{"id":"1fbceae0-b414-4cd4-820a-40b4a9146af2","arxiv_id":"2411.17946","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The higher Taylor coefficients of the logarithmic derivative of a Dedekind zeta function at s=1 satisfy new arithmetic formulas, GRH-conditional bounds, and an Ihara-type limit.","lead":"This paper defines and studies the higher Euler-Kronecker constants, the higher-order coefficients in the Laurent expansion of the logarithmic derivative of a number field's Dedekind zeta function at s=1. It proves arithmetic formulas for these constants, conditional bounds under the generalized Riemann hypothesis, and an unconditional generalization of Ihara's limit formula.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4's proof relies on Proposition 2.2's hypothesis E(x)=O(x^b), b<1, but for zeta_K only O(x exp(-c sqrt(log x))) is known unconditionally; the arithmetic formula is not proven as stated.","rationale":"The reader's weakest_assumption focused on GRH for Theorem 1.5 and on Siegel-zero delicacy in Lemma 5.5. The most load-bearing concern, in my judgment, is different: the proof of the unconditional arithmetic formula Theorem 1.4 is not complete as written because Proposition 2.2's hypothesis E(x)=O(x^b), b<1, is not satisfied by psi_K(x)-x under any known unconditional zero-free region. This does not mean the formula is false; it is likely true and repairable by a more general Tauberian argument using the standard error O(x exp(-c sqrt(log x))). But the paper does not provide that argument, so the central identity (4) is not established by the given proof. Since the verdict CONDITIONAL already requires corrections before the paper is fully trusted, and my concern is another such correction rather than a reason to reject outright, the recommended verdict is unchanged. I disagree with the reader's identification of the weakest assumption: the unconditional proof gap in Theorem 1.4 is more fundamental than the GRH-conditional bounds or the Siegel-zero subtlety in Lemma 5.5, because Theorem 1.4 is stated and used without conditions. The sign inconsistency in Proposition 2.2's statement versus its proof is secondary, as the limit (13) is unaffected, but it should be fixed for correctness of the displayed identity (12).","tokens_in":16652,"tokens_out":22391,"duration_ms":180786,"concrete_test":"Independently re-derive Theorem 1.4 directly from (12) using the unconditional bound psi_K(x)-x = O(x exp(-c sqrt(log x))). Concretely, verify that, for each fixed r>=0, E(x) log^r x / x -> 0 and the tail integral I_r(x)=int_x^infty (log^r t + r log^{r-1}t) E(t)/t^2 dt -> 0 as x->infty. If both vanish, then the limit (13) follows from the weaker condition and the gap is fixable; if either fails, the arithmetic formula (4) lacks support as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.2 (Section 2) is the engine for Theorem 1.4. Its hypothesis is E(x):=B(x)-Cx=O(x^b) for some 0<=b<1. For zeta_K, B(x)=sum_{n<=x} Lambda_K(n)=psi_K(x), C=1, so E(x)=psi_K(x)-x. The best unconditional prime ideal theorem gives psi_K(x)-x = O(x exp(-c sqrt(log x))), which is not O(x^b) for any b<1 because (1-b) log x eventually exceeds c sqrt(log x). Thus the proof of Theorem 1.4 invokes a proposition whose hypothesis is not known to hold; the displayed limit (4) is not justified by the cited argument. The gap is load-bearing because (4) is one of the two central identities from which all later bounds and the Ihara-type limit are claimed. The same issue recurs implicitly in Corollary 3.1. A likely repair is to weaken the hypothesis to E(x)=o(x) plus integrability of |E(t)| log^r t / t^2, which the known zero-free region supplies, but the paper does not state or prove such a version. There is also a sign inconsistency in the integral term between statement (12) and the proof of Proposition 2.2, though this does not affect the limit (13).