/teal-sea
teal-sea / zeta-labstate of record · compiled 28 Sep 2026 · revision e4945c4 · source

Library · hunts/prime_zeta_rightmost/SOURCE.md

SOURCE.md: pinned source texts (WP1)

4,924 words · 589 lines · source

This file pins the texts the hunt attacks and the literature it leans on, so that every later step works against the served words rather than a paraphrase. Nothing in this file is a result of this hunt. Constants that appear here are quoted claims of the cited sources, not values decided by this hunt; decided values belong to WP2 and are stated there with backend and precision.

Fetch log:

1. OEIS A107311, verbatim as served

The complete JSON response body, reproduced byte for byte from the 2026-08-16 fetch (the server escapes < and > as \u003c and \u003e inside the link fields; the indentation is the server's):

[
	{
		"number": 107311,
		"data": "1,7,2,8,6,4,7,2,3,8,9,9,8,1,8,3,6,1,8,1,3,5,1,0,3,0,1,0,2,9,7,6,9,1,4,6,4,2,3,4,1,0,9,8,4,9,3,3,5,0,3,5,7,3,2,3,2,1,2,8,5,9,0,8,4,2,3,1,7,8,5,9,6,5,3,5,7,1,0,0,8,6,7,7,2,7,4,6,0,8,1,0,8,8,9,8,2,6,4,4,0,1",
		"name": "Decimal expansion of the solution to zeta(x) = 2.",
		"comment": [
			"From _Artur Jasinski_, Dec 21 2024: (Start)",
			"Borwein et al. (2007) proved (Theorem 3.1) that the real parts of the zeros of the partials sums of the Riemman zeta functions are not greater than this constant.",
			"Conjecture 1: the real parts of the zeros of the prime zeta function are not greater than this constant.",
			"Conjecture 2: the real parts of the zeros of the anyone subset of the prime zeta function are not greater than this constant. (End)"
		],
		"reference": [
			"Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5 Kalmar's Composition Constant, p. 293."
		],
		"link": [
			"Peter Borwein, Greg Fee, Ron Ferguson, and Alexa van der Waal, \u003ca href=\"https://projecteuclid.org/journals/experimental-mathematics/volume-16/issue-1/Zeros-of-Partial-Summs-of-the-Riemann-Zeta-Function/em/1175789799.full\"\u003eZeros of Partial Sums of the Riemann Zeta Function\u003c/a\u003e, Experiment. Math. 16(1) (2007), pp. 21-40. See p. 25.",
			"Einar Hille, \u003ca href=\"http://matwbn.icm.edu.pl/ksiazki/aa/aa2/aa215.pdf\"\u003eA problem in \"factorisatio numerorum\"\u003c/a\u003e, Acta Arithmetica, 2(1);134-144, 1936.",
			"Hsien-Kuei Hwang, \u003ca href=\"https://doi.org/10.1006/jnth.1999.2467\"\u003eDistribution of the number of factors in random ordered factorizations of integers\u003c/a\u003e, Journal of Number Theory 81:1 (2000), pp. 61-92.",
			"M. Klazar and F. Luca, \u003ca href=\"https://arxiv.org/abs/math/0505352\"\u003eOn the maximal order of numbers in the \"factorisatio numerorum\" problem\u003c/a\u003e, arXiv:math/0505352 [math.NT], 2005-2006."
		],
		"example": [
			"zeta(1.72864723899818361813510301...) = 2."
		],
		"mathematica": [
			"x /. FindRoot[ Zeta[x] == 2, {x, 2}, WorkingPrecision -\u003e 102] // RealDigits // First (* _Jean-François Alcover_, Mar 19 2013 *)"
		],
		"program": [
			"(PARI) solve(X=1.5,2,zeta(X)-2)"
		],
		"xref": [
			"Cf. A129374, A247667."
		],
		"keyword": "nonn,cons",
		"offset": "1,2",
		"author": "_Ralf Stephan_, May 20 2005",
		"references": 23,
		"revision": 55,
		"time": "2024-12-29T23:50:41-05:00",
		"created": "2005-07-19T03:00:00-04:00"
	}
]

