# Verification report — AMR-022-7031 (Hayman–Lingham, Problem 7.31: for which functions f does Σ f(c_n/a_n) converge?)

Verification date: 2026-10-11.

**Verdict.** The note answers both questions of Problem 7.31 for the class of all admissible sequences
(a_1 > 0, 0 ≤ a_n ≤ n; b_n = a_1 + … + a_n, c_n = b_1 + … + b_n). Every statement it makes is proved, without
computation. The scope is as follows.
- **Proved.**
  - Theorem 1.1: if √x·f(x) is non-negative and non-increasing for x ≥ x_0 and Σ f(n²) < ∞, then Σ f(c_n/a_n)
    converges for every admissible sequence; for x_0 = 1,
    Σ f(c_n/a_n) ≤ 6·Σ f(n²) + (2 log(1/a_1) + 2 + 5 log 2)·f(1). If Σ f(n²) = ∞, the series diverges for a_n = n
    and for a_n = 1.
  - Theorem 1.2: for an arbitrary real function f on [1, ∞), the series converges for every admissible sequence
    if and only if ∫^∞ sup_{t ≥ u} √t·|f(t)| du/u < ∞; equivalently, |f| ≤ F near infinity with √x·F(x)
    non-increasing and Σ F(n²) < ∞. The same criterion holds for the classes 0 < a_n ≤ 1 and b_n ≤ n.
  - Theorem 1.3: without a condition the comparison with Σ f(n²) fails (a non-increasing C^∞ function; a
    non-negative C^∞ function vanishing at all squares); "x^θ·f(x) non-increasing" with θ < 1/2 is not enough;
    for a_n ≤ n·ψ(n) with ψ unbounded even Σ (a_n/c_n)^α can diverge for every α.
  - Lemma 2.2 (the key estimate): Σ over the n with a_n > 0 and c_n ≤ X·a_n of (a_n/c_n)^{1/2} is smaller than
    3 log X + 2 log(1/a_1) + 2 + 2 log 2. Proposition 7.1: in a bound A log X + B log(1/a_1) + C one needs
    A ≥ √6/2 and B ≥ 1; the best constants are not known.
  - Remark 5.4(4): for a single sequence the series can converge conditionally; if it converges for every
    admissible sequence, it converges absolutely for every admissible sequence (Theorem 1.2(b)).
  - Theorem 8.2: the statement that the problem quotes for the ratio c_n/b_n holds for all admissible sequences.
- **Quoted in the problem as known, and proved again here.** The convergence of Σ (a_n/c_n)^α for α > 1/2 (no
  proof or reference in the collection; origin not identified) and the result for c_n/b_n attributed to
  D. Borwein. Borwein's note (1965) proves the latter for positive non-decreasing sequences b_n ≤ K·n (as a case
  of a weighted theorem for b_n ≤ n·k_n); Theorem 8.2 extends the unweighted statement to 0 ≤ a_n ≤ K·n.
- **Not treated.** Subclasses of sequences (for instance monotone ones), for which the necessity part is not
  known; sums iterated more than twice; the best constants in Lemma 2.2.
- **Limits of the verification.** Two independent verification runs (A and B) examined the first written
  version of the results, and a further independent verification run (the final run) examined the final text
  of the note; all three were AI-assisted, and none is a review by a human expert. Several sources were not
  read (below): Monthly problem 5167, the problem list of 1977 in which the problem was proposed, the later
  lists, and Update 7.31 of the Springer edition. Novelty is not certified: a search that finds nothing is not
  a proof of novelty, and no priority is claimed.

The note is unrefereed.

