{
  "schema_version": 1,
  "problem_number": "AMR-022-7031",
  "title": "On Hayman's Problem on the Series Σ f(c_n/a_n) Formed with Second Partial Sums: A Comparison Theorem and a Characterization",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Problem 7.31 of Hayman and Lingham's collection Research Problems in Function Theory, proposed by W. K. Hayman, concerns sequences with a_1 > 0 and 0 ≤ a_n ≤ n and their partial sums b_n = a_1 + … + a_n, c_n = b_1 + … + b_n. It states that Σ (a_n/c_n)^α < ∞ for α > 1/2, and asks for which functions f the series Σ f(c_n/a_n) converges, and whether, under a regularity condition on f, it converges whenever Σ f(n²) does. We prove: if √x·f(x) is non-negative and non-increasing and Σ f(n²) < ∞, then Σ f(c_n/a_n) ≤ 6 Σ f(n²) + (2 log(1/a_1) + 2 + 5 log 2) f(1) for every such sequence, and if Σ f(n²) = ∞, the series diverges for a_n = n. For an arbitrary real function f, the series converges for all such sequences if and only if ∫^∞ sup_{t ≥ u} √t·|f(t)| du/u < ∞; the answer is the same under the normalizations a_n ≤ 1 and b_n ≤ n. The proofs rest on the bound Σ_{c_n ≤ X a_n} (a_n/c_n)^{1/2} < 3 log X + 2 log(1/a_1) + 2 + 2 log 2 and on explicit sequences along which c_n/a_n is piecewise constant. Without a condition the comparison fails: there are a non-increasing C^∞ function, and a non-negative C^∞ function vanishing at all squares, with Σ f(n²) < ∞ and Σ f(c_n/a_n) = ∞; the exponent 1/2 cannot be lowered, and a_n ≤ n cannot be relaxed to a_n ≤ n·ψ(n) with ψ unbounded. The problem quotes as known the case f(x) = x^{-α} and an analogue for c_n/b_n attributed to D. Borwein. Borwein's note of 1965, which we read, concerns positive non-decreasing sequences b_n ≤ n·k_n with k_n ≥ 1: if x·f(x) is positive and non-increasing and ∫^∞ f < ∞, then Σ k_n f(k_n c_n/b_n) < ∞, hence Σ f(c_n/b_n) < ∞ if b_n ≤ K·n. We extend the latter statement to 0 ≤ a_n ≤ K·n. We did not read Monthly problem 5167 of Barry and Hayman (1964–65), the problem list of Anderson, Barth and Brannan (1977), in which the problem was proposed according to the tables of the collection, the later lists, or Update 7.31 of the 2019 edition of the collection (those we tried were not accessible to us), and the origin of the statement for α > 1/2 is not identified. Our literature search found no earlier answer; a search that finds nothing is not a proof of novelty, no priority is claimed, and the arguments are elementary and may be known to specialists. The computations reported are tests and are not used in the proofs. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CA"
  ],
  "keywords": [
    "series of positive terms",
    "second partial sums",
    "iterated partial sums",
    "convergence criteria",
    "comparison of series",
    "characterization",
    "counterexamples",
    "Hayman",
    "Borwein",
    "Research Problems in Function Theory",
    "Hayman-Lingham Problem 7.31",
    "UnsolvedMath",
    "AMR-022-7031",
    "math.CA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-022-7031/",
  "pdf_url": "https://eulersolve.org/papers/amr-022-7031/paper.pdf?v=225bc104f64c",
  "doi": "10.5281/zenodo.23292270",
  "zenodo_record_url": "https://zenodo.org/records/23292270",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers both questions of Problem 7.31 (W. K. Hayman) of Hayman and Lingham, Research Problems in Function Theory, for the class of all admissible sequences. Not read, because they were not accessible: Monthly problem 5167 of Barry and Hayman, the problem list of Anderson, Barth and Brannan (1977) in which the problem was proposed, the later lists, and Update 7.31 of the 2019 edition; the origin of the statement for α > 1/2 is not identified. The arguments are elementary and may be known to specialists; no priority is claimed.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}