Field notes, no content changed:

2. The two conjectures, verbatim, and their precise reading

The comment field in full, word for word as served (spelling as served: "partials sums", "Riemman" and "the anyone subset" are in the source):

From Artur Jasinski, Dec 21 2024: (Start) Borwein et al. (2007) proved (Theorem 3.1) that the real parts of the zeros of the partials sums of the Riemman zeta functions are not greater than this constant. Conjecture 1: the real parts of the zeros of the prime zeta function are not greater than this constant. Conjecture 2: the real parts of the zeros of the anyone subset of the prime zeta function are not greater than this constant. (End)

What Conjecture 1 asserts, quantified

For every s at which the prime zeta function P(s) = sum_p p^{-s} vanishes, Re s <= x*. "Not greater than" is <=, so a refutation requires a zero with Re s > x* strictly.

Domain of the statement. The series converges absolutely for Re s > 1. The classical continuation (Glaisher's inversion, treated in detail by Froberg

  1. is

P(s) = sum_{k>=1} (mu(k)/k) log zeta(ks),

analytic on Re s > 0 except at s = 1/k for squarefree k (from the pole of zeta at 1) and at s = rho/k for nontrivial zeros rho of zeta (logarithmic branch points); the line Re s = 0 is a natural boundary (Landau and Walfisz 1920; Froberg 1968). The entry does not restrict the domain, and the standard reading is: the zeros of P wherever P is defined.

Reading-independence of the planned refutation. Since x* = 1.7286... > 1, any zero with Re s > x* found in the half-plane of absolute convergence Re s > 1 is a zero of the defining series itself, where every branch of every continuation agrees with the series. Zeros there are zeros under every reading of "the prime zeta function", so a refutation built from them refutes all readings at once. This is why the hunt's target strip (x*, sigma_c) sits entirely inside Re s > 1 (MISSION.md).

What Conjecture 2 asserts, quantified

For every subset S of the primes (the evident intent of "the anyone subset"), every zero of P_S(s) = sum_{p in S} p^{-s} satisfies Re s <= x*. For finite S this is an exponential polynomial, entire; for infinite S the series is analytic at least on Re s > 1. The same reading-independence holds: subset zeros with Re s > 1 refute every reading. Taking S = all primes recovers Conjecture 1; the extra exposure of Conjecture 2 is the universal quantifier, which must survive every S, including the tails S = {p >= p_k}.

Continuation sources, pinned

C.-E. Froberg, "On the prime zeta function", BIT (Nordisk Tidskr. Informationsbehandling) 8 (1968), no. 3, pp. 187-202, doi:10.1007/BF01933420, Zbl 0167.04201. Continuation to Re s > 0, natural boundary at Re s = 0, numerical tables, and the remark on roots quoted in section 4.

E. Landau and A. Walfisz, "Uber die Nichtfortsetzbarkeit einiger durch Dirichletsche Reihen definierter Funktionen", Rend. Circ. Mat. Palermo 44 (1920), pp. 82-86. Natural boundary at Re s = 0. (The reference list of Froberg's paper, as transcribed by zbMATH, dates it 1919; MathWorld dates it 1920.)

3. The partial-sums provenance of x*

The entry's own attribution

The first comment line (verbatim in section 2) plus the entry's link: "Peter Borwein, Greg Fee, Ron Ferguson, and Alexa van der Waal, Zeros of Partial Sums of the Riemann Zeta Function, Experiment. Math. 16(1) (2007), pp. 21-40. See p. 25."

The paper, pinned

P. Borwein, G. Fee, R. Ferguson, A. van der Waall, "Zeros of Partial Sums of the Riemann Zeta Function", Experimental Mathematics 16 (2007), no. 1. Pages 21-40 per the Project Euclid metadata and the OEIS link, 21-39 per the Platt-Trudgian bibliography below. Stable URL: https://projecteuclid.org/euclid.em/1175789799 (the served page title carries the typo "Partial Summs").

