{"id":"126f4433-f4da-4283-afc1-500d130c7142","arxiv_id":"1909.02943","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper gives closed-form evaluations of families of Euler-Apéry-type series in terms of alternating multiple zeta values, with a Maple package for automation, but Theorem 2.4 contains a sign and factor error that contradicts the paper's own examples.","lead":"This mathematics paper derives explicit formulas that rewrite certain infinite series containing binomial coefficients and harmonic numbers as alternating multiple zeta values. A reader interested in exact evaluations of Euler-sum-like constants would use these formulas, but one of the central stated formulas is internally inconsistent.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.4's Eq (2.17) is false: for m=1 it gives -4ζ(2), while the same series equals 2ζ(2) by Corollary 2.5/Example 2.3, so a central stated formula is wrong.","rationale":"The reader's weakest assumption correctly identifies Eq (2.17) as the location of the failure. My independent check confirms that the displayed formula is not merely missing a displayed cancellation but is numerically and analytically false, with the root cause a factor of (m!)^2 and a missing (1-2^{-m}) in the proof's reduction of the integral from Eq (2.16). This invalidates a central theorem of Section 2 and, with it, the paper's stated claim to have established explicit formulas for the family S⋆_{m,p}. The remaining formulas may be salvageable, but the preprint as written cannot be accepted. Since the reader already recommends REJECT, my verdict is unchanged.","tokens_in":22959,"tokens_out":16760,"duration_ms":144915,"concrete_test":"Set m=1 in Theorem 2.4 and compare with Corollary 2.5/Example 2.3. Independently evaluate S⋆_{1,1} = ∑_{n=1}∞ H_n (2n n)/(4^n n) using Eq (2.16): with (1/√(1-t)-1)/t = ∑_{n≥1} (2n n)/4^n t^{n-1}, one obtains S⋆_{1,1} = -∫_0^1 (1/√(1-t)-1) ln(1-t)/t dt = 2ζ(2), whereas Eq (2.17) states -4ζ(2). This single instance, or a direct partial sum of the first 10^4 terms, settles the failure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.4, Theorem 2.4 asserts S⋆_{m,1} = -2^{m+1}ζ(m+1). This is false as stated. For m=1, the left side is ∑_{n≥1} H_n (2n n)/(4^n n), which has positive terms and, by the paper's own Corollary 2.5 and Example 2.3, equals 2ζ(2), not -4ζ(2). For m=0, the right side is the divergent -2ζ(1), while Theorem 2.2 gives S_1 = 2 ln 2. The proof of (2.17) contains a concrete algebraic error: Eq (2.16) gives ∫_0^1 t^{n-1} ln^m(1-t) dt = (-1)^m m! ζ⋆_n({1}^m)/n, so after inserting the generating function (1/√(1-t)-1)/t = ∑_{n≥1} (2n n)/4^n t^{n-1}, the sum equals (-1)^m/m! times the integral, not (-1)^m m! times it. A correct reduction of the log-integral yields 2^{m+1}(1-2^{-m})ζ(m+1), not -2^{m+1}ζ(m+1). Because (2.17) is a central formula for the S⋆_{m,p} family, the paper's central claim, as stated, is not established; the related formula (2.18) uses the same proof technique and also needs independent verification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops explicit evaluations for Euler-Apéry-type series involving central binomial coefficients and (generalized) harmonic numbers. Section 2 uses iterated integrals and alternating multiple zeta values (MZVs), while Section 3 uses contour integration with gamma, polygamma, and trigonometric kernels. The authors claim that the families in (4.1) and (4.2) reduce to (alternating) MZVs, with S_p reducible to ln(2) and zeta values, and they provide a Maple package and many worked examples. The paper's central message is seriously undermined by a false formula in Theorem 2.4, Eq. (2.17), which is inconsistent with the paper's own Corollary 2.5 and Example 2.3.","tokens_in":23344,"tokens_out":13135,"duration_ms":114760,"significance":"If the remaining formulas are correct, the paper would be a useful reference collection of explicit evaluations and would support a uniform MZV-reducibility statement for these Euler-Apéry-type series. The strengths are the large number of examples, the downloadable Maple package EASum, and the systematic organization of contour-integral and iterated-integral methods. No free parameters are fitted, and the identities are checkable numerically. The novelty is incremental relative to existing algorithmic packages such as HarmonicSums, and the main obstacle to acceptance is the correctness of Theorem 2.4.","major_comments":[{"comment":"Equation (2.17) is false as stated. For m=1, the left-hand side is S*_{1,1} = sum_{n>=1} H_n (2n choose n)/4^n/n, which is positive; by the paper's own Corollary 2.5 and Example 2.3 this sum equals 2 zeta(2), whereas the right-hand side is -4 zeta(2). For m=0, the right-hand side is the divergent quantity -2 zeta(1), while the series equals S_1 = 2 ln(2) by Theorem 2.2. The error is visible in the proof: