{"id":"d66260f3-ec9f-4cfd-9822-8806f7872262","arxiv_id":"2411.13255","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper derives an asymptotic for S_T(a,δ) = Σ ζ'(ρ_a+iδ) X^{ρ_a} over a-points, generalizing and correcting earlier formulas by Fujii, Garunkštis-Steuding, and Jakhlouti-Mazhouda.","lead":"This paper proves new asymptotic formulas for sums of zeta derivatives at the a-points of the Riemann zeta function, weighted by X raised to the a-point, and corrects an earlier published formula. The results refine the known picture of how the zeta function's values and derivatives distribute near its nontrivial zeros, a topic with connections to prime number theory and random matrix models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.3 is stated for all a∈C, but the defining equation for the coefficients c_a(r) fails at a=1 and the proof only covers a≠1; the central claim is unsupported for a=1.","rationale":"The paper's headline result, Theorem 2.3, is asserted for every fixed a∈C. The proof, however, is carried out only for a≠1, and the exceptional case a=1 is dismissed in a short paragraph with no derivation of the final formula. The internal inconsistency in (2.5) at a=1 is decisive: the defining equation for the coefficients c_a(r) cannot hold when a=1, because the two sides have different limits as σ→∞. This makes the theorem's statement not merely unproven for a=1 but undefined. A referee would require either explicitly excluding a=1 or providing a separate statement with a different coefficient system, such as one based on f(s)=2^s(ζ(s)-1). The reader's flagged concern about Lemma 4.8 is legitimate—the partial fraction expansion is quoted from Garunkštis-Steuding and its proof is not fully reproduced—but the a=1 issue is a demonstrated failure of the statement itself and therefore more load-bearing. The a≠1 portion of the proof appears plausible and the overall conditional verdict remains appropriate; the needed condition is to repair the a=1 case or exclude it from the theorem.","tokens_in":16626,"tokens_out":21766,"duration_ms":215850,"concrete_test":"Set a=1 in (2.5) and let σ→∞ along the real axis: compute lim_{σ→∞} ζ′(σ)/(ζ(σ)−1) = −log 2, while the claimed Dirichlet series Σ_{r≥2} c_1(r) r^{−σ} tends to 0 because c_1(1)=0. This contradiction settles that c_1(r) is not defined by (2.5). Then, to test the a=1 case of Theorem 2.3, repeat the Section 7 computation with f(s)=2^s(ζ(s)−1), derive the analogue of (2.4), and check whether the resulting main terms match the claimed expression; if they differ, Theorem 2.3 must be restricted to a≠1 or restated with a separate formula for a=1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.3 is stated for all a∈C, but its proof and the coefficients it uses are only valid for a≠1. Equation (2.5) defines c_a(r) by ζ′(s)/(ζ(s)-a) = Σ_{r≥2} c_a(r) r^{-s} for ℜ(s)>1, with c_a(1)=0. For a=1 this identity is impossible: as σ→+∞, the left-hand side tends to −log 2 (since ζ(s)-1 = 2^{-s}+3^{-s}+··· and ζ′(s) = −2^{-s}log 2 − 3^{-s}log 3 − ···), whereas the right-hand side, being a Dirichlet series with c_1(1)=0, tends to 0. Thus c_1(r) is not defined by (2.5). Moreover, the proof of Theorem 2.3 explicitly restricts to a∈C\\{1} after only a one-sentence comment for a=1, and the subsequent derivation of S_R, S_L, and the final formula is not carried out for a=1. Consequently, the theorem as stated is either false or ill-posed for a=1; the claimed generality over all complex a is unsupported. This is an internal inconsistency in the statement, not merely a gap in a peripheral lemma.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a-points of the Riemann zeta function, i.e., solutions of ζ(s)=a. The main result (Theorem 2.3) is an asymptotic formula for the weighted sum S_T(a,δ)=Σ_{τ<γ_a≤T} ζ'(ρ_a+iδ) X^{ρ_a}, valid uniformly in α where δ=2πα/log(T/(2πX)). The formula expresses S_T in terms of the analogous sum over ordinary zeros, a correction term involving the arithmetic coefficients c_a(r), and a further term K_δ. The paper also proves a δ-shifted version of Fujii's mean-value formula for ordinary zeros (Theorem 2.1), derives a corollary for integer X (Corollary 2.2), and obtains a corrected version of a result of Jakhlouti and Mazhouda (Corollary 