Access note (2026-08-16): the Euclid abstract page is reachable, the PDF is behind a bot wall (Incapsula interstitial), and no open copy was found in this session. The opening sentence of the abstract as served by Euclid: "The semiperiodic behavior of the zeta function zeta(s) and its partial sums zeta_N(s) as a function of the imaginary coordinate has been long established." (Greek transcribed to ASCII.) Theorem 3.1's exact wording is therefore pinned through the entry's paraphrase above plus two independent secondary statements, quoted next from peer-reviewed papers whose authors read the original.

Secondary statement A: Platt and Trudgian

D. J. Platt and T. S. Trudgian, "Zeroes of partial sums of the zeta-function", arXiv:1507.01340v2 (15 Dec 2015), published in LMS J. Comput. Math. 19 (2016). They write zeta_N(s) = sum_{n<=N} n^{-s}. Quotes transcribed to ASCII from the typeset PDF:

Spira [13, Thm. 1] proved that all zeroes of zeta_N(s) must have real part less than 1.85; this was sharpened in [3, Theorem 3.1] to 1.73.

with [3] the Borwein-Fee-Ferguson-van der Waall paper ("Experiment. Math., 16(1):21-39, 2007" in their list) and [13] R. Spira, "Zeros of sections of the zeta function I", Math. Comp., 20:542-550, 1966.

Turan [16] showed that the Riemann hypothesis would follow if for all N sufficiently large zeta_N(s) had no zero in sigma > 1. Let psi_N be the supremum over all values of sigma for which zeta_N(s) = 0. Montgomery [9] showed that for all N sufficiently large,

psi_N = 1 + (4/pi - 1 - o(1)) (log log N / log N),

where the constant 4/pi - 1 is best possible. Therefore for N sufficiently large, zeta_N(s) has zeroes in sigma > 1.

with [9] H. L. Montgomery, "Zeros of approximations to the zeta function", in P. Erdos, editor, Studies in Pure Mathematics: to the Memory of Paul Turan, pp. 497-506, Birkhauser, Basel, 1983, and [16] P. Turan, "On some approximative Dirichlet-polynomials in the theory of the zeta-function of Riemann", Danske Vid. Selsk. Mat.-Fys. Medd., 24(17):1-36, 1948.

Secondary statement B: Gonek and Ledoan

S. M. Gonek and A. H. Ledoan, "Zeros of partial sums of the Riemann zeta-function", arXiv:0807.0019v2, published in Int. Math. Res. Not. IMRN (2010), no. 10, pp. 1775-1791 (journal data per the Platt-Trudgian bibliography). They write F_X(s) = sum_{n<=X} n^{-s}. Their Theorem 1, transcribed to ASCII from the typeset PDF:

Theorem 1. The zeros of F_X(s) lie in the strip alpha < sigma < beta, where alpha and beta are the unique solutions of the equations 1 + 2^{-sigma} + ... + (X-1)^{-sigma} = X^{-sigma} and 2^{-sigma} + 3^{-sigma} + ... + X^{-sigma} = 1, respectively. In particular, alpha > -X and beta < 1.72865. For X sufficiently large F_X(s) has no zeros in the half-plane sigma >= 1 + 2 log log X / log X. Moreover, for any constant c with c > 4/pi - 1 there exists a number X_0(c) such that if X >= X_0(c), then F_X(s) has at most a finite number of zeros in the half-plane sigma > 1 + c log log X / log X.