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the coefficients γ_{K,r} in the Laurent expansion of ζ'_K/ζ_K at s=1, calling them higher Euler-Kronecker constants. It claims an arithmetic limit formula for these coefficients in terms of weighted sums of the number-field von Mangoldt function (Theorem 1.4), GRH-conditional lower and upper bounds for odd and even r (Theorem 1.5), unconditional bounds depending on a possible Siegel zero (Theorem 4.3), and an Ihara-type limit formula involving the function Φ_K(r,x) (Theorem 1.8). The proofs use partial summation, the Hadamard factorization of the Dedekind zeta function, and contour integrals over Dirichlet series.","tokens_in":16813,"tokens_out":42500,"duration_ms":348404,"significance":"If fully proved, the results would give a systematic extension of Ihara's Euler-Kronecker theory to all Laurent coefficients. The cleanest contribution is Theorem 1.8, whose contour/residue computation is a genuine generalization of Ihara's method and whose final expression does not require explicit knowledge of the Dirichlet coefficients. The paper also connects the higher constants to the error term in the prime ideal theorem. However, the proof of the central arithmetic formula currently rests on an auxiliary proposition whose main hypothesis is not known unconditionally for Dedekind zeta functions and whose displayed identities contain sign errors; a careful revision is required. The paper is not machine-checked and contains no code, but the analytic arguments are standard and the final formulas are plausible.","major_comments":[{"comment":"The hypothesis E(x)=O(x^b) with 0≤b<1 is not satisfied by the known unconditional error term for the Dedekind zeta function. The prime ideal theorem gives only ψ_K(x)-x = O(x exp(-c√(log x))), which is not O(x^b) for any fixed b<1 because (1-b)log x eventually exceeds c√(log x). Since Theorem 1.4, Eq. (4), is proved by direct appeal to Proposition 2.2, the proof of the arithmetic formula is incomplete as written. The gap is repairable: the argument only needs E(x)=o(x), E(x)log^r x/x → 0, and convergence of the tail integral ∫_x^∞ |E(t)| log^{r-1} t / t^2 dt (or of the analogous tail in Proposition 2.2), all of which follow from the standard zero-free region for zeta_K. But this weaker version is neither stated nor proved in the paper.","section":"Section 2, Proposition 2.2; Section 3, proof of Theorem 1.4"},{"comment":"Equation (12) and the surrounding proof contain sign errors. The correct identities from Proposition 2.1 with u=-1 are C_0 = ∑_{n≤x} b_n/n - C log x - E(x)/x + ∫_x^∞ E(t)/t^2 dt and, for r≥1, C_r = ∑_{n≤x} b_n log^r n/n - C/(r+1) log^{r+1}x - E(x)log^r x/x + ∫_x^∞ (log^r t - r log^{r-1}t)E(t)/t^2 dt, not the expression with -∫(log^r + r log^{r-1})E/t^2. As a consequence, Corollary 3.1 is wrong as stated: it should read γ_{K,r} = (-1)^{r+1}/r! ∫_1^∞ (log^r t - r log^{r-1}t)Δ_K(t)/t^2 dt. Remark 3.2 is also incorrect; the r=0 identity is γ_{K,0} = -1 - ∫_1^∞ Δ_K(t)/t^2 dt. The final limit formula (13) is unaffected because the E(x)/x and tail-integral terms vanish in the limit, but Theorem 1.5 uses the displayed identity and must be rewritten with the correct signs.","section":"Section 2, Eq. (12); Section 3, Corollary 3.1 and Remark 3.2"}],"minor_comments":[{"comment":"Theorem 1.5 is stated for all number fields but uses log_2|d_K| and log_3|d_K|; for K=Q these are undefined. The statement should exclude |d_K|=1 or handle it separately.","section":"Section 1.2, Theorem 1.5"},{"comment":"The function Λ_K(n) is used in Theorem 1.4 and Section 3 but is only defined implicitly through the Dirichlet series. It should be defined explicitly as Λ_K(n) = ∑_{N(P)^k = n} log N(P) before it is used.","section":"Section 1.2, Theorem 1.4"},{"comment":"The statement of Proposition 2.1 writes log^{r-1}t in the integrand, which is not meaningful for r=0. The r=0 case should be stated separately or the notation should be adjusted.","section":"Section 2, Proposition 2.1"},{"comment":"Several contour integrals are written with c+∞ and c-∞ in place of c+i∞ and c-i∞ (for example in Lemma 5.1, Eq. (28), and the proof of Lemma 5.4). These appear to be typesetting errors and should be corrected.","section":"Section 5, Lemmas 5.1 and 5.4"},{"comment":"In the proof of Lemma 5.5, the sentence that excluding finitely many low-lying zeros has no effect should explicitly address the possible Siegel zero: for a fixed field, 1-β_0 is a positive constant, so x^{β_0-1}(log x)^r tends to 0. The current wording is terse but the underlying point is valid.","section":"Section 5, Lemma 5.5"},{"comment":"The lower bound for odd r has a main term -1/(r+1) plus an error term; when the error term is not small the displayed bound is negative. The paper should state the intended asymptotic regime, e.g. fixed r