2.4). The proofs use contour integration over rectangles, the partial fraction expansion of ζ'(s)/(ζ(s)-a), and known estimates for ζ(s).","tokens_in":16932,"tokens_out":9220,"duration_ms":88970,"significance":"If correct, Theorem 2.3 is a new weighted, δ-shifted discrete mean-value formula for a-points, and it corrects an earlier formula (1.5). It also reveals a dependence on X through Δ(X) and Δ(X^{-1}), giving a richer structure than previous results. The computations are detailed, the error terms are explicit, and the paper carefully restates known results on the distribution of trivial a-points. However, the a=1 case of the main theorem is not treated correctly, and the proof has a gap concerning the choice of τ; these issues must be resolved before the main claim can be accepted.","major_comments":[{"comment":"The defining equation for c_a(r) fails at a=1. Since ζ'(σ)/(ζ(σ)-1) tends to -log 2 as σ→∞ along the real axis, whereas a Dirichlet series with c_1(1)=0 tends to 0, no coefficients satisfying (2.5) with c_a(1)=0 exist for a=1. The proof of Theorem 2.3 explicitly assumes a≠1 after a one-sentence note on f(s)=2^s(ζ(s)-1), and the subsequent derivation of S_L, S_R, and the final formula is only for a≠1. Consequently, Theorem 2.3 as stated for all a∈C is ill-posed and unsupported; the statement should exclude a=1 or give a correct separate treatment.","section":"§7, Eq. (2.5) and Theorem 2.3"},{"comment":"The text reads \"we may take τ large enough such that there are no a-points on the boundary ∂R\", but τ is fixed at the start of the theorem. The proof as written covers only τ chosen to avoid the (discrete) set of ordinates of a-points; either the theorem must include this condition on τ, or a limiting/indentation argument is needed for arbitrary fixed τ.","section":"§7, after Eq. (7.1)"}],"minor_comments":[{"comment":"In the formula for S_R, the subscript \"Σ_{mr=k}\" should be \"Σ_{mr=X}\"; the summation variable k is not defined in that context.","section":"§7.1, display after definition of a_k"},{"comment":"Theorem 8.1 is stated without proof; either supply the promised \"repeating the proof\" details or clearly mark it as a remark/sketch, since unproved theorems cannot be part of the formal results.","section":"§8, Theorem 8.1"},{"comment":"Lemma 4.8 is quoted from [4, p. 8] and is essential for the S_T estimate in Theorem 2.3; please include at least a sketch of the proof or an explicit verification of the O(log(|t|+1)) bound.","section":"§4, Lemma 4.8"},{"comment":"The notation \"τ /greaterorequalslant|δ| + 1\" is a rendering artifact; please ensure all inequalities are typeset correctly in the final version.","section":"Throughout"},{"comment":"The remark that S_T(a,δ) is \"analytic in δ\" and, as a function of a, \"analytic at a=0 and discontinuous elsewhere\" is not substantiated; please clarify or remove.","section":"§8, Remark 8.1"}],"recommendation":"major_revision","confidential_remarks":"The main result is plausible for a≠1, but the a=1 case as stated is untenable because the coefficient expansion (2.5) does not exist for a=1. The authors should be asked to revise the statement and proof accordingly, and to address the τ-avoidance gap. The paper is otherwise within the scope of the journal and contains substantial computational content."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely useful: it extends Fujii's discrete mean value formula to a δ-shifted version for ordinary zeros (Theorem 2.1), and it gives the first δ-shifted weighted mean-value formula for a-points (Theorem 2.3), correcting a mistake in Jakhlouti-Mazhouda's (1.5). The contour-integral proof is detailed and, for a ≠ 1, the main line of argument is plausible. The authors also carefully reformulate Garunkštis–Steuding's results on trivial a-points; Lemma 4.8 is cited from that work but at least they provide the context. I give credit for the honesty about error terms and the explicit removal of the |T−γ| restriction.