And from their proof of Theorem 1:

That the zeros all lie in a strip follows immediately from the fact that |1 + 2^{-s} + ... + X^{-s}| > 0 if 1 + 2^{-sigma} + ... + (X-1)^{-sigma} < X^{-sigma} or if 2^{-sigma} + ... + X^{-sigma} < 1. The estimates for alpha and beta may be found in Borwein et al. [1]. The last two assertions are due to Turan [9] and Montgomery [4], respectively.

with their [1] the Borwein-Fee-Ferguson-van der Waall paper, their [9] Turan 1948 and their [4] Montgomery 1983, as pinned above.

The mechanism, and the exact sense in which x* belongs to this family

Observation (elementary, this file's own; no computation is being decided here): beta_X in Gonek-Ledoan's Theorem 1 is the unique real root of sum_{n=2}^X n^{-sigma} = 1, the wall past which the leading term 1 outweighs the rest of the sum by the triangle inequality. The left side increases with X, pointwise in sigma, toward zeta(sigma) - 1; hence beta_X increases with X and converges upward to the unique real root of zeta(sigma) = 2, which is the entry's x*. So:

This is the provenance of x*: the limit of the triangle-inequality walls for the family of partial sums of zeta.

What approaches x*, and what does not

The walls approach x*. The zeros do not: by Montgomery's theorem quoted above, the supremum psi_N of real parts of actual zeros of zeta_N is 1 + (4/pi - 1 - o(1)) log log N / log N, which tends to 1. The wall is asymptotically far from sharp for partial sums; the frequencies log n for 2 <= n <= N are rationally dependent (log 4 = 2 log 2, and so on), and the phase steering that saturates a wall needs independent frequencies. For P(s) the frequencies log p are linearly independent over Q, which is exactly why MISSION.md expects the corresponding wall sigma_c to be approached by actual zeros of P. The conjecture therefore transplants a constant from a family where the wall is never attained, and is not even the limit of the real parts of zeros, onto a family with a different wall that plausibly is attained. Both halves of that last sentence are the hunt's business to establish (WP2, WP3); this file only fixes what the sources say.

4. Prior literature on zeros of P(s) itself

Corrected 2026-08-16, after the first version of this section was found to be wrong. The first version ended with the sentence "No source was found naming any rightmost-zero threshold for P(s), so nothing found here trips kill condition 2 of MISSION.md." That sentence is false. Two published sources own the threshold and the mechanism, and both were located by a later adjudication sweep on the same day. The false sentence is removed rather than softened, and what replaces it is below. Everything that was found by the first sweep (Froberg, the zbMATH review, the search log) is kept unchanged, because the queries were really run and what came back is really what came back; only the conclusion drawn from their emptiness was wrong.

4.0 Kill condition 2 has fired

MISSION.md's kill condition 2 reads:

a literature source is found proving the sigma_c threshold for P, in which case the finding is reclassified as a rediscovery and the OEIS correction cites that source instead of this work

Such a source exists, was published on 3 June 2025, and states the same constant sigma_c = 1.77954465354699... by the same triangle-inequality proof. A second, older source owns the general theorem of which the whole hunt is a specialization, and it is stronger than what the hunt proved. The condition has fired on both counts. Consequences, applied throughout this directory:

4.1 The source that owns the threshold: Belovas, Cepaityte and Sabaliauskas (2025)

Full pin: Igoris Belovas, Rugile Cepaityte and Martynas Sabaliauskas, "On the zero-free region and the distribution of zeros of the prime zeta function", Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica 33(2) (2025), 27-44. DOI 10.2478/auom-2025-0017. Received 07.10.2024, accepted 28.02.2025. Open access, CC BY-NC-ND. PDF: https://www.anstuocmath.ro/mathematics/anale2025v2/2_Igoris_Belovas_et_al.pdf (2.2 MB, image-scanned; the text used here was obtained by downloading the file and running pdftotext, not by fetching the page). Bibliographic note: the PDF's own running header says 27-43 while external indexing and the last printed folio give 27-44; cite 27-44.

Their Theorem 1, verbatim:

Theorem 1. The prime zeta function has no zeros in the half-plane sigma > sigma_0. Here sigma_0 = 1.77954465354699... is the zero of the function U(sigma) = 2^{1-sigma} - zeta_P(sigma).