and |d_K| → ∞, or explain how the bound is meaningful uniformly.","section":"Section 1.2, Theorem 1.5 and Remark 1.6"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be an early-stage draft with several sign and notational errors, but the main ideas are sound and the final identities are consistent with standard residue calculus. The most urgent repair is the statement and proof of Proposition 2.2: both the hypothesis on E(x) and the signs in Eq. (12) need correction, because Theorem 1.4 and Theorem 1.5 depend on it. The Section 5 contour computation is the strongest part of the paper and deserves a careful referee check after the auxiliary lemmas are fixed. The novelty relative to Ihara's work is moderate but real, especially Theorem 1.8."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper defines higher Euler-Kronecker constants and proves three genuinely new things for r ≥ 1: an arithmetic formula (4), an Ihara-type limit (7), and conditional/unconditional bounds. The r = 0 case is due to Ihara and Dixit–Murty, and the author says so plainly. The methods are a mixture of elementary partial summation and contour integration, and the final expressions look consistent with standard residue calculus. The novelty is real, though it is an extension of an existing program rather than a breakthrough.\n\nThe soft spots are real but mostly fixable. The main issue is that Theorem 1.4 is proved by invoking Proposition 2.2, whose hypothesis is E(x) = O(x^b) for some b < 1. For the Dedekind zeta function, the best unconditional prime ideal theorem gives ψ_K(x) − x = O(x exp(−c√(log x))), which is not O(x^b) for any b < 1. So the proposition, as stated, does not apply to the key case. The limit (4) is very likely true — the tail integral converges because of the standard zero-free region — but the paper does not prove the needed weakened version. That gap is load-bearing in the sense that the later bounds depend on (4).\n\nThere is also a sign inconsistency in the integral term of Proposition 2.2 between the statement and the proof. It does not affect the limit (13), but it does make the integral representation in Corollary 3.1 incorrect as written. Theorem 1.5 needs an explicit exclusion of K = Q, since log_2 |d_K| is not defined when |d_K| = 1. The residue display near Lemma 5.5 has a notational slip, but the computations around it are detailed and the final limit appears correct.\n\nOn the positive side, nothing is circular, no data are fitted, and the author is honest about what is new and what is borrowed. The contour arguments in Section 5 are thorough enough to follow, and the treatment of the zero contributions through standard zero-free regions is acceptable, even though a Siegel zero requires care.\n\nMy bottom line: the paper deserves a serious referee. It should not be desk-rejected, but it needs revision before it is fully reliable. A referee should ask for a restated Proposition 2.2 with hypotheses that actually hold for Dedekind zeta functions, fix the sign errors, and handle K = Q explicitly.","headline":"A real extension of Ihara's program with mostly sound results, but the proof of the central arithmetic formula has a repairable gap in the stated hypothesis.","tokens_in":17468,"tokens_out":14730,"would_cite":true,"duration_ms":112645,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R42","11M06","11M38","11M20","11M36"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves explicit arithmetic limit formulas for all higher Euler-Kronecker constants of a number field, and derives both conditional and unconditional bounds for them.","keywords":["Euler-Kronecker constants","Dedekind zeta function","Laurent series coefficients","von Mangoldt function","explicit formula","Li's criterion","generalized Riemann hypothesis","Stieltjes constants"],"falsifier":"Compute $\\gamma_{\\mathbb{Q},1}$ directly from the Laurent expansion of $\\zeta'(s)/\\zeta(s)$ at $s=1$, then evaluate the right side of equation (4) numerically with the usual von Mangoldt function for increasing $x$; if the two values do not converge to the same number, Theorem 1.4 fails. Independently, testing equation (7) for $K=\\mathbb{Q}$, $r=1$ with $\\Phi_{\\mathbb{Q}}(1,x)$ and $f(1,x)$ would settle Theorem 1.8.","tokens_in":16323,"feed_emoji":"🧮","tokens_out":5365,"duration_ms":45929,"temperature":0.7,"pith_summary":"The