## Statement checked
- **Primary source.** W. K. Hayman and E. F. Lingham, "Research Problems in Function Theory", arXiv:1809.07200,
  version 2 of 21 September 2018 (the latest arXiv version, checked with the arXiv interface on 2026-10-11),
  Problem 7.31, printed page 169; proposer W. K. Hayman.
  - Read in the arXiv PDF of version 2 (1,706,228 bytes, sha256
    `8e28fd4403a07e4e19a9816b7efaafddf9f475d59cf8c34a8255cb03833ed4f0`), page 170 of the file. Verification run A
    compared the TeX sources of versions 1 and 2: the lines of Problem 7.31 and of Update 7.31 are identical.
    The final run fetched the TeX source and the PDF of version 2 again and compared the note with both.
  - The tables at the end of this text (printed pages 252–255): Table 1 (abbreviations for the earlier
    problem lists), Table 2 (list of problems proposed), Table 3 (comments on problems). According to Table 2
    the problems 7.23–7.32 were proposed in the list C = J. M. Anderson, K. F. Barth, D. A. Brannan and
    W. K. Hayman, "Research problems in complex analysis", Bull. London Math. Soc. 9 (1977) 129–162 (Crossref,
    doi:10.1112/blms/9.2.129, gives the first three names). Table 3 has no entry for Problem 7.31. The same
    tables are Tables A.1–A.3 of the Springer edition (2019), whose front matter, appendix and list of
    references are freely accessible on the publisher's site and were read by the final run; in that list of
    references Borwein's note is the only entry marked as cited in 7.31.
  - Content, in the notation of the note: a_1 > 0, 0 ≤ a_n ≤ n, b_n = Σ_{ν≤n} a_ν, c_n = Σ_{ν≤n} b_ν. It is stated
    that Σ (a_n/c_n)^α < ∞ if α > 1/2. Question 1: for what other functions f is Σ f(c_n/a_n) < ∞? Question 2:
    is it true, for instance, that under some smoothness condition on f the series Σ f(c_n/a_n) converges with
    Σ f(n²)? Then: the analogous result for c_n/b_n was obtained by Borwein: if x·f(x) is positive and
    non-increasing for x ≥ a > 0 and Σ f(n) < ∞, then Σ f(c_n/b_n) < ∞.
  - Update 7.31 in this version: "No progress on this problem has been reported to us."
- **Corpus record.** ulamai/UnsolvedMath, record AMR-022-7031 (dataset version 1.6.0; upstream status `open`).
  Its statement agrees with the source; only the citation mark after "Borwein" is missing.
- **Borwein's note.** D. Borwein, "On a class of convergent series of positive terms", J. London Math. Soc. 40
  (1965) 587–588, doi:10.1112/jlms/s1-40.1.587. Read completely in the scan linked from the bibliography page
  "David Borwein at Ninety" of the CARMA centre (2 pages, 727,131 bytes, sha256
  `9655cf6df9f643f1eec0f280e930f67372b638ef0efe803664645b1f8b7ce1fb`), by verification run A, at the writing,
  and by the final run, which compared every sentence of the note about it with the scan; each time with the
  same result:
  - standing assumption 0 < u_1 ≤ u_2 ≤ …, s_n = u_1 + … + u_n (in the notation of the problem u_n = b_n,
    s_n = c_n);
  - it cites the problem of Hayman and Barry in the American Mathematical Monthly (1964, p. 99), without its
    number: if u_n ≤ n and α > 1, then Σ (u_n/s_n)^α < ∞;
  - Theorem: if u_n ≤ n·k_n with k_n ≥ 1, if x·f(x) is positive and non-increasing for x ≥ a > 0, and if
    ∫_a^∞ f < ∞, then Σ_{n ≥ n_0} k_n·f(k_n·s_n/u_n) < ∞;
  - Remark: for bounded (k_n) the conclusion can be replaced by Σ_{n ≥ n_0} f(s_n/u_n) < ∞; for every unbounded
    (k_n) with n·k_n non-decreasing there is such a sequence with s_n/u_n ≤ 2 for infinitely many n;
  - nothing on a ratio of the type c_n/a_n.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Quantifier: the series is to converge for every admissible sequence (the sense in which α > 1/2 is the threshold) | used throughout | Section 1.1 |
| Question 2: under a regularity condition, does Σ f(n²) < ∞ imply Σ f(c_n/a_n) < ∞? | yes if √x·f(x) is non-negative and non-increasing near infinity, with an explicit bound; false for merely non-increasing f and for merely smooth f | Theorem 1.1(a); Theorem 1.3(a), (b) |
| "Converges with" read in both directions | the converse holds for the class: if Σ f(n²) = ∞, the series diverges for a_n = n. It is false sequence by sequence (a_n = 2^{-n}) | Theorem 1.1(b); Theorem 1.2; Section 1.1 |
| Question 1: for which f does the series converge for every admissible sequence? | if and only if ∫^∞ sup_{t ≥ u} √t·\|f(t)\| du/u < ∞; f arbitrary and real | Theorem 1.2 |
| Terms with a_n = 0 (c_n/a_n undefined) | omitted, as in the series of the problem; the admissible sequences for which divergence is proved have a_n > 0 for all n | Section 1.1 |
| Conditional convergence | possible for a single sequence; convergence for all admissible sequences is absolute | Theorem 1.2(b); Remark 5.4(4) |
| Normalization a_n ≤ 1, or b_n ≤ n as in Borwein's note, instead of a_n ≤ n | the same answer | Theorem 1.2 (a′), (a″) |
| a_n ≤ K·n | the same answer, by scaling | Remark 5.4(1) |
| a_n ≤ n·ψ(n) with ψ unbounded (for instance a_n ≤ n^p, p > 1) | no statement of this kind holds | Theorem 1.3(c) |
| Convergence for one given sequence | not the question; not treated | — |
| The ratio c_n/b_n under a_n ≤ n | the quoted statement holds | Theorem 8.2 |
| Monotone sequences only; sums iterated more than twice | not treated | "Scope and priority" |