Eq. (2.16) gives the integral as (-1)^m m! S*_{m,1}, so the prefactor in the displayed proof should be inverted, and the known evaluation integral_0^1 ln^m(t)/(1+t) dt = (-1)^m m! (1-2^{-m}) zeta(m+1) introduces an additional factor (1-2^{-m}). A corrected derivation yields S*_{m,1} = 2(2^m-1) zeta(m+1) for m>=1. Since S*_{m,1} is one of the target families in (1.3), this repair is necessary before the theorem can be used.","section":"Section 2.4, Theorem 2.4, proof of Eq. (2.18)"},{"comment":"The proof of Eq. (2.18) is only sketched as similar to the proof of Eq. (2.10), and it relies on the same sign-string telescoping whose neighboring derivation in Eq. (2.17) contains the inversion error described above. The current manuscript therefore does not provide enough detail to certify Eq. (2.18) as it stands. The authors should display the intermediate cancellations explicitly and verify the formula numerically for small cases, for example m=1, p=0 against Example 2.3, before the theorem is accepted.","section":"Section 2.4, Theorem 2.4, proof of Eq. (2.18)"},{"comment":"The statement of Theorem 2.4 for all m,p>=0 is not compatible with Eq. (2.17): for m=0 the right-hand side is -2 zeta(1), which is divergent, whereas the left-hand series is S_1 = 2 ln(2). The range of parameters for Eq. (2.17) must be corrected to m>=1, or the m=0 case must be stated separately.","section":"Section 2.4, Theorem 2.4, m=0 case"}],"minor_comments":[{"comment":"The theorem statement should also mention that zeta(m+1) is only meaningful in the usual sense when m>=1, and that the m=0 case requires a separate limiting or explicit evaluation.","section":"Section 2.4, Theorem 2.4, Eq. (2.17)"},{"comment":"In Eq. (2.5), the range of the index j in the sum over sigma_j in {+1,-1} is not made explicit in the first displayed formula; the convention is clear from the following display but should be stated once.","section":"Section 2.1, Eq. (2.5)"},{"comment":"The representative formula for the series involving zeta*_n({1}^m) uses the symbols tilde and hat before their definitions are introduced in Section 2.1; the authors should define these notations earlier or move the display.","section":"Section 1, Introduction, unnumbered display after Eq. (1.3)"}],"recommendation":"major_revision","confidential_remarks":"The inconsistency at Eq. (2.17) is elementary and is contradicted by the paper's own Example 2.3. I would ask the authors not only to correct that formula and its proof, but also to certify the remaining Theorem 2.4 identities by independent numerical checks, since the neighboring telescoping steps are not fully displayed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe thing you should know about arXiv:1909.02943 is that it is a serious, technically dense paper with one false central formula. The main contribution is a set of explicit evaluations of Euler-Apéry-type series—central binomial coefficients times harmonic numbers—in terms of alternating MZVs, plus a Maple package. The contour-integral part (Section 3) gives formulas for single sums S_p and tilde S_q, linear sums tilde S_{1,q}, etc., in terms of ln(2), zeta values, and alternating MZVs. That part is largely independent of the Section 2 machinery and appears sound; I checked S_1, tilde S_2, and the tilde S_3 evaluations against known values, and they match.\n\nThe problem is Theorem 2.4's Eq (2.17), which states S*_{m,1} = -2^{m+1}ζ(m+1). This is false. Using the paper's own integral identity (2.16) with the generating function 1/√(1-t), a correct reduction gives S*_{m,1} = 2^{m+1}(1 - 2^{-m})ζ(m+1). For m=1, the left side is the positive sum ∑ H_n (2n choose n)/(4^n n), which the paper's own Example 2.3 evaluates as 2ζ(2); the printed formula gives -4ζ(2). This is not a matter of convention—it is a factor and sign error in the proof. Eq (2.18) is described as 'similar' and needs independent verification; given (2.17) is wrong, I would not trust the prefactor in (2.18) without careful checking. This weakens the central Section 2 family S*_{m,p} and the corollaries that rely on it.\n\nWhat is genuinely good: the paper is honest about overlap—Section 3 explicitly notes Chen's formula for tilde S_q and cites prior special cases. The Maple package EASum is a real deliverable. The contour-integral method is clean, and the examples are reproducible. The paper also makes a fair conjecture about all series in (1.1) being reducible to alternating MZVs.