\n\nThe soft spots are real, and one is load-bearing. The stress-test note is right: Theorem 2.3 is stated for all a ∈ C, but the coefficients c_a(r) defined by (2.5) do not exist at a = 1. Let σ→∞; the left-hand side of (2.5) tends to −log 2, while a Dirichlet series with c_1(1) = 0 tends to 0. So (2.5) cannot define c_1(r). The proof explicitly assumes a ≠ 1, and the one-sentence workaround with f(s) = 2^s(ζ(s)−1) does not supply a definition for the c_1(r) appearing in the statement of the theorem. Either the theorem should be restricted to a ≠ 1, or the authors need to define c_1(r) through a limiting procedure or a different expansion. As written, the a = 1 case is unsupported.\n\nTwo smaller issues. First, Theorem 8.1 is stated without proof; the text says repeating the proof of Theorem 2.3 would give it, but that is not a proof in the paper. Second, Lemma 4.8 (the partial fraction expansion) is taken from Garunkštis–Steuding and depends on delicate lower bounds for |ζ(s)| in the left half-plane; the authors reformulate the key lemmas but only sketch the proof. If that decomposition fails for some a, the bound on the horizontal integral S_T and hence the main term in Theorem 2.3 could be incomplete. This is a known difficult part and deserves a careful check. The choice of τ avoiding a-points on the boundary is also asserted without full justification, but that is a minor fix.\n\nWho is this for? Analytic number theorists working on discrete mean values of ζ and its a-points. The paper deserves a serious referee, not a desk reject. My recommendation: send it to review, but the referee should insist that the a = 1 case be fixed or excluded, and should verify the Garunkštis–Steuding input. With that, the a ≠ 1 results are likely to stand.","headline":"Useful δ-shifted a-point mean value formula for a≠1, but Theorem 2.3 is ill-posed at a=1 and needs fixing before it is complete.","tokens_in":17475,"tokens_out":2986,"would_cite":false,"duration_ms":32128,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","41A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a shifted, weighted mean-value formula for the a-points of the zeta function, with explicit arithmetic correction terms.","keywords":["Riemann zeta function","a-points","value distribution","discrete mean value theorem","higher derivatives","partial fraction expansion","von Mangoldt function","Möbius function"],"falsifier":"Take a concrete non-integer, non-reciprocal case such as $a=1$, $X=\\sqrt2$, $\\alpha=1$, $\\tau=2$, and compute both sides of (2.4) numerically at $T=10^5$ by locating the $a$-points of $\\zeta(s)=1$ in the rectangle. Since $\\Delta(\\sqrt2)=\\Delta(1/\\sqrt2)=0$, the theorem predicts that the $a$-point sum equals the ordinary zero sum within $O(T^{1/2}\\log^7 T)$; a difference with a term of size comparable to $T$ would disprove the formula or the partial-fraction expansion behind it.","tokens_in":16447,"feed_emoji":"🧮","tokens_out":14502,"duration_ms":130786,"temperature":0.7,"pith_summary":"This paper establishes an asymptotic formula for sums of the derivative of the zeta function over its $a$-points, the points $\\rho_a$ where $\\zeta(\\rho_a)=a$, shifted by a small imaginary amount $\\delta$ and weighted by $X^{\\rho_a}$. The central result, Theorem 2.3, expresses such a sum as the corresponding sum over ordinary zeros minus an explicit arithmetic correction built from the divisor sum $\\sum_{mr=X}(\\Lambda(r)+c_a(r))m^{-i\\delta}\\log m$, minus an $a$-dependent $K$-term, plus an error of size $O(T^{1/2}\\log^7 T)$, uniformly in the shift parameter $\\alpha$. This corrects a previously published formula, which omitted the $c_a(r)$-type and $a$-dependent contributions. The result matters because it supplies a $\\delta$-shifted, $X$-weighted mean-value theorem for $a$-points, and differentiating the formula in $\\alpha$ yields explicit expansions for sums of higher derivatives of $\\zeta$ at $a$-points, extending known results for ordinary zeros.","feed_headline":"Zeta a-point sums equal zero sums plus corrections","feed_subtitle":"A δ-shifted, X-weighted average over points where ζ(s)=a reduces to ordinary zero sums plus explicit arithmetic terms.","key_machinery":"The load-bearing tool is the partial-fraction expansion $$\\frac{\\zeta'(s)}{\\zeta(s)-a}=\\sum_{|t-\\gamma_a|\\le1}\\frac{1}{s-\\rho_a}+O(\\log(|t|+1)),\\qquad -1\\le\\$\\sigma$\\le2,$$ which represents the ratio as a sum of simple pole terms over nearby $a$-points. This expansion, quoted from earlier work and here buttressed by a reformulated