Bombieri and Ghosh, *Around the Davenport-Heilbronn function*: read in full

RISK STATUS: CLOSED, and it is a mixed verdict.

The paper was retrieved and read in full this session (50 pages, complete English translation, not an abstract or a fragment). The three questions the gate asked have definite answers:

The short form: the hunt's de Bruijn-Newman bracket is not anticipated, and the novelty sentence survives untouched. The hunt's claim to own the zero strip does not. Bombieri and Ghosh determined the exact least upper bound of the real parts of the zeros of the Davenport-Heilbronn function in 2011, and it is smaller than this hunt's sigma_0. That is simultaneously bad news for the originality of sigma_0 and good news for the upper bound, which their constant improves by a factor of 2.08.

1. Retrieval record

Earlier sessions recorded this source as unreadable ("IOPscience subscription, mathnet.ru returned 503 on every attempt, no preprint or mirror found"). Two things were wrong with that attempt, and both are worth recording so the route is reproducible:

• The mathnet identifier was wrong. The tried URLs were mi.mathnet.ru/umn9410 and mi.mathnet.ru/eng/umn9410. The paper's mathnet id is rm9410, not umn9410 (umn is the Russian journal abbreviation, not the id namespace).
• The host mi.mathnet.ru returns 503; www.mathnet.ru serves normally.

The route that worked, in full:

https://www.mathnet.ru/eng/rm9410                                     (record page)
https://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=9410&what=fullteng&option_lang=eng

The second URL redirects to https://www.mathnet.ru/links/86b2d05efd18c606e9f632710771a092/rm9410_eng.pdf and serves the full English translation, free, no login: 1,610,005 bytes, Pages: 50, Creator: LaTeX with hyperref, Author: E. Bombieri, A. Ghosh, title Around the Davenport-Heilbronn function. A Russian original PDF (1747 kB) is at the same endpoint with what=fullt&option_lang=rus.

Text was extracted with pdftotext -layout. Everything quoted below is from that extraction. Every other route confirmed the paper is otherwise closed: Unpaywall reports oa_status: closed, has_repository_copy: false, oa_locations: []; OpenAlex reports is_oa: False, any_repository_has_fulltext: False; Semantic Scholar reports openAccessPdf.status: CLOSED. No sci-hub-style mirror was used or needed.

The PDF is left in the session scratchpad and is deliberately not committed: it is a copyrighted translation, and the retrieval route above is reproducible in one command.

Bibliographic facts, confirmed against the article itself: Russian Math. Surveys 66:2 (2011), 221-270 = Uspekhi Mat. Nauk 66:2, 15-66; DOI 10.1070/RM2011v066n02ABEH004740; MSC Primary 11M41, 11M26, Secondary 11E45; received 04.10.2010; dedicated to the memory of A. A. Karatsuba; bibliography of 28 titles. Both authors at the IAS at the time (Ghosh's affiliation is given as Oklahoma State).

2. It is the same function, defined the same way

Section 5.1, verbatim:

5.1. The Davenport-Heilbronn series. Let xi be a complex number and define the Dirichlet series f(s, xi) = sum_{n>=1} a(n, xi) / n^s where a(n, xi) = 1 if n = 1 mod 5, xi if n = 2 mod 5, -xi if n = 3 mod 5, -1 if n = 4 mod 5, 0 if n = 0 mod 5.

and, further down the same page:

These series were introduced by Titchmarsh ([10], Chap. X, 10.25), who noted that for xi = (sqrt(10 - 2 sqrt 5) - 2)/(sqrt 5 - 1) the function f(s; xi) satisfies the functional equation (pi/5)^{-s/2} Gamma((1+s)/2) f(s; xi) = (pi/5)^{-(1-s)/2} Gamma((1+(1-s))/2) f(1-s; xi), which is analogous to the functional equation of Dirichlet series for an odd character mod 5, except for the fact that now the root number is 1. This function is usually called the Davenport-Heilbronn series. In what follows we will write tau_+ for the above value of xi.

This is the hunt's object exactly: the same period-5 coefficient pattern (1, kappa, -kappa, -1, 0), the same gamma factor (pi/5)^{-(s+1)/2} Gamma((s+1)/2), the same root number 1. Their tau_+ is the hunt's kappa. Measured, this session, at dps 40:

tau_+ = -phi + sqrt(1 + phi^2)  with phi = (1+sqrt5)/2
      = 0.2840790438404122960282918323931261690911
repo   zeta.epstein kappa
      = 0.2840790438404122960282918323931265126188
difference = -3.4e-34   (the repo call's own precision, not a discrepancy)

They also name the other root, tau_- = -phi - sqrt(1+phi^2) = -3.520147021340..., which they call "the second Davenport-Heilbronn function". That is Righetti's function, and section 4 below settles who owns which constant.