\n\nMy recommendation: this deserves a serious referee, but not acceptance as is. The authors should fix (2.17) and re-derive (2.18), then the paper is probably solid. If you work on Apéry-type series, keep an eye on the corrected version, but don't cite the current one for S*_{m,p}.","headline":"A serious, technically dense evaluation paper with a concrete false central formula in Theorem 2.4 (Eq 2.17); Section 3 is largely independent and looks sound, but the current version should not be used without correction.","tokens_in":23930,"tokens_out":16393,"would_cite":false,"duration_ms":118841,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65B10","11B65","11M32"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that every series in the families (4.1) and (4.2) has an explicit closed form in alternating multiple zeta values, with the single sums reducing to $\\ln 2$ and zeta values.","keywords":["alternating multiple zeta values","central binomial coefficients","generalized harmonic numbers","iterated integrals","contour integration","Riemann zeta values","log-sine integrals","inverse binomial sums"],"falsifier":"Check Eq. (2.17) at $m=1$: the series $S^\\star_{1,1} = \\sum_{n\\ge1} H_n \\binom{2n}{n}/(4^n n)$ has positive terms, while the displayed right-hand side is $-4\\zeta(2)$; resolving this discrepancy against the value $2\\zeta(2)$ quoted in Example 2.3 settles whether the hidden sign-string telescoping step is being carried out correctly.","tokens_in":22742,"feed_emoji":"🧮","tokens_out":11396,"duration_ms":107105,"temperature":0.7,"pith_summary":"The paper's project is to show that a natural family of infinite sums--terms with a central binomial coefficient divided by $4^n$, multiplied by generalized harmonic numbers--has closed forms rather than merely numerical values. The central claim is that every sum in the families (4.1) and (4.2) is an explicit finite rational combination of alternating multiple zeta values; within that, the single sums $S_p$ land in the smaller algebra $\\mathbb{Q}[\\ln 2,\\zeta(2),\\zeta(3),\\ldots]$. If this is right, dozens of known and conjectural evaluations collapse into one bookkeeping rule, and the question of which constants appear is settled by the parity and sign structure encoded in the zeta arguments. The paper also conjectures that the wider family (1.1), with arbitrary products of harmonic numbers, admits the same reduction.","feed_headline":"Central-binomial sums reduce to alternating zeta values","feed_subtitle":"A uniform method evaluates dozens of these infinite series exactly, through weight 6.","key_machinery":"The central object is the alternating multiple zeta value with barred and hatted entries: a multiple sum whose numerators carry powers of $(-1)^n$ and parity filters $1\\pm(-1)^n$. Section 2 works through iterated integrals: the generating function $\\sum_{n\\ge0} \\binom{2n}{n}4^{-n}t^n=(1-t)^{-1/2}$ and the harmonic-number analogue (2.19) convert each series into a simplex integral, while identity (2.5) expands parity-restricted multiple harmonic sums into $2^m$ sign choices, which become the barred arguments of the final zeta values. Section 3 uses contour integrals whose kernels are $\\psi(-z)+\\gamma$, $\\pi^2\\cot^2(\\pi z)$, and related functions; the gamma-function expansion of Lemma 3.3 in polynomial coefficients turns residues into weighted products of zeta values, yielding the $\\ln 2$-and-zeta reductions.","core_discovery":"On its own terms the discovery is a pair of reduction theorems. Theorem 2.2 states $S_{p+1} = -2\\,\\zeta(\\bar 1,\\{\\hat 1\\}_p)$. Theorem 2.3 states $$S_{m+1,p+1} = 4\\,\\zeta(\\bar 1,\\{\\hat 1\\}_p,\\hat 2,\\{\\hat 1\\}_{m-1}) - 2\\,\\zeta(m+1)\\zeta(\\bar 1,\\{\\hat 1\\}_p),$$ where the barred and hatted arguments record sign and parity choices inside the alternating multiple zeta value. The contour-integral half of the paper proves the companion statements for the inverse binomial sums $\\tilde S_q,\\tilde S_{1,q},\\tilde T_{1,q},\\tilde U_{1,q}$, and in particular that each single sum $S_q$ is a polynomial in $\\ln 2$ and Riemann zeta values. The final section packages these identities with an evaluation program, listing all series in (4.1)--(4.2) through weight 6.","pith_inferences":["The paper leaves implicit that its iterated-integral construction does not need the index string $\\{1\\}^m$; the same mechanism should extend to arbitrary harmonic index strings, giving the closing conjecture a proof path.","The contour-residue route computes residues through the polynomial coefficients of Lemma 3.2, so a purely algebraic reduction algorithm for inverse binomial sums at arbitrary weight should be possible.","A high-precision numerical check of one weight-7 series of type (1.1) with two distinct harmonic indices would test the closing conjecture before a proof is attempted."],"forward_implications":["$S_{p+1}=-2\\zeta(\\bar 1,\\{\\hat 1\\}_p)$ turns each single sum into one alternating MZV of weight $p+1$, so all low-weight values are immediately available.","The single sums $S_p$ are reduced constructively to a finite sum of products of $\\ln 2$ and Riemann zeta values, not merely shown to lie in that algebra.","The inverse binomial sums $\\tilde S_q,\\tilde S_{1,q},\\tilde T_{1,q},\\tilde U_{1,q}$ inherit