Their proof, verbatim, which is the hunt's Theorem A(a) argument:

First we note that |zeta_P(s)| = |1/2^s + 1/3^s + 1/5^s + ...| > 1/2^sigma - 1/3^sigma - 1/5^sigma - ... = 2^{1-sigma} - zeta_P(sigma) = U(sigma).

Their Lemma 1, verbatim, which is the hunt's Lemma 1 (uniqueness of the root):

Lemma 1. Let the function U(sigma) be defined as above and sigma_1 = 2.18, then U'(sigma) > 0 if 1 < sigma <= sigma_1, and U(sigma) > 0 if sigma >= sigma_1.

Their Remark 1 and Conjecture 1, verbatim (the printed Remark writes zeta_P(sigma) = 0 where it means zeta_P(s) = 0 with s = sigma + it; the typo is in the source and the meaning is unambiguous):

Remark 1. Let M = 200000 and define

sigma_T = max_{|t| < T} { sigma | zeta_P(sigma) = 0 }, (2)

then we receive sigma_M = 1.682628788045196... . Thus, the result of Theorem 1 can not be refined by more than Delta = 0.097.

Conjecture 1. The estimate for the zero-free plane given by Theorem 1 can not be improved, that is, if sigma_T is defined as above (see (2)), then

lim_{T -> infinity} sigma_T = sigma_0. (3)

What their paper does not contain, established by word-level search over the extracted text: no occurrence of A107311, Jasinski or 1.72864, and no occurrence of almost period, Moreno, Sepulcre, Vidal, Kronecker, Bohr, exponential polynomial or subset. There is one incidental oeis.org URL, in reference [2], hosting Cohen's Hardy-Littlewood-constants preprint; it is not a reference to A107311. So the paper does not engage the OEIS conjectures, does not refute them (its bound 1.7795 is weaker than the conjectured 1.72864, and its numerics stop at 1.6826, below x*), and does not treat prime subsets.

4.2 The source that owns the mechanism: Sepulcre and Vidal (2022, preprint 2018)

Full pin: J. M. Sepulcre and T. Vidal, "On the real projections of zeros of analytic almost periodic functions", Carpathian Journal of Mathematics 38 (2022), no. 2, 489-501. MSC 30B50, 30D20, 30Axx, 11J72. Preprint: arXiv:1805.02041 [math.CV], 5 May 2018, titled "On the real projections of zeros of almost periodic functions" (the journal title adds "analytic"). Preprint Theorem 6 = journal Theorem 4.3, statements word for word identical apart from the label.

Their Theorem 4.3, verbatim from the journal text, with R_f defined at their equation (1.4) as R_f := closure{Re s : f(s) = 0, s in U} intersect (alpha, beta):

Theorem 4.3. Let f(s) be an almost periodic function in a vertical strip U = {s = sigma + it : alpha < sigma < beta} whose Dirichlet series is given by sum_{n >= 1} a_n e^{lambda_n s} with {lambda_1, lambda_2, ..., lambda_k, ...} Q-linearly independent and k > 2. Let sigma_0 in (alpha, beta). Then sigma_0 in R_f if and only if

|a_j| e^{sigma_0 lambda_j} <= sum_{i >= 1, i != j} |a_i| e^{sigma_0 lambda_i} (j = 1, 2, ..., k, ...). (4.8)

Specialized to P(s) = sum_p p^{-s} (a_n = 1, lambda_n = -log p_n), condition (4.8) becomes 2 p_j^{-sigma_0} <= P(sigma_0), whose binding instance is j = 1, that is U(sigma_0) = 2^{1-sigma_0} - P(sigma_0) <= 0: exactly Belovas et al.'s U. The specialization is worked out hypothesis by hypothesis in PRIOR-ART.md section 6. The conclusion it yields, closure{Re s : P(s) = 0, Re s > 1} intersect (1, infinity) = (1, sigma_c], is strictly stronger than the hunt's Theorem B.