3. Question (i): the zero-strip abscissa. The hunt's number is absent; a sharper one is present

3.1 What they define and prove

They define (section 5, p. 240):

We define sigma(xi, q) to be the supremum of the numbers sigma > 1 for which the series sum_{(n,q)=1} a(n, xi) n^{-s} vanishes at s = sigma + it for some t in R.

So sigma(xi, 1) is the least upper bound of the real parts of the zeros of the Davenport-Heilbronn function. Righetti (arXiv:1506.05716) uses the same object under the name sigma* and defines it in the same words, "the least upper bound of the real parts of such zeros", which corroborates the reading.

They then prove it exactly. Theorem 7, verbatim:

Theorem 7. 1) Suppose that xi is real. Then sigma(xi, q) is the value of sigma > 1 satisfying the equation sum_{p = 2,3 mod 5, (p,q)=1} arctan(p^{-sigma}) = pi/2 - |theta|.

with xi = tan(theta). The proof route is Bohr's method plus Kronecker's theorem, legitimate here because f(s, xi) is a linear combination of exactly two L-functions with Euler products (they flag this dependence explicitly in section 10). Section 6 evaluates the prime sum through log L(., chi_0) and log L(., chi_2) decomposed into Hurwitz zeta functions.

3.2 The published value for the Davenport-Heilbronn function

Section 6, p. 246, verbatim:

For the Titchmarsh value tau_+ = -phi + sqrt(1 + phi^2) we find

sigma(tau_+, 2) = 1.046712, sigma(tau_+, 3) = 1.063679, sigma(tau_+, 7) = 1.093482, sigma(tau_+, 13) = 1.106200, ..., sigma(tau_+, 1) = 1.120362.

That is a published zero-strip abscissa for this hunt's function, fifteen years old, and it is sharper than sigma_0:

B-G   sigma(tau_+, 1) = 1.120362...   exact least upper bound of Re rho
hunt  sigma_0         = 1.395136...   coefficient-domination upper bound on it

The two are not the same quantity. sigma_0 solves sum_{n>=2} |a_n| n^{-sigma} = 1 and is an upper bound for the sup of real parts obtained by the triangle inequality; sigma(tau_+, 1) is the sup of real parts. The hunt's number is a cruder bound on the exact quantity Bombieri and Ghosh determined.

3.3 Digit searches, so the negative half is on the record

The strings 1.395, 1.39, 1.3951, 0.2840, .28407 and 2.4779 return zero hits in the full text. sigma_0 = 1.39513615823510972106135889... is not in this paper, in any precision, and neither is the coefficient-domination abscissa 2.477958... for tau_-. The mechanism the keyword list advertised ("reciprocals of Dirichlet series, estimates of coefficients") is genuinely there, but it is used for a different purpose: sections 3, 4 and 7 study the growth and distribution of the coefficients b_n of 1/f(s), following Landau and Hille, and relate sup Re(rho) to that growth (their Theorem 3 defines alpha(f) = sup Re(rho); their Corollary 6 gives sum_{n<=X} |b_n|^2 / n >> X^{2 sigma* - 1 + o(1)}). They never write down the elementary abscissa at which sum_{n>=2} |a_n| n^{-sigma} crosses 1.

So the narrow reading of the risk is negative and the substantive reading is positive. The hunt did not duplicate a published decimal. It did compute, by a weaker method, a bound on a quantity these authors published exactly.

3.4 Verification I ran

Two checks, both this session, both graded.

Check A (independent, finite, exact arithmetic). Section 9 makes a self-contained claim that uses none of their Hurwitz-zeta machinery:

The smallest such set P with this property [sum_{p in P} arctan(p^{-1}) > pi/2] consists of all primes p = 2, 3 mod 5 with p <= 6323, so |P| = 420.

Recomputed by direct prime summation (sympy primes, mpmath dps 30), with no shared code: threshold prime 6323, cardinality 420, both exact matches. Their quoted arctan(1/2) = 0.463648 and arctan(1/3) = 0.321751 also match, and their quoted prime-sum target 0.2767872 for tau_- reproduces as pi/2 - |arctan(tau_-)| = 0.276787179448522625754266365045. Grade: decided (finite exact prime set, integer answers).

