Searched 2026-10-26, before any DH correction term was written (DH.md, step 3). The question: is there a published Riemann-Siegel-type or other effective approximation, with explicit error terms, for H_t^DH, or for DH itself as c L(s, chi) + conj(c) L(s, conj chi) with chi mod 5?
Queries and what they returned
Web search (standard mode), 2026-10-26. Hits are listed as the search returned them; "read" means more than the abstract or snippet was read, and almost nothing was.
| query | relevant hits | what they settle |
|---|---|---|
| Riemann-Siegel formula Dirichlet L-functions explicit error term | Hiary, An alternative to Riemann-Siegel type formulas, arXiv:1403.0317 (Math. Comp. 85 (2016), no. 298); Arb's acb_dirichlet documentation; a Proc. Japan Acad. note correcting an earlier L-function RS formula | Hiary (snippet, introduction): for the Riemann-Siegel type formulas for L(1/2+it, chi) of Davies, Deuring, Lavrik and others, "it does not seem that we have an analogue of Gabcke's explicit estimate for the remainder terms", nor an easy a posteriori estimate. Arb's explicit RS bound (Arias de Reyna, Theorem 4.2) covers zeta only. |
| Davenport-Heilbronn function zeros computation Riemann-Siegel type formula | Gonek-Ivic and Balanzario-type DH papers; Gritsenko, On the zeros of the Davenport-Heilbronn function (Trudy MIAN 296, 2017); Zetafast, arXiv:1703.01414 | no RS formula for DH specifically. Balanzario and Sanchez-Ortiz compute DH's off-line zeros by deforming a series with known zeros, not with an RS formula. |
| de Bruijn-Newman constant Davenport-Heilbronn heat flow effective approximation | Polymath 15 (arXiv:1904.12438), Rodgers-Tao, the Wikipedia entry | every effective heat-flow approximation found is zeta's (Polymath 15). |
| Rubinstein smoothed approximate functional equation incomplete gamma | Booker et al., Computing L-functions: a survey (JTNB, doi 10.5802/jtnb.920); Johansson, arXiv:2110.10583; Rubinstein's method restated in arXiv:1211.4181 | the smoothed approximate functional equation: L-values as rapidly converging sums of incomplete gamma functions, exact up to truncation, with a free complex parameter that controls cancellation. This is what step 3 uses. |
| Siegel 1943, Contributions to the theory of the Dirichlet L-series and the Epstein zeta-functions | the citation only (Ann. of Math. 44 (1943), 143-172, doi 10.2307/1968761) | not read; its role in L-function RS formulas is not confirmed by the hits. |
| Davies 1965, approximate functional equation for Dirichlet L-functions | D. Davies, Proc. Roy. Soc. A 284 (1965), 224-236, doi 10.1098/rspa.1965.0060, through Hiary's description | an RS-type formula for L(s, chi), with no explicit remainder bound (Hiary). |
| Balanzario Sanchez-Ortiz 2007 | Math. Comp. 76 (2007), no. 260, 2045-2049 | off-line zeros by deformation, as above. |
| Dobner arXiv:2005.05142 | Acta Arith. 201 (2021), 29-62 | Lambda_F >= 0 for the extended Selberg class, so DH has a de Bruijn-Newman constant (hunt #61 uses this). No approximation. |
Hunt #61's own search record (hunts/lambda_dh_bounds/NOVELTY.md, "Searches run", 2026-08-15 to 2026-08-18) covers the de Bruijn-Newman side in depth: ANTEDB chapter 18 and a three-index forward-citation sweep of Dobner. It found no heat-flow computation attached to DH, and its note is not repeated here.
Conclusion, at the level of the search
- No published effective approximation of H_t^DH was found, and no Riemann-Siegel formula for DH or for L(s, chi) mod 5 with an explicit remainder. One such source would have been the analogue of Polymath's Theorem 1.3. Hiary (2016) states the gap for L-functions in general.
- What can be used: Rubinstein's smoothed approximate functional equation. It is exact up to a truncation that can be bounded, and its two sums play the roles of Polymath's A and B. The correction C is then the difference between the sharp sums and the smoothed ones (DH.md, step 3). Hunt #61's ball instruments (
instrument.H_ball,winding.second_derivative_bound,winding.winding_rectangle) enclose H_t^DH directly and are reused for t > 0 and as ground truth. - Not searched: MathSciNet and zbMATH (no access from this session), and the full texts of Davies, Deuring and Lavrik. A remainder bound may be hidden in one of them. Hiary's statement is the evidence against it, not a proof of absence.
Spira's DH zeros used by the gate (0.650830 + 114.163343i, 0.574356 + 166.479306i, 0.724258 + 176.702461i, besides the pinned 0.808517 + 85.699348i) were recalled from Spira (Math. Comp. 63 (1994), 747-748; the citation is zeta.epstein's). Each was then located here with mpmath's findroot on F. It moved by at most 6e-7 from the recalled digits (measured, 2026-10-26).