## Results in the paper
- **Lemma 2.1, Lemma 2.2.** Elementary facts; the key lemma with parts (i) b_n < 2X for n ≤ m/2 and m good,
  (ii) n·a_1 < 2X² for two good indices n, m with m ≥ 2n, (iii) finiteness of the set G_X of good indices,
  (iv) the bound for W(X), (v) |G_X| ≤ √X·W(X). The proof of (iv) splits G_X at half of its largest element and
  telescopes (a_n/c_n)^{1/2} ≤ ½ log(b_n/b_{n-1}) + ½ log(c_n/c_{n-1}) over each half.
- **Theorem 3.1, Theorem 1.1.** A weighted inequality for non-increasing g (summation by parts over dyadic
  layers of c_n/a_n), and the comparison theorem.
- **Lemma 4.1, Proposition 4.2.** Sequences with a_n = (b_{n-1} + c_{n-1})/(Y_l − 1) in phases: c_n/a_n = Y_l
  exactly, 0 < a_n ≤ 5/16 for n ≥ 2 (so 0 < a_n ≤ 1 and b_n ≤ n), and phase lengths of order √Y_l·log(Y_l/Y_{l-1}).
- **Lemma 5.1, Theorem 1.2, Corollary 5.3, Remark 5.4.** A selection lemma; the characterization; the case of
  almost decreasing √x·f(x); scaling, two examples for the supremum in the criterion, and an example of
  conditional convergence for a single sequence.
- **Propositions 6.1 and 6.2.** The explicit functions and sequences of Theorem 1.3.
- **Proposition 7.1.** The constants 2 in Lemma 2.2(i), (ii) are best possible; lower bounds √6/2 and 1 for the
  coefficients of log X and log(1/a_1).
- **Lemma 8.1, Theorem 8.2.** The ratio c_n/b_n: Σ over the n with c_n ≤ X·b_n of b_n/c_n is smaller than
  5 log X + 2 log(1/a_1) + 2 + 4 log 2; if x·f(x) is non-negative and non-increasing and Σ f(n) < ∞, then
  Σ f(c_n/b_n) converges; divergence for a_n = 1 otherwise.