Check B (reproduction of their Theorem 7 root). I solved their Theorem 7 equation by bisection at mpmath dps 30, evaluating the prime sum through arctan expansion plus sum_k mu(k)/k log L(ks, chi^k) over the characters mod 5:

Grade: measured, and honestly labelled a reproduction rather than an independent route. After running it I read their section 6 and found that their equation (6.1) is the same decomposition I had used, so this check confirms their arithmetic but not their method. Check A is the genuinely independent leg, and it passed exactly.

4. Question (ii): no de Bruijn-Newman content, none at all

Case-insensitive counts over the full 50-page extraction:

Bruijn 0    Newman 0    heat 0    Polya 0    Turan 0
Lambda 0    deformation 0         zero-free 0

The 28-item bibliography contains no de Bruijn, no Newman, no Polya, no Csordas-Norfolk-Varga, and nothing on entire functions of Laguerre-Polya type. It runs Hamburger, Potter-Titchmarsh, Davenport-Heilbronn I and II, Cassels, Stark, Bombieri, Selberg, Voronin, Titchmarsh, Karatsuba, Bombieri-Hejhal, Bohr, Davenport collected works, Jessen-Tornehave, Borchsenius-Jessen, Gonek, Bombieri-Mueller, Lee, Voronin, Karatsuba-Voronin, Voronin, Laurincikas, Landau (twice), Hille (twice), Carlson.

Nothing in this paper bears on Lambda_DH, on H_t, or on the backward heat flow. The hunt's headline result is not anticipated here, and the adopted novelty sentence in NOVELTY.md needs no change on account of this source.

5. Question (iii): other things in it that bear on the hunt

5.1 Righetti's 2.3822861089 is Bombieri and Ghosh's number. It appears in their section 6 as sigma(tau_-, 1) = 2.3822861089..., and their section 9 adds that the extreme zero found in a search to height 10,000 is rho = 2.37474435 + 1649.8708285 i among 2,479 zeros of f(s, tau_-) with Re rho >= 0.5. NOVELTY.md currently attributes this constant to Righetti with a note that he calls the function "of the Davenport-Heilbronn type studied by Bombieri and Ghosh". The attribution should be corrected: the constant is theirs, and Righetti is quoting it.

5.2 Zeros of the Davenport-Heilbronn function with Re s > 1 are very rare. Section 9 explains why, for xi small, such zeros are hard to find: reaching one requires aligning the arguments of p^{it} near 0 mod pi for hundreds of primes simultaneously. For xi = 0 they searched the rectangle 1/2 <= sigma < 1.2, 0 <= t < 10000, found 5,358 zeros, and found no zero with real part greater than 1 (their Figure 4). They contrast this with tau_-, where the required prime-sum margin is only 0.2767872 and such zeros are plentiful. The Davenport-Heilbronn function proper (tau_+, prime-sum target 1.2940091) sits at the hard end of that spectrum. This is context the hunt should carry: the strip Delta is set by zeros that are real but extraordinarily sparse, which is consistent with 0.4006 (and even 0.19242) being visibly loose against the deepest measured DH zeros at |Im z| = 0.347, as GATE.md already notes.

5.3 Zero counting on the critical line. Section 2.1 surveys Voronin, Karatsuba's cT (log T)^{1/2 - eps}, and reports that Selberg filled the gap to c T log T in an unpublished 1998 IAS lecture, with N_0(f; T) > c n^{-1} T log T for a combination of n Dirichlet L-functions. Relevant background for census.py, not a threat to anything.

5.4 They say what their method does not cover. Section 10: their sharp results depend on f being a linear combination of just two Euler products, and they note that Davenport-Heilbronn and Landau already showed such combinations "always have zeros with real part greater than 1". No claim in the hunt collides with this.

6. What this obliges the hunt to change

Recorded here; not yet applied to the other artifacts.

1. NOVELTY.md must stop calling sigma_0 a new number. The current text says the "genuinely new numbers are the Davenport-Heilbronn zero-strip constant sigma_0" and that "the strip constant sigma_0 = 1.39513615823510972... is what this work supplies". Both sentences must go. The accurate replacement: Bombieri and Ghosh (2011) determine the least upper bound of the real parts of the zeros of this function exactly, sigma(tau_+, 1) = 1.120362, by their Theorem 7; the hunt independently derived a weaker elementary bound sigma_0 = 1.395136... on the same quantity, and the only thing the hunt's version has that theirs does not is an enclosure-carrying derivation at rung 2 from a two-line triangle-inequality argument that does not invoke Bohr-Kronecker theory.

1. The upper bound can be improved by a factor of 2.08, by citation. Feeding their constant into the same de Bruijn Theorem 13 / Newman-Wu Theorem 7 engine:

ratio of the two upper bounds = 2.08203069740495884852.

