{"id":"73b154f2-e5c5-4926-8f94-ee33d879fc68","arxiv_id":"1908.03044","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A mild bound |γ_K| ≪ exp((log log |d_K|)^m) on Euler-Kronecker constants along a tower implies the generalized Brauer-Siegel conjecture for that tower; the paper also proves unconditional |γ_K| bounds for almost normal and solvable-closure fields.","lead":"This number theory paper proves that a mild bound on the Euler-Kronecker constant in a tower of number fields forces the generalized Brauer-Siegel conjecture to hold in that tower, and gives new unconditional bounds on that constant. It also links known bounds on the constant for cyclotomic fields to the error in the Dedekind zeta zero count, though the author states these estimates are weaker than existing ones.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unconditional status of the [18] limsup inequality is unstated and load-bearing; if it is only GRH-conditional, Theorem 2.3 lacks its upper bound.","rationale":"The reader's weakest_assumption already identified the unverified [18] inequality as a primary risk. I agree that this is the most load-bearing concern: without an unconditional upper bound on limsup log ρ/g, the proof of Theorem 2.3 only establishes liminf ≥ RHS, not equality. The other flagged issues (the uniform convergence of prime-power sums and the O(1) bound (25)) are real but appear repairable: the uniform convergence follows from dominated convergence with N_q/g_i bounded, and (25) can likely be replaced by a weaker polynomial-in-g bound that still forces θ|γ|/g → 0 for the chosen θ. The [18] citation, in contrast, cannot be repaired from the manuscript's own arguments. The reader's verdict of CONDITIONAL is appropriate; if the check shows the inequality is unconditional, the concern disappears and the paper may merit ACCEPT, but without that verification the conditional verdict stands.","tokens_in":14023,"tokens_out":46256,"duration_ms":513453,"concrete_test":"Consult [18] (Tsfasman–Vlăduţ, Moscow Math. J. 2 (2002)) or the survey [10]: locate the inequality limsup log ρ/g ≤ Σ_q φ_q log(q/(q-1)) and determine whether it is stated as an unconditional theorem for all asymptotically exact families or as a step inside the GRH proof. If it is only conditional, check whether a separate unconditional proof exists for towers; if not, Theorem 2.3 must either assume this inequality or add GRH. This is a literature verification, not a numerical experiment, and it settles the status of the upper bound directly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.3's proof (Section 2.3.3, paragraph before (22)) imports the upper bound limsup log ρ_Ki/g_Ki ≤ Σ_q φ_q log(q/(q-1)) from [18] without stating the hypotheses under which it is proved. The manuscript itself notes that [18] proves GBS under GRH for all asymptotically exact families and unconditionally only for asymptotically good towers of almost normal fields. The γ-bound hypothesis supplies only the liminf half of the limit, via (22)-(27); no alternative upper bound is derived. If the quoted inequality is part of the GRH-conditional argument or carries extra hypotheses, then the claimed unconditional GBS for arbitrary towers has no proof. The central equality requires both one-sided bounds, so this concern is decisive if it lands.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Euler-Kronecker constants γ_K and their relation to the generalized Brauer-Siegel (GBS) conjecture. The main result (Theorem 2.3) states that for any tower of number fields K={K_i} satisfying |γ_{K_i}| ≪ exp((log log |d_{K_i}|)^m) for any fixed large m, the GBS conjecture holds for K; the stated sufficient condition is that |γ_{K_i}| ≪ exp(α_i) with α_i = o(g_{K_i}/log g_{K_i}). The proof chooses θ_{K_i} = exp(-(log g_{K_i})^{m+2}) and splits the desired limit into an upper bound for limsup log ρ/g, imported from [18], and a lower bound obtained by bounding the factor F_K(1+θ) in terms of |γ_K| and then passing to the Euler-product sum. The paper also proves unconditional upper bounds on |γ_K| for almost normal fields (Theorem 2.1) and for fields with solvable normal closure (Theorem 2.2), and in Section 3 uses known bounds on γ for cyclotomic fields to refine the error term in the zero-counting function N_K(T) (Proposition 