## Computations (tests; programs and outputs in reproducibility/)
No proof depends on a computation. In all programs the sequences are exact (integers over a common denominator)
and memberships are decided exactly; floating point enters only through square roots and logarithms of the final
comparisons.
- **The program of the note** (`writing_stage/check_note.py`, standard library, about half a minute; TOTAL
  failures: 0). It tests the statements with the constants as printed:
  - Lemma 2.2 at every threshold of 69,442 sequences (588,035 pairs of a sequence and a threshold), among them all
    65,536 sequences of length 8 on a four-value grid, sequences with a_1 down to 10^-30, single terms after up to
    10^12 zeros, late bursts and exact ties; largest W/bound 0.4978; the two partial sums of the proof reach 0.8502
    and 0.8156 of their bound;
  - Theorem 3.1 (1,016 tests) and the inequality of Theorem 1.1(a) (14,253 tests) on 254 sequences; largest
    quotients 0.3890 and 0.4530;
  - Lemma 4.1 and Proposition 4.2 exactly for 403 lists (98,036 indices); the numbers of Propositions 6.1, 6.2
    and 7.1;
  - Lemma 8.1 at every threshold of 12,752 sequences extended by zeros (347,759 pairs; largest quotient 0.6002)
    and the inequality of Theorem 8.2(b) (38,256 tests).
- **The programs with which the results were first obtained** (`original/`). They test the first form of the
  statements (bound (9/2) log X + 3 log(1/a_1) + 2 + 2 log 2 in Lemma 2.2(iv); 9·Σ f(n²) + (7 + 3 log(1/a_1))·f(1)
  in Theorem 1.1(a); 7 log X + 3 log(1/a_1) + 2 + 5 log 2 in Lemma 8.1): 53,900 pairs of a sequence and one of 14
  fixed values of X (largest quotient 0.3346); 1,835,008 pairs from all 262,144 sequences of length 9 on a grid;
  5,096 and 6,000 tests of the sufficiency theorem; Proposition 4.2 exactly for 48 lists (53,689 indices).
- **An earlier check** (`earlier_check/`), AI-assisted, made with the first version: in its recorded run
  2,094,005 sequences for Lemma 2.2 (largest quotient 0.3362), 3,280,027 and 922,280 tests of the sufficiency
  theorem, 10,441 lists for Proposition 4.2, 358,186 sequences for Lemma 8.1; no violation. Six of its nine
  programs run for a fixed time; their counts are those of the recorded run and change from run to run.
- **Verification run A** (`verification_run_A/`): Lemma 2.2 (first form) at 18.6 million pairs of a sequence and a
  threshold (2,411 structured and 2,193,265 short sequences; largest quotient 0.3346) and in 7,680
  simulated-annealing searches (0.3335); 20,041 tests of the sufficiency theorem; Proposition 4.2 exactly on 803
  lists (271,840 indices); the numbers of Propositions 6.1 and 6.2; Lemma 8.1 (first form); five functions chosen
  to separate the criterion of Theorem 1.2 from the truth; algebraic identities with a computer algebra system.
- **Verification run B** (`verification_run_B/`): both forms of the bound of Lemma 2.2(iv) on 865,376,700
  sequences in 18 exhaustive enumerations, on 9,194 structured and random sequences (11,068,550 terms, a_1 down
  to 10^-300), on 864 configurations with a late burst (150,176 good indices beyond 2X²/a_1) and in a beam search
  for large W(X); no violation; largest quotients 0.337 (first form) and 0.501 (present form). The sequences of
  Proposition 6.1 rebuilt exactly; Theorem 3.1, Theorem 1.1(a) and Lemma 8.1 in their first forms in two runs
  of 1,599 sequences each. The six original programs were run again: outputs identical up to running times.
- **The final run** (`independent_run_2/`), with the constants as printed, at every threshold, zero tail
  included:
  - Lemma 2.2 and Lemma 8.1 on all 3,155,623 sequences of nine grids (27,024,902 and 43,059,200 pairs of a
    sequence and a threshold; largest quotients 0.3968 and 0.4959);
  - Lemma 2.2 on 6,104 structured and random sequences of ten families, with a_1 down to 10^-300 and single terms
    after up to 10^15 zeros (largest quotient 0.4977; the two partial sums of the proof reach 0.7318 and 0.8521
    of their bound), and Lemma 8.1 on 4,615 of them (largest quotient 0.6157, larger than the value 0.6002 of the
    program of the note);
  - a randomized search with 150 configurations of 40,000 steps (largest quotients 0.4827 and 0.5980);
  - Lemma 4.1 (20,000 steps); Proposition 4.2 (362 lists, 38,439 indices); Proposition 6.1 (the numbers m_l; the
    first three phases index by index; a_n ≤ 5/16 at n = 639,572 in exact integer arithmetic; the partial sums;
    parts (b) and (c)); Proposition 6.2 (four functions ψ, one of them not monotone); Proposition 7.1; the
    examples of the introduction and of the remarks;
  - inequality (2), Theorem 3.1 and Theorem 8.2 in 607,935 tests on 5,268 sequences (largest quotient 0.4809);
  - seven functions with point support chosen to separate the criterion of Theorem 1.2 from the truth (for four
    the criterion fails, among them one of variable sign and one with unbounded √x·f(x); for three it holds),
    and the example of Remark 5.4(4).
  No violation was found.
- **Weak points of the tests**, as found by run B: the first original program uses 14 fixed values of X instead
  of the thresholds, and the upper part of the first form of the proof is non-empty for only 243 of its 53,900
  pairs (run B added a targeted test); two of the eight hill climbs of the original programs are too short to
  carry information; the original test of Lemma 8.1 ignores the zero tail (run B, the program of the note and
  the final run include it); the searches are heuristic; Proposition 6.1(c) was tested numerically only
  through the two inequalities of its proof, by the final run for one function η.
- **Re-runs.** At the writing the quick programs of the package and most of the slow ones were run again from
  the layout of the package. Their outputs agree with the recorded ones up to fields that record running times.
  The final run repeated the quick part from an extracted copy of the archive, with the same result, and again
  after its own programs had been added (32 quick programs, and its two slow programs).
  Details are in `reproducibility/README.md` and `reproducibility/RERUN_LOG.txt`; `reproducibility/run_quick.sh`
  repeats the quick part.