The proof of the "if" direction also contains, in print, the hunt's aggregated-tail device: the infinite tail is collapsed into a single polygon side of length r := sum_{j >= n_0} |a_j| e^{sigma_0 lambda_j}, with the polygon step attributed to Moreno at [14, p.71].

4.3 Moreno (1973)

C. J. Moreno, "The zeros of exponential polynomials (I)", Compositio Mathematica 26 (1973), no. 1, 69-78. Not fetched; cited here on the strength of Sepulcre and Vidal's citations at [14, p.71] for the polygon construction (the hunt's Lemma 2) and [14, Lemma, p.73] elsewhere. The adjudication further records that Moreno himself applies the Geometric Principle to sum_{p <= M} p^{-s}; that statement is accepted from the adjudication and was not verified against the paper.

Also named by the adjudication and not checked in this directory: Sepulcre and Vidal, "On the non-isolation of the real projections of the zeros of exponential polynomials", J. Math. Anal. Appl. 437 (2016), no. 1, 513-525 (Proposition 5 and Corollary 6: supremum of real parts equals the balance root, and is not attained, for finite sums), and a Math.StackExchange comment (question 3894479, K. Conrad, 2020-11-05) giving the balance equation and the triangle-inequality argument publicly.

4.4 Froberg 1968, the one source the first sweep found

Full pin: C.-E. Froberg, "On the prime zeta function", BIT 8 (1968), no. 3, pp. 187-202, doi:10.1007/BF01933420, Zbl 0167.04201. Access attempt on 2026-08-16: the Springer landing page redirects to a login wall; the paper text was not reachable in this session. Two independent witnesses to its content on roots were captured instead.

The zbMATH review (Zbl 0167.04201, reviewer Edgar Karst, in German), fetched 2026-08-16 via the zbMATH Open API, quoted verbatim including the review's opening slip (it says "Riemanns Zeta-Funktion" while defining the prime zeta function):

Riemanns Zeta-Funktion sei P(s) = sum_{(p)} p^{-s}, s = sigma + i tau, über alle Primzahlen p. Die Schwierigkeiten der Berechnung werden außerordentlich groß in der Nähe der imaginären Achse. Das Haselgrove-Miller-Verfahren, das auf halbkonvergierenden, passend zugestutzten Reihen beruht, findet hier Anwendung. Sehr wenig ist über Lösungen von P(s) = 0 bekannt.

Tafel I enthält vier davon; ziemlich weit entfernt von der imaginären Achse, außerdem je eine von P(s) = -1 und P(s) = 1.

(TeX markup dropped and math transcribed to ASCII; the German text and its punctuation are kept.) Translation, this file's own: "Very little is known about solutions of P(s) = 0. Table I contains four of them; rather far away from the imaginary axis, in addition one each of P(s) = -1 and P(s) = 1."

So Froberg's paper contains, besides the continuation treatment, four numerically computed roots of P(s) = 0 in its Table I (float grade, 1968 methods), located "rather far from the imaginary axis"; the review does not give their coordinates. Operator note: pulling the paper through a library would recover the four roots; if any had Re s > x*, the conjecture would contradict literature it postdates. Nothing in the review suggests the roots sit near the right edge, and the review gives no real-part bound either way.

MathWorld's sentence (Prime Zeta Function page, fetched 2026-08-16), verbatim including its dropped word:

According to Fröberg (1968), very little is known about the roots P(s).

4.5 Search log, first sweep

Queries run 2026-08-16 through this session's web search tool, with what came back. This is the sweep that missed both sources in 4.1 and 4.2; it is kept verbatim because the record of a failed search is the evidence for the lesson in 4.6.