In the wide frame the hunt's decided bracket would become 0.2304 < Lambda_DH <= 0.7697, against Lambda_zeta <= 0.22, which is a materially better result than the one currently written up.

Grade of that improved bound today: cited plus measured, not decided. Their 1.120362 is a six-decimal Mathematica value, and their Theorem 7 rests on Bohr-Kronecker machinery this tree has not verified. To carry it at rung 2 the hunt would have to (a) check Theorem 7's hypotheses in-tree, and (b) re-solve the arctan equation with outward-rounded enclosures, which is straightforward because the prime sum is monotone decreasing in sigma. Until then the decided headline stays 0.4006343708899557, with the sharper number quoted as an improvement available from the literature. Do not silently swap the headline for 0.19242.

Update 2026-08-18: conditions (a) and (b) are both met, and the headline moved by derivation rather than by swap. STRIP2.md and strip2.py derive the necessary half of their Theorem 7 in-tree, from the Euler products of L(s, chi) and L(s, conj chi) plus one Moebius image, with no Bohr theory and no Kronecker theorem, and re-solve the arctan equation with outward-rounded enclosures on both backends (python-flint 192 bits and mpmath.iv dps 40, sieve limit P = 10^5, 4814 class primes; a flint-only deep point at 320 bits and P = 10^7). The decided abscissa is the exact rational sigma_0' = 1.12036249819, so the headline is now Delta^2/2 = 0.19242481458026887663805 narrow and 0.7696992583210755065522 wide, decided, and nothing was swapped silently: the superseded value is printed beside the new one in RESULTS.md section 0, FRAME.md section 6 and GATE.md. Their converse, which turns the abscissa into an exact supremum, is neither used nor claimed, so the number is still theirs and only the grade is this hunt's. Two of their published constants now serve as controls on the new instrument, by machinery their Theorem 7 does not share: the section 9 finite claim (threshold prime 6323, cardinality 420) and sigma(tau_-, 1) = 2.38228610898712387152... against their published ten digits.

A correction this forces against check B of this file. At P = 10^7 and 320 bits, strip2.py decides that both of check B's 29-digit re-solves sit on the wrong side of their own root: sigma(tau_+, 1) is high by about 1.2e-17 and sigma(tau_-, 1) is low by about 6e-18. This corrects two in-tree re-solves and not the published paper, which prints 1.120362 and 2.3822861089 and which this instrument reproduces exactly. What it does touch is the 18-digit figure 0.192424814576128011... in the table just above, derived from the tau_+ re-solve: the decided replacement from the deep point is 0.1924248145761280190 narrow and 0.7696992583045120759956154 wide, agreeing to 17 digits.

1. STRIP.md should note that the quantity sigma_0 bounds is not merely a classical named object (Titchmarsh section 9.41, already noted) but has been computed exactly for this very function by Bombieri and Ghosh, and cite Theorem 7.

1. Attribution fix per section 5.1 above.

1. GATE.md known assumption 9 ("the novelty line assumes Bombieri-Ghosh 2011 ... contain neither sigma_0 nor a bound on this constant. Neither has been read.") is now half discharged. Bombieri-Ghosh has been read. It contains no bound on Lambda_DH. It does contain a sharper strip constant. The academia.edu preprint 166936409 remains unread.

7. Residual risk after this reading

• On Lambda_DH itself: closed by this source. Bombieri-Ghosh cannot contain a de Bruijn-Newman bound, because the concept does not appear in it.
• On the strip constant: closed, and adverse. The exact constant is published. The hunt's originality claim on sigma_0 does not survive; its enclosure-carrying elementary derivation still stands as work, but the number it bounds was known.
• Still open, unchanged by this session: academia.edu preprint 166936409, and the single-source forward-citation sweep on Dobner (Semantic Scholar only, OpenAlex returned 429) that NOVELTY.md flags as needing repetition against Google Scholar or MathSciNet.
• One thing this reading newly suggests checking: Bombieri-Mueller, "On the zeros of certain Epstein zeta functions", Forum Math. 20:2 (2008), 359-385 (their reference [18]), which is where the sigma(xi, q) computation method comes from and which the hunt has not consulted. It is about Epstein zeta functions, so it is unlikely to carry heat-flow content, but it is the parent of the constant that just displaced sigma_0 and it is one citation away.

Retrieved, read and verified 2026-08-16. Full text: 50 pages, mathnet.ru rm9410, English translation, free. Verification scripts for checks A and B are in this session's scratchpad (verify_primes.py, verify_bg2.py); check A is cheap and worth folding into the hunt's own test set if the improved bound is ever adopted.