## Independent verification runs
The results were first obtained with the constants 9/2 and 3 in Lemma 2.2(iv). An earlier AI-assisted check of
this first version followed. Then two independent verification runs, both AI-assisted, examined the first
version, each with its own programs (2026-10-11). Run A examined the statement and every proof line by line;
run B made independent computations, examined the original programs and searched the literature. After the
note had been written, a further independent verification run, also AI-assisted and with programs of its own,
examined the final text (2026-10-11); it is called the final run here and in the note (second round of
verification, folder `independent_run_2/`).

| Item | Run A | Run B |
|---|---|---|
| Statement of the problem, hypotheses, conventions, attribution to Borwein | CONFIRMED_WITH_FIXES: text compared with the arXiv TeX of versions 1 and 2; Borwein's note read; the description of the context had to be rewritten, no theorem affected | agrees with the source (arXiv versions 1 and 2) |
| Key lemma, first form (9/2, 3) | CONFIRMED: re-derived line by line; 18.6 million exact threshold checks | CONFIRMED by tests; a simpler proof with the constants (3, 2); sharpness of parts (i), (ii); lower bounds √6/2 and 1 |
| Theorem 3.1 and Theorem 1.1, first form | CONFIRMED: re-derived; 20,041 tests | tested (47,970 tests); read, no problem seen |
| Proposition 4.2, Lemma 5.1, Theorem 1.2, Corollary 5.3 | CONFIRMED: re-derived; 803 lists exactly; attempts to separate the criterion from the truth failed | constructions CONFIRMED (rebuilt exactly); the step from (a″) to (d) re-derived |
| Propositions 6.1, 6.2; Lemma 8.1 (first form) with its consequence | CONFIRMED_WITH_FIXES: mathematics correct; the relation to Borwein's note had to be stated correctly | re-derived and rebuilt exactly; all numbers as stated |
| The original programs | outputs read and compared with the quoted numbers | CONFIRMED_WITH_FIXES: outputs reproduced exactly; no vacuous test; descriptions of coverage corrected |
| Novelty | not its part; Borwein's note contains nothing on c_n/a_n | CONFIRMED_WITH_FIXES: no prior solution found; novelty not certified; two bibliographic leads added |

No run found a wrong theorem, proposition or lemma, a gap or a wrong constant.