3.1).","tokens_in":14063,"tokens_out":24533,"duration_ms":262756,"significance":"If Theorem 2.3 is fully established, it is a substantial new result: an extremely weak upper bound on |γ_K| along a tower forces the full generalized Brauer-Siegel limit. The construction of the sequence θ_K and the reduction of the F-term to a bound on |γ_K| are elegant and potentially influential. The paper also contributes new unconditional bounds on |γ_K| for two classes of fields and illustrates a novel connection between Euler-Kronecker constants and zero-counting error terms. These strengths are conditional, however, on repairing the several load-bearing gaps described below; the current written proof is not complete.","major_comments":[{"comment":"The upper-bound half of Theorem 2.3 rests on the quoted result from [18] that limsup log ρ_Ki/g_Ki ≤ Σ_q φ_q log(q/(q-1)) for any asymptotically exact family, but the manuscript never states the hypotheses under which this inequality is proved in [18]. The introduction describes [18] as proving GBS under GRH for all asymptotically exact families and unconditionally only for asymptotically good towers of almost normal fields. Since Theorem 2.3 is an unconditional claim for arbitrary towers, the author must either prove the inequality directly (for example, from the monotonicity of (s-1)ζ_K(s) for s>1) or give a precise reference with its unconditional status. Without that, the unconditional claim is not justified.","section":"Section 2.3.3, paragraph before Eq. (22)"},{"comment":"The assertion that the higher-prime-power sums in the Euler product converge uniformly for θ > -δ is unproved. This uniformity is necessary to justify passing to the limit inside the infinite sum in Eq. (22); without it, the liminf inequality is not established. A dominated-convergence argument using N_q(K_i) ≤ n_Ki and a uniform bound n_K/g_K ≤ C should be supplied, and the range θ < 0 must be handled explicitly.","section":"Section 2.3.3, after Eq. (25)"},{"comment":"The proof of Lemma 2.4 is not correct as written. The displayed inequality Σ_{m≤n} mN_{p^m}(L) ≤ [L:K] Σ_{m≤n} mN_{p^m}(K) has the direction opposite to what is needed for monotonicity; combined with the standard relation g(K) ≥ [L:K]g(L) (from d_K = d_L^{[K:L]} N_{L/Q}(D_{K/L})), it does not imply that Σ mN_{p^m}(K_i)/g(K_i) is non-increasing. The correct inequality is N_{p^m}(K) ≤ [L:K]N_{p^m}(L) for L⊂K, since each prime ideal of L has at most [L:K] extensions in K. This lemma is used in the proof of (22) to assert φ_p ≤ N_p(K_i)/g_i, so the proof must be repaired.","section":"Section 2.2.3, Lemma 2.4"},{"comment":"The proof of the O(1) bound for Z_K(1+θ)-Z_K(1) is too terse to be verifiable. In particular, the estimate θΣ_ρ 1/|θ+ρ|² ≪ θ n_K Σ log n/n² only applies to zeros with |Im ρ| ≥ 1 after using N_K(T+1)-N_K(T) ≪ n_K log T; the contribution of zeros with |Im ρ| < 1, and of a possible Siegel zero, must be bounded separately. This is load-bearing because (25), together with the subsequent bound log F_K(1+θ)/g_K ≪ θ|γ_K|/g_K, feeds directly into (23) and hence into the limsup part of the proof of Theorem 2.3.","section":"Section 2.3.3, proof of Eq. (25)"}],"minor_comments":[{"comment":"Equation (22) and the following display are missing the logarithm before ζ_K; the surrounding text and the Euler-product formula show that log ζ_K is intended. The same omission appears in the sentence 'Note that ζKi(1+θ)/gKi = ...'.","section":"Section 2.3.3, display before Eq. (22)"},{"comment":"The sign in the integral expression following Eq. (25) is inconsistent with the definition Z_K(s) = -F_K'/F_K(s); the absolute-value estimate is unaffected, but the displayed formula should be corrected for consistency.","section":"Section 2.3.3, Eq. (27)"},{"comment":"The manuscript contains numerous OCR-type typographical errors (for example, 'CONST ANTS' in the title, malformed variables and braces throughout, and broken symbols in displayed equations), which materially reduce readability; a careful copyedit is required.","section":"Throughout"},{"comment":"The relation between g(L) and g(K) should be stated explicitly: for L⊂K with [K:L]=d, one has g(K) ≥ d g(L), a consequence of d_K = d_L^d N_{L/Q}(D_{K/L}); this is the key fact needed in the monotonicity argument.","section":"Section 