description of the trivial $a$-points via a zero-counting argument, allows the proof to turn the $a$-point sum into a contour integral of $\\frac{\\zeta'(s)}{\\zeta(s)-a}\\zeta'(s+i\\delta)X^s$ around a rectangle. The right vertical side is evaluated by expanding the ratio into the Dirichlet series $\\sum_{r\\ge2}c_a(r)r^{-s}$; the left vertical side uses the expansion of $1/(\\zeta(s)-a)$ in powers of $a/\\zeta(s)$; and the horizontal sides are controlled by the partial-fraction bound. The coefficients $c_a(r)$ and the auxiliary function $K_\\delta^{(1)}$ carry the arithmetic content of the correction terms.","core_discovery":"On the paper's own terms, the discovery is Theorem 2.3: for fixed $a\\in\\mathbb{C}$, $X>0$, and $\\tau\\ge|\\delta|+1$, with $0\\ne\\delta=2\\pi\\alpha/\\log(T/(2\\pi X))\\ll 1$, as $T\\to\\infty$, $$\\sum_{\\tau<\\gamma_a\\le T}\\zeta'(\\rho_a+i\\delta)$X^{{\\rho_a}}$ = \\sum_{\\tau<\\gamma\\le T}\\zeta'(\\rho+i\\delta)$X^{{\\rho}}$ - \\$\\Delta$(X)\\frac{T}{2\\pi}\\sum_{mr=X}(\\Lambda(r)+c_a(r))$m^{{-i\\delta}}$\\log m - aK_\\$delta^{{(1)}}$\\left(\\frac{T}{2\\pi}\\right) + O\\left($T^{{1/2}}$\\$log^{7}$ T\\right),$$ uniformly in $\\alpha$. Here $\\Lambda$ is the von Mangoldt function, the coefficients $c_a(r)$ are defined by the Dirichlet series $\\zeta'(s)/(\\zeta(s)-a)=\\sum_{r\\ge2}c_a(r)r^{-s}$, and $K_\\delta^{(1)}$ is an explicit combination of Möbius- and von Mangoldt-weighted sums over divisors of $1/X$. In words: the average of $\\zeta'$ at shifted $a$-points differs from the same average at ordinary zeros only through arithmetic terms supported on the divisor pair $mr=X$, plus a term that vanishes unless $1/X$ is an integer. The paper also derives the integer-$X$ specialization, the $\\delta\\to0$ limit, and the $a=0$ case, recovering and correcting earlier results.","pith_inferences":["A clean numerical check of the $X=\\sqrt2$, $a=1$ case would isolate the theorem's strongest new prediction: because both $\\Delta$-terms vanish, the $a$-point and zero sums should agree to leading order, testing the partial-fraction expansion without needing the $c_a(r)$ coefficients.","Repeating the contour proof with $\\zeta^{(n+1)}$ in the integrand should give explicit formulas for $\\sum \\zeta^{(n)}(\\rho_a+i\\delta)X^{\\rho_a}$ for all $n$, with the same arithmetic corrections expressed through higher von Mangoldt functions.","Letting $a$ tend to $0$ with $T$ should produce a transition formula interpolating between the $a$-point and zero sums; the shape of that interpolation may reveal how trivial $a$-points relate to the usual trivial zeros."],"forward_implications":["Differentiating (2.4) with respect to $\\alpha$ yields explicit asymptotic expansions for sums of higher derivatives $\\zeta^{(n)}(\\rho_a)$ at $a$-points, as stated in Theorem 8.1; setting $a=0$ recovers known higher-derivative mean-value formulas for zeros.","For positive integer $X$, Corollary 2.4 corrects the published evaluation of $\\sum_{1<\\gamma_a\\le T}\\zeta'(\\rho_a)X^{\\rho_a}$, adding the previously missing $c_a(r)$ and $a$-dependent terms.","The uniformity in $\\alpha$ means the formula holds simultaneously for every allowed shift $\\delta$, so it can be differentiated or integrated in $\\alpha$ to generate families of related mean values.","The factors $\\Delta(X)$ and $\\Delta(X^{-1})$ show that the asymptotic takes different shapes according to whether $X$, $1/X$, or neither is an integer; for a generic non-integer $X$, the $a$-point sum equals the zero sum up to the stated error."],"supporting_citations":[{"why":"supplies the Riemann–von Mangoldt-type count $N_a(T)$ for $a$-points, used throughout to bound the number of terms and to remove ordinate restrictions.","marker":"[1]"},{"why":"evaluates the ordinary zero sum $\\sum\\zeta'(\\rho)X^\\rho$, the comparison target and the base case from which the shifted formula is built.","marker":"[3]"},{"why":"gives the trivial $a$-point structure and the partial-fraction expansion of $\\zeta'(s)/(\\zeta(s)-a)$ that anchors the contour proof.","marker":"[4]"},{"why":"provides the higher-derivative zero-sum expansion and the auxiliary estimates (Lemmas 4.1–4.3) used in the contour integrals and in Theorem 8.1.","marker":"[7]"},{"why":"states the earlier $a$-point sum formula that the paper identifies as containing a minor mistake and corrects.","marker":"[10]"},{"why":"supplies the $\\chi(1-s)$ exponential-integral lemma and the $\\xi'/\\xi$ expansion used to evaluate the vertical and horizontal integrals.","marker":"[14]"},{"why":"defines the coefficients $c_a(r)$ by the Dirichlet series expansion of $\\zeta'(s)/(\\zeta(s)-a)$, which appear in the main correction term.","marker":"[15]"},{"why":"provides the lower bounds for $|\\zeta(s)|$ in the left half-plane used to justify the trivial $a$-point description.","marker":"[16]"}],"fun_headline_variants":["a-Point zeta sums match zero sums plus explicit corrections","Zeta a-point average equals zero sum plus divisor corrections","a-Point zeta average: zero sum plus von Mangoldt arithmetic terms","Shifted zeta at a-points differs from zeros by arithmetic term","a-Point zeta sums: zero sum plus explicit arithmetic factors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on a formula that rewrites the logarithmic-derivative-like ratio of the zeta function near each $a$-point as a sum of simple pole terms plus a controlled error; if that formula fails for some $a$, the main result collapses.","fun_headline_variants_meta":{"raw":{"variants":["a-Point zeta sums match zero sums plus explicit corrections","Zeta a-point average equals zero sum plus divisor corrections","a-Point zeta average: zero sum plus von Mangoldt arithmetic terms","Shifted zeta at a-points differs from zeros by arithmetic term","a-Point zeta sums: zero sum plus explicit arithmetic factors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000216,"raw_usage":{"total_tokens":1494,"prompt_tokens":1069,"completion_tokens":425,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":685,"completion_tokens_details":{"reasoning_tokens":334}},"tokens_in":685,"tokens_out":425,"duration_ms":4377,"temperature":1.0,"reasoning_tokens":334,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:38:38.740770+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete non-integer, non-reciprocal case such as $a=1$, $X=\\sqrt2$, $\\alpha=1$, $\\tau=2$, and compute both sides of (2.4) numerically at $T=10^5$ by locating the $a$-points of $\\zeta(s)=1$ in the rectangle. Since $\\Delta(\\sqrt2)=\\Delta(1/\\sqrt2)=0$, the theorem predicts that the $a$-point sum equals the ordinary zero sum within $O(T^{1/2}\\log^7 T)$; a difference with a term of size comparable to $T$ would disprove the formula or the partial-fraction expansion behind it.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the Riemann–von Mangoldt-type count $N_a(T)$ for $a$-points, used throughout to bound the number of terms and to remove ordinate restrictions."},{"cited_title":"Fujii, On the distribution of values of the derivative of the Riemann zeta function at its zeros, I, Proc","cited_arxiv_id":null,"evidence_quote":"evaluates the ordinary zero sum $\\sum\\zeta'(\\rho)X^\\rho$, the comparison target and the base case from which the shifted formula is built."},{"cited_title":"Garunkˇ stis and J","cited_arxiv_id":null,"evidence_quote":"gives the trivial $a$-point structure and the partial-fraction expansion of $\\zeta'(s)/(\\zeta(s)-a)$ that anchors the contour proof."},{"cited_title":"Hughes and A","cited_arxiv_id":null,"evidence_quote":"provides the higher-derivative zero-sum expansion and the auxiliary estimates (Lemmas 4.1–4.3) used in the contour integrals and in Theorem 8.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"states the earlier $a$-point sum formula that the paper identifies as containing a minor mistake and corrects."},{"cited_title":"Pearce-Crump, A further generalization of sums of hi gher derivatives of the Riemann zeta function, Int","cited_arxiv_id":null,"evidence_quote":"supplies the $\\chi(1-s)$ exponential-integral lemma and the $\\xi'/\\xi$ expansion used to evaluate the vertical and horizontal integrals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the coefficients $c_a(r)$ by the Dirichlet series expansion of $\\zeta'(s)/(\\zeta(s)-a)$, which appear in the main correction term."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the lower bounds for $|\\zeta(s)|$ in the left half-plane used to justify the trivial $a$-point description."}],"review_version":1}