the same closed-form status, yielding identities such as $\\tilde S_2=3\\zeta(2)$.","All series in (4.1)--(4.2) through weight 6 are evaluated in the companion tables, making the reduction usable without re-deriving integrals."],"supporting_citations":[{"why":"supplies the harmonic-number generating function (2.19) and the integral reductions that Section 2.5 builds on.","marker":"[16]"},{"why":"gives the logarithmic generating function (2.8) for the central-binomial single sums used in Theorems 2.2--2.4.","marker":"[31]"},{"why":"provides the residue lemma and kernel expansions for $\\psi(-z)+\\gamma$, cotangent, and polygamma functions used throughout Section 3.","marker":"[23]"},{"why":"furnishes the precomputed alternating-MZV relations that let the companion program convert the formulas into evaluated constants.","marker":"[10]"},{"why":"supplies the comparison formula $\\tilde S_q = 2^q\\,t(2,\\{1\\}_{q-2})$, tying the contour results to earlier reductions.","marker":"[17]"},{"why":"furnishes the multiple-harmonic-sum identities (2.3)--(2.5), including the parity-to-sign expansion that creates alternating arguments.","marker":"[47]"},{"why":"gives the generating function (2.3) used to expand the iterated integrals.","marker":"[30]"},{"why":"provides the gamma-function asymptotic expansion that justifies the vanishing of the large-circle contour integrals in Theorem 3.4.","marker":"[33]"}],"fun_headline_variants":["Exact formulas for Euler-Apery sums via alternating zeta values","Central binomial sums reduce to alternating multiple zeta values","Uniform reduction of binomial series to zeta and ln2","Dozens of Euler-Apery series evaluated exactly via reductions","Inverse binomial sums become polynomials in zeta and ln2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is an unstated cancellation among sums over sign choices; if that cancellation is not valid, the Section 2 formulas do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Exact formulas for Euler-Apery sums via alternating zeta values","Central binomial sums reduce to alternating multiple zeta values","Uniform reduction of binomial series to zeta and ln2","Dozens of Euler-Apery series evaluated exactly via reductions","Inverse binomial sums become polynomials in zeta and ln2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000214,"raw_usage":{"total_tokens":1381,"prompt_tokens":857,"completion_tokens":524,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":438}},"tokens_in":473,"tokens_out":524,"duration_ms":5544,"temperature":1.0,"reasoning_tokens":438,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:22:14.289686+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check Eq. (2.17) at $m=1$: the series $S^\\star_{1,1} = \\sum_{n\\ge1} H_n \\binom{2n}{n}/(4^n n)$ has positive terms, while the displayed right-hand side is $-4\\zeta(2)$; resolving this discrepancy against the value $2\\zeta(2)$ quoted in Example 2.3 settles whether the hidden sign-string telescoping step is being carried out correctly.","supporting_citations":[{"cited_title":"Chen, Interesting series associated with central bi nomial coefficients, Catalan numbers and harmonic numbers, J","cited_arxiv_id":null,"evidence_quote":"supplies the harmonic-number generating function (2.19) and the integral reductions that Section 2.5 builds on."},{"cited_title":"Lehmer, Interesting series involving the central binomial coefficient, Amer","cited_arxiv_id":null,"evidence_quote":"gives the logarithmic generating function (2.8) for the central-binomial single sums used in Theorems 2.2--2.4."},{"cited_title":"Flajolet, B","cited_arxiv_id":null,"evidence_quote":"provides the residue lemma and kernel expansions for $\\psi(-z)+\\gamma$, cotangent, and polygamma functions used throughout Section 3."},{"cited_title":"Blümlein, D.J","cited_arxiv_id":null,"evidence_quote":"furnishes the precomputed alternating-MZV relations that let the companion program convert the formulas into evaluated constants."},{"cited_title":"Chen, Generalized Arakawa-Kaneko zeta function s, Integral Transforms Spec","cited_arxiv_id":null,"evidence_quote":"supplies the comparison formula $\\tilde S_q = 2^q\\,t(2,\\{1\\}_{q-2})$, tying the contour results to earlier reductions."},{"cited_title":"Xu, Multiple zeta values and Euler sums, J","cited_arxiv_id":null,"evidence_quote":"furnishes the multiple-harmonic-sum identities (2.3)--(2.5), including the parity-to-sign expansion that creates alternating arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the generating function (2.3) used to expand the iterated integrals."},{"cited_title":"Luke, The Special Functions and Their Approximati ons, Vol","cited_arxiv_id":null,"evidence_quote":"provides the gamma-function asymptotic expansion that justifies the vanishing of the large-circle contour integrals in Theorem 3.4."}],"review_version":1}