**Corrections required by runs A and B**, all applied in the note:
1. (Run A) What Borwein's note proves is stated with its hypotheses (Section 1.3, abstract, "Scope and
   priority"); the statement that the note could not be read was removed.
2. (Run A) The section on c_n/b_n is presented as an extension of the unweighted form of Borwein's theorem from
   b_n ≤ K·n to 0 ≤ a_n ≤ K·n, not as a new proof of that theorem (Section 1.3, Remark 8.3(1)).
3. (Run A) The normalization b_n ≤ n was added: the classes {0 < a_n ≤ 1} ⊂ {b_n ≤ n} ⊂ {a_n ≤ n} give the same
   answer (Section 1.1; Theorem 1.2 (a′); Proposition 4.2(b)).
4. (Run A) Proposition 6.2 is presented as the analogue of the counterexample in Borwein's remark.
5. (Run A) The Monthly problem of Hayman and Barry is, by Borwein's citation, the statement on b_n/c_n with
   α > 1; the origin of the statement for α > 1/2 remains unidentified; the Monthly pages and the Springer
   edition are listed as not read.
6. (Run A, optional) a_1 ≤ 1 is stated in Lemma 2.1(a); the direct proof that (c) implies (d) is Remark 5.2.
7. (Run B) The problem list of Barth, Brannan and Hayman (1984), which one citation index lists as the only work
   citing Borwein's note, and the proposal of Monthly problem 5167 are in the bibliography, as not read; the
   databases that list no citing work are named. (The inference that the problem first appeared in the list of
   1984 was corrected by the final run, see below.)
8. (Run B) The sources that could not be obtained are disclosed in the abstract, in Section 1.3 and in "Scope
   and priority".
9. (Run B) The weak points of the original tests are described in Section 9 and in the README of the package.
10. (Run B) The values of W(X) found by run B replace the understated range of the first version; the two
    uninformative hill climbs are named.
11. (Run B) It is said that the programs of the earlier check are time-budgeted.
12. (Run B, optional) Lemma 2.2(iv) is stated with the constants 3 and 2 and proved by the split at half of the
    largest good index; Theorem 1.1(a) has the constants 6 and 2 log(1/a_1) + 2 + 5 log 2; Proposition 7.1
    contains the sharpness statements.

**Changes made when the note was written, after the two runs.**
- Lemma 2.2(iv), Theorem 3.1 and Theorem 1.1(a) carry the constants of item 12; the proof of Lemma 2.2(iv) is
  the one given by run B, written out in full.
- Lemma 8.1(iv) is proved by the same split, which gives 5 log X + 2 log(1/a_1) + 2 + 4 log 2 instead of the bound
  7 log X + 3 log(1/a_1) + 2 + 5 log 2 confirmed by the runs. The explicit inequality of Theorem 8.2(b) and the
  divergence statement for a_n = 1 were added.
- The proofs of Proposition 7.1 were written out; in Proposition 6.2 the function ψ is only assumed unbounded.
- The identities for one greedy step were separated as Lemma 4.1.
- These changes were examined by the final run (next paragraph) and are tested by parts A, B, F and G of the
  program of the note and by the programs of the final run.

**The final run (second round, final text).**

| Item | Result |
|---|---|
| Statement of the problem | CONFIRMED: compared again with the TeX source and the PDF of arXiv version 2; the reading "for every admissible sequence" is the literal one |
| Lemma 2.1; Lemma 2.2 with the constants 3, 2, 2 + 2 log 2 and the split at N/2, as written | CONFIRMED: re-derived line by line, including the cases T = {N}, E = ∅, E = {1}; tested with programs of its own |
| Theorem 3.1, Theorem 1.1 with the constants 6 and 2 log(1/a_1) + 2 + 5 log 2 | CONFIRMED: re-derived; tested |
| Lemma 4.1, Proposition 4.2, Lemma 5.1, Theorem 1.2, Corollary 5.3, Remark 5.4 | CONFIRMED: re-derived, also for functions of variable sign; constructions rebuilt exactly |
| Propositions 6.1, 6.2 | CONFIRMED: re-derived; all numbers rebuilt |
| Proposition 7.1 as written | CONFIRMED: re-derived; (a) rebuilt for 2,001 values of X, (b) tested for all steps up to 2·10^6, (c) for 38 pairs |
| Lemma 8.1 with the constants 5, 2, 2 + 4 log 2; Theorem 8.2 with the explicit inequality and the divergence for a_n = 1 | CONFIRMED: re-derived line by line; tested with the zero tail |
| Borwein's note | CONFIRMED: read again; every sentence of the note about it agrees with the scan |
| Where the problem was proposed | CORRECTED: the list of 1977, not that of 1984 (tables of the collection) |
| Bibliography | CONFIRMED through Crossref, zbMATH Open and the arXiv interface; one entry added |
| Novelty | no earlier answer found; novelty not certified |
| Package | the quick part re-run from the archive: all 28 programs reproduce the recorded outputs; after the programs of the final run had been added, all 32 quick programs and the two slow programs of the final run do so |