paper treats the coefficients in the Laurent expansion of the logarithmic derivative of a Dedekind zeta function at $s=1$ as a full sequence of arithmetic invariants. It proves an explicit arithmetic formula expressing each higher Euler-Kronecker constant $\\gamma_{K,r}$ as a limit of weighted prime-power sums, generalizing the classical formulas for Euler's constant and Ihara's $r=0$ case. It also proves bounds under the generalized Riemann hypothesis and unconditional bounds, together with a second formula in the spirit of Ihara's counting function. If these identities are correct, the entire Taylor expansion of $\\zeta_K'(s)/\\zeta_K(s)$ at $s=1$ is arithmetic information carried by the distribution of prime powers.","feed_headline":"Every higher Euler-Kronecker constant gets an explicit arithmetic formula","feed_subtitle":"A number field's logarithmic-derivative coefficients are computable from prime-power sums and a recursive subtraction.","key_machinery":"The machinery is the Laurent expansion itself: writing $-\\zeta_K'(s)/\\zeta_K(s) = \\sum_{n\\ge 1} \\Lambda_K(n)/n^s$, a partial-summation identity for Dirichlet series converts the Taylor coefficient $C_r$ of the analytic part into a limit of finite sums of $\\Lambda_K(n)(\\log n)^r/n$, with the smooth term $(\\log x)^{r+1}/(r+1)$ removed. For the Ihara-style formula, the machinery is a contour integral of $x^s$ against the $r$-th derivative of the logarithmic derivative of $\\zeta_K$; residues at $s=1$, $s=0$, the nontrivial zeros, and the Gamma-factor poles produce respectively the recursive subtraction $f(r,x)$, the explicit zero sum, and the Gamma term.","core_discovery":"The paper's central claim is that for every number field $K$ and every $r \\ge 0$, the $r$-th coefficient $\\gamma_{K,r}$ in the Laurent expansion of $\\zeta_K'(s)/\\zeta_K(s)$ at $s=1$ satisfies the arithmetic identity $\\gamma_{K,r} = \\frac{(-1)^{r+1}}{r!} \\lim_{x\\to\\infty} \\left( \\sum_{n\\le x} \\frac{\\Lambda_K(n)(\\log n)^r}{n} - \\frac{(\\log x)^{r+1}}{r+1} \\right)$, with $\\Lambda_K(n)$ the von Mangoldt function of the number field. The same coefficients also satisfy a second expression, $\\gamma_{K,r} + (-1)^r = \\frac{(-1)^{r+1}}{r!} \\lim_{x\\to\\infty} (\\Phi_K(r,x) - f(r,x))$, where $\\Phi_K(r,x)$ is an Ihara-type weighted sum over prime powers and $f(r,x)$ is recursively defined. From the first identity the paper derives GRH-conditional lower and upper bounds for odd and even $r$, and from the Hadamard factorization it derives unconditional bounds whose main term is polynomial in $\\log|d_K|$ or, in the Siegel-zero case, dominated by the exceptional zero.","pith_inferences":["If the limit identities are numerically stable, they suggest a direct computational route to the higher Euler-Kronecker constants from prime powers alone, avoiding any search for zeros of $\\zeta_K$.","The parity asymmetry in the GRH bounds hints that the signs of the higher Euler-Kronecker constants may follow a parity pattern; the paper proves bounds but does not settle the sign question.","Since the difference $\\Phi_K(r,x)-f(r,x)$ converges to a sum over nontrivial zeros, its rate of convergence could be sensitive to a nearby Siegel zero, offering a possible numerical probe for exceptional zeros.","The same contour method may adapt to other $L$-functions with a functional equation, producing higher analogues of Li's coefficients and related zero-counting information; the paper does not make this extension."],"forward_implications":["For every $r$, $\\gamma_{K,r}$ is determined by the prime-power distribution, so the full Laurent expansion at $s=1$ is a consequence of the error term in the prime ideal theorem.","Corollary 3.1 expresses $\\gamma_{K,r}$ as an integral of the error term $\\Delta_K(x)$, generalizing Ihara's integral formula for the ordinary Euler-Kronecker constant.","Under GRH, odd $r$ constants have explicit lower bounds and even $r$ constants have explicit upper bounds depending on $\\log|d_K|$ and lower iterated logarithms, with the bounds strongest for small $r$.","Unconditionally, unless a Siegel zero exists, $\\gamma_{K,r} = O((4\\log|d_K|)^{r+2})$; if a Siegel zero $\\beta_0$ exists, the dominant contribution is $(-1)^r/(1-\\beta_0)^{r+1}$.","The Ihara-type formula gives a way to compute $\\gamma_{K,r}$ from prime-power counts and a