, by exact rational comparison of the outward endpoints:

• Delta^2/2 < 0.4007 (against 4007/10000; flint margin ~6.6e-5). This is the number WP2's upper bound runs through: with de Bruijn 1950 Theorem 13 (hypotheses per NOVELTY.md, confirmation against the original text still a standing task and a kill condition), all zeros of H_0 in |Im z| <= Delta implies H_t real-rooted for t >= Delta^2/2, so Lambda_DH <= Delta^2/2 < 0.4007.
• g(2) < 0: the in-tree coarse strip re-derived as a decided inequality. zeta/epstein.py pins f away from 0 for Re s >= 2 by sum_{n>=2} |a_n| n^{-2} = 0.2666... < 1 (numerically verified in the tests); here that becomes a two-backend enclosure of g(2) with a decided sign, and with the reflection of section 3(d) it gives the coarse strip -1 < Re s < 2 for the zeros of F. The present sigma_0 tightens that to width 2 sigma_0 - 1, about 1.79.

kappa (section 1) is decided at 500 bits on flint; the iv leg consumes the interval KAPPA_REF +/- 1e-39, which is an enclosure because the flint ball is decided to lie inside it (same construction as validate.py check 2).

Sign decisions: 79 (flint bisection, bracket [1.39, 1.40]) and 46 (iv). Every bisection endpoint is an exact rational; every sign was decided by an enclosure excluding 0, with an undecided sign a loud abort, never a guess. Timings (strip_results.json): the whole run is about 1 s.

5. Method note on the iv backend

mpmath's iv context has no Hurwitz zeta. The plain integral-test tail enclosure [0, N^{1-sigma}/(sigma-1) + N^{-sigma}] is valid but its width decays like N^{-0.395} near sigma_0: about 1.7e-3 at N = 1e8, hopeless for a 1e-15 bisection target at any feasible N. The iv leg therefore keeps the integral-test skeleton and adds Euler-Maclaurin correction terms: each residue-class sum sum_{j>=0} (5j+c)^{-sigma} is a finite sum (M = 100 terms) plus the tail

(5M+c)^{1-sigma}/(5(sigma-1)) + (5M+c)^{-sigma}/2

• sum_{j=1}^{8} B_{2j}/(2j)! (sigma)_{2j-1} 5^{2j-1} (5M+c)^{-sigma-2j+1}
• R,

with |R| enclosed symmetrically by twice the first omitted correction term. That remainder statement holds when the relevant derivative of the integrand keeps one sign (DLMF 2.10(i)); x^{-sigma} is completely monotone, so it applies. Per the house rule the convention was not recalled but checked: check_em_tail requires each iv class-sum enclosure to contain an independent mpmath.mp Hurwitz-zeta reference at dps 60, over all four classes and five sigma values spanning the bracket (20/20 contained, enclosure widths ~1e-36). The two-backend overlap of the sigma_0 intervals is a second, independent check of the same arithmetic.

6. Honest scope

• The strip argument conditions on: the period-5 coefficient pattern with 0 < kappa < 1 (decided, flint, 500 bits); absolute convergence for Re s > 1; F entire with F(s) = F(1-s) (classical structure, in-tree measured defect, test-pinned); and the classical nonvanishing of Gamma. It uses no Euler product and no positivity, which is why it applies to an RH-violating function at all.
• The strip is not vacuous padding. The literature records DH zeros with Re s > 1 (discussed in Ferry et al., arXiv:1602.06328; see NOVELTY.md), and by the present result any such zero has Re s < sigma_0, so the shoulder region (1, sigma_0) is genuinely occupied territory, not slack: no argument can shrink the strip to Re s <= 1.
• Nothing here says anything about RH or about zeta. DH violates RH inside this strip (the pinned off-line zero of zeta/epstein.py has Re s ~ 0.8085, comfortably interior); the strip merely caps how far off the line its zeros can sit, which is what de Bruijn's theorem consumes.
• Grades, per the vocabulary contract: sigma_0, Delta, Delta^2/2 < 0.4007, g(2) < 0, and kappa are decided (backends and precisions above). The strip statement itself is exact analysis on top of the decided constants plus the cited structural facts; the composite claim "Lambda_DH <= Delta^2/2" additionally rides on de Bruijn 1950 Theorem 13 and is withdrawn if the pinning of that theorem's exact form fails (kill condition in MISSION.md). The identity and remainder checks of sections 2 and 5 are measured cross-checks, and say so.