**Corrections required by the final run**, all applied:
1. History of the problem: according to the tables of the collection, Problem 7.31 was proposed in the list of
   Anderson, Barth and Brannan (1977), and no later comment on it is recorded; the statement that it probably
   first appeared in the list of 1984 was removed everywhere, the list of 1977 was added to the bibliography as
   not read, and the record of the citation index is described as it is (identifier of the list of 1984,
   authors of the list of 1977).
2. Springer edition: the note says what was read (front matter, appendix with the tables, list of references)
   and what was not (the chapter with Update 7.31).
3. Wording on the sources that were not accessible and on the order of the names of the proposers of the
   Monthly problem (Barry and Hayman in the Crossref record, Hayman and Barry in Borwein's note).
4. The paragraph "Verification" of the note, this report and the README of the package describe the final
   state.
5. The remark "the constants in (3) are not sharp", which the note did not prove, was replaced by the statement
   that the best constants are not known.
6. The sentence on conditional convergence after Theorem 1.2 was made exact, and Remark 5.4(4) was added
   (an example of conditional convergence for a single sequence; its proof was checked line by line and its
   numbers were computed by the final run).
7. Four inaccuracies of wording: which divergent sequences have no zeros (Section 1.1); the length of a greedy
   phase (at least (3/16)·√Y·log(Y/Y′)); the numbers m_l of Proposition 6.1(a) do not depend on θ; the converse
   in "Scope and priority".
8. Section 9 describes the programs of the final run, with the larger quotient 0.6157 for Lemma 8.1.
9. Package: folder `independent_run_2/`, re-run log, metadata.

## Relation to the literature, novelty and scope
- **Searches (10 and 11 October 2026).** When the results were first obtained, by runs A and B, at the
  writing, and by the final run: queries to the arXiv interface, to zbMATH Open, to OpenAlex (among them the
  works citing the Springer edition of the collection and the lists of 1977, 1984 and 1989, filtered by title
  and abstract), to Crossref and to Semantic Scholar, and ten web searches.
  - No paper on Problem 7.31, on series Σ f(c_n/a_n) or on estimates of the type of Lemma 2.2 was found.
  - The closest general work found: G. Bennett and K.-G. Grosse-Erdmann, "On series of positive terms", Houston
    J. Math. 31 (2005) 541–586; only its zbMATH review was read, and the review does not mention the problem.
- **What was read.** Problem 7.31, its update and the three tables in the arXiv version 2 of the collection;
  Borwein's note, completely (see above); of the Springer edition (2019), the front matter, the appendix with
  the tables and the list of references.
- **What was not read.**
  - Monthly problem 5167 of P. D. Barry and W. K. Hayman (the names in the order of the Crossref record): the
    proposal (Amer. Math. Monthly 71 (1964), no. 1, in "Problems for Solution: 5161–5170", pp. 98–99,
    doi:10.2307/2311331) and the solution (72 (1965), no. 6, p. 675, doi:10.2307/2313885; the Crossref record
    lists Barry, Hayman and Borwein). They were not accessible to us. That the problem cited by Borwein
    carries the number 5167 is taken from these records.
  - J. M. Anderson, K. F. Barth and D. A. Brannan, "Research problems in complex analysis", Bull. London Math.
    Soc. 9 (1977) 129–162, doi:10.1112/blms/9.2.129, in which the problem was proposed according to the tables
    of the collection. It was not accessible to us. So the original wording of the problem, and what this list
    says about the statement for α > 1/2, are not known to us.
  - K. F. Barth, D. A. Brannan and W. K. Hayman, "Research problems in complex analysis", Bull. London Math. Soc.
    16 (1984) 490–517. It was not accessible to us; the other later lists were not consulted. The table of
    comments of the collection has no entry for Problem 7.31. The Semantic Scholar index lists one work citing
    Borwein's note; its record carries the identifier of the list of 1984 and the authors of the list of 1977.
    Crossref and OpenAlex list no citing work.
  - The chapter of the Springer edition (2019) of the collection with Update 7.31. It was not accessible to us.
- **Caveats.** The origin of the statement for α > 1/2 is not identified. The literature on series of this kind
  from the years 1965 to 1990 is poorly covered by the indexes used. A search that finds nothing is not a proof
  of novelty. No priority is claimed; the arguments are elementary and may be known to specialists.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/