recursively defined subtraction, without explicit knowledge of the individual Dirichlet coefficients of $\\zeta_K' / \\zeta_K$."],"supporting_citations":[{"why":"Introduced the Euler-Kronecker constant and proved the $r=0$ Ihara limit formula that Theorem 1.8 generalizes.","marker":"[9]"},{"why":"Provided an elementary proof of Ihara's formulas, whose partial-summation method extends to Proposition 2.2 used in Theorem 1.4.","marker":"[6]"},{"why":"Supplies the Dirichlet-series coefficient formula underlying Propositions 2.1 and 2.2.","marker":"[2]"},{"why":"Supplies the GRH-conditional bound on $\\Delta_K(x)$ used in the proof of Theorem 1.5.","marker":"[14]"},{"why":"Supplies the zero-free region used in Lemma 5.5 to justify passing the limit through the sum over nontrivial zeros.","marker":"[11]"},{"why":"Supplies the Hadamard-factorization formula, the Siegel-zero lemma, and zero-counting estimates used in Theorem 4.3 and the residue computations.","marker":"[15]"},{"why":"Records the analogous Stieltjes-constant formula for the Dedekind zeta function itself, providing the pattern that Theorem 1.4 generalizes to logarithmic derivatives.","marker":"[7]"}],"fun_headline_variants":["Explicit arithmetic formulas for all higher Euler-Kronecker constants","Higher Euler-Kronecker constants expressed via prime-power sums","Every Euler-Kronecker coefficient now has a closed arithmetic form","Number-field zeta coefficients exactly given by recursive subtraction","All higher Euler-Kronecker constants obey new prime-sum identities"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise for the main bounds is the generalized Riemann hypothesis for the Dedekind zeta function; the unconditional Ihara-type formula relies on the standard zero-free region being strong enough to interchange a limit with the sum over nontrivial zeros.","fun_headline_variants_meta":{"raw":{"variants":["Explicit arithmetic formulas for all higher Euler-Kronecker constants","Higher Euler-Kronecker constants expressed via prime-power sums","Every Euler-Kronecker coefficient now has a closed arithmetic form","Number-field zeta coefficients exactly given by recursive subtraction","All higher Euler-Kronecker constants obey new prime-sum identities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000233,"raw_usage":{"total_tokens":1458,"prompt_tokens":876,"completion_tokens":582,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":499}},"tokens_in":492,"tokens_out":582,"duration_ms":5800,"temperature":1.0,"reasoning_tokens":499,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:42:43.112388+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\gamma_{\\mathbb{Q},1}$ directly from the Laurent expansion of $\\zeta'(s)/\\zeta(s)$ at $s=1$, then evaluate the right side of equation (4) numerically with the usual von Mangoldt function for increasing $x$; if the two values do not converge to the same number, Theorem 1.4 fails. Independently, testing equation (7) for $K=\\mathbb{Q}$, $r=1$ with $\\Phi_{\\mathbb{Q}}(1,x)$ and $f(1,x)$ would settle Theorem 1.8.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the Euler-Kronecker constant and proved the $r=0$ Ihara limit formula that Theorem 1.8 generalizes."},{"cited_title":"Murty , On Ihara’s conjectures for Euler-Kronecker constants , Acta Arithmetica 210 (2023), 95–123","cited_arxiv_id":null,"evidence_quote":"Provided an elementary proof of Ihara's formulas, whose partial-summation method extends to Proposition 2.2 used in Theorem 1.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Dirichlet-series coefficient formula underlying Propositions 2.1 and 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the GRH-conditional bound on $\\Delta_K(x)$ used in the proof of Theorem 1.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the zero-free region used in Lemma 5.5 to justify passing the limit through the sum over nontrivial zeros."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Hadamard-factorization formula, the Siegel-zero lemma, and zero-counting estimates used in Theorem 4.3 and the residue computations."},{"cited_title":"Japan Acad., 94, Ser","cited_arxiv_id":null,"evidence_quote":"Records the analogous Stieltjes-constant formula for the Dedekind zeta function itself, providing the pattern that Theorem 1.4 generalizes to logarithmic derivatives."}],"review_version":1}