  1. Froberg 1968 "On the prime zeta function" BIT 8 zeros roots analytic continuation: the MathWorld page, the Semantic Scholar record (doi:10.1007/BF01933420 confirmed; abstract withheld by the publisher), HandWiki's copy of the Wikipedia material. Continuation facts and the roots remark; nothing further on zeros.
  2. "zeros of the prime zeta function" (exact phrase): no paper with that subject in the results. Hits concern zeros of zeta(s), or P on the critical line as a statistical object (G. Chavez and A. Allawala, "Prime zeta function statistics and Riemann zero-difference repulsion", arXiv:2102.02280, which studies P(1/2+it) distributionally and says nothing about zeros of P; its abstract page was checked directly).
  3. "prime zeta" function zeros "sigma > 1" OR "Re(s) > 1" zero-free region Dirichlet series over primes: zero-free-region literature for zeta(s) only; nothing for P.
  4. Jasinski conjecture prime zeta function A107311 zeros 1.728647: nothing engaging the conjectures; no citation of the entry's conjecture lines found anywhere.
  5. zbMATH Open API, title search "prime zeta function" with author Froberg: the Zbl 0167.04201 record quoted above, whose linked OEIS cross-references (A008480, A143524, A341444) concern factorization counts, not zeros.

Adjacent but not on point, recorded so the neighborhood is visibly looked at: a retracted note claiming results about P (M. Vassilev-Missana, Notes on Number Theory and Discrete Mathematics 22 (2016), no. 4, retracted) and a published refutation of it (arXiv:2103.09418) concern identities relating P to zeta and the Riemann hypothesis, not zero locations of P.

4.6 Why the first sweep missed both sources

Two distinct failures, needing two distinct countermeasures. Neither is a failure of effort, which is what makes them worth recording.

Failure 1, the venue gap. The hunt's exact-phrase query "zeros of the prime zeta function" (query 2 above) is a literal substring of the Belovas et al. title, "On the zero-free region and the distribution of zeros of the prime zeta function". The query was re-run during the adjudication and the failure reproduces exactly: ten results, MathWorld's prime zeta page among them, and the Belovas et al. paper not among them. Contributing causes, each of which alone defeats a standard sweep: the paper was never preprinted on arXiv; it lives in a Sciendo/DOAJ journal that an arXiv-plus-MathWorld-plus-zbMATH title sweep does not reach; and its PDF is image-scanned, so a retrieval step that reads landing pages rather than downloading and extracting files yields nothing even after finding it.

Countermeasure: an exact-title-substring query that returns nothing proves nothing. Search the DOI registries and open-access aggregators (Crossref, DOAJ, Sciendo) by title, not only the engines and arXiv, and treat an image-scanned PDF as a document that must be downloaded and extracted.

Failure 2, the classification gap. Sepulcre and Vidal Theorem 4.3 is filed under almost periodic functions, MSC 30B50 and 30D20, not number theory, and the paper never names a prime. No query about prime zeta functions can reach it. It was reachable only by searching for the device rather than the application.

Countermeasure: when an argument's engine is a general device (here, rational independence of frequencies plus a polygon construction), search for the device under its own name in its own field. The hunt's own Lemma 2 and Lemma 4 were the signposts, and a query for "real projections of zeros", "exponential polynomials" or "almost periodic Dirichlet series" lands on the right MSC class immediately.

The transferable corollary: ontology/knownness.py defaults to "the literature was not consulted", and this episode is the argument for that default. Four independent searches returned nothing on the core result while two published papers owned it outright. An unrun or unsuccessful search is not evidence of absence.

4.7 Finding, corrected

As of 2026-08-16, after the adjudication sweep:

Even now this is a record of queries run on the dates above, not a completed search of record: MathSciNet was not available, zbMATH full-text search was not available beyond the single API record quoted, and the density statement behind the hunt's Lemma 4 sits in classical almost-periodic territory (Jessen and Tornehave) that has not been searched at all. Do not read the corrected section as a completed audit; read it as a sweep that found what it found after a first sweep found nothing.

5. What this file settles for the other work packages