2.2.3, Lemma 2.4"},{"comment":"The proof defers to 'the proof of Lemma 2.5 in [2]' without indicating which modifications are needed to incorporate Murty's zero-free region from Eq. (12); a short indication of the changes would improve readability, especially since [2] is a self-citation.","section":"Section 2.3.2, proof of Theorem 2.2"}],"recommendation":"major_revision","confidential_remarks":"The central theorem appears likely to be correct, but the written proof is not complete: the imported upper bound from [18] needs a precise unconditional statement, the uniform convergence assertion needs a proof, Lemma 2.4's proof has a direction error, and the proof of Eq. (25) omits low-lying zeros. If the [18] inequality is in fact only GRH-conditional, Theorem 2.3 would need a different upper-bound argument. The paper would also benefit from a full copyedit to remove the many typesetting artifacts."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the quick take: if Theorem 2.3 is right, it's a real step—those very weak bounds on |γ_K| along a tower force the full Tsfasman–Vladut limit, and the tower monotonicity trick is genuinely nice. The two unconditional bounds on |γ_K| for almost normal fields and for fields with solvable normal closure are also solid new inputs. But the proof as written doesn't establish the hypotheses under which the quoted [18] limsup inequality holds, and that inequality supplies the load-bearing upper half of the limit. I'd send it to a referee with a specific request to check that citation.\n\nWhat I like: the paper finds a genuinely new sufficient condition for GBS, not a repackaging. The choice θ = exp(-(log g)^{m+2}) is the right scale, and the argument that a weak pointwise bound on γ_K controls log F through integration of Z is clean. The self-citation in Theorem 2.2 is fine—it's the author's own Lemma 2.5 in [2], and deferring there is normal.\n\nWhere I'd push: (1) The sentence \"In [18], it is shown that for any asymptotically exact family, limsup log ρ/g ≤ Σ φ_q log(q/(q-1))\" is used without stating what assumptions [18] actually requires. The paper's own survey says [18] proves GBS unconditionally only for almost normal towers; that makes the unconditional-sounding import suspicious. If that inequality is only GRH-conditional, Theorem 2.3 lacks its upper bound. This is the first question for the referee. (2) Equation (22) is missing a log—should be liminf log ζ..., and the prose clarifies it, so that's just a typo. (3) The uniform convergence of the higher prime-power sums in the liminf argument is asserted, not proved; probably fixable, but it needs a line. (4) The abstract's \"finer estimate\" for zero-counting is an overclaim; Section 3 explicitly says its estimates are weaker than Trudgian's. That section is an illustration, not a record.\n\nOverall: this is a serious paper with a fixable-looking gap. The central implication is structurally sound; the missing hypothesis on the [18] result is the kind of thing that gets sorted out in one referee round. The zero-counting application is not a reason to accept, but it's fine as context. If I were editing, I'd send it out.","headline":"A genuinely new sufficient condition for GBS, but the main theorem's upper bound rests on an imported inequality whose hypotheses are not stated; worth a referee round to pin that down.","tokens_in":705,"tokens_out":940,"would_cite":false,"duration_ms":57804,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R42","11R18","11R29"],"pacs":[],"model":"deepseek-v4-flash","headline":"A weak bound on Euler-Kronecker constants in a tower of number fields forces the generalized Brauer-Siegel conjecture.","keywords":["Euler-Kronecker constants","Brauer-Siegel theorem","generalized Brauer-Siegel conjecture","asymptotically exact families","towers of number fields","Dedekind zeta functions","cyclotomic fields","exceptional zeros"],"falsifier":"Compute, for an explicit tower satisfying the gamma-bound of Theorem 2.3, the quantity $\\lim_i \\log\\rho_{K_i}/g_{K_i}$ and compare it with $\\sum_q \\varphi_q \\log(q/(q-1))$; any value strictly below the predicted sum, or any tower where the higher prime-power sums fail to converge uniformly in a neighborhood of $\\theta=0$, would refute Theorem 2.3.","tokens_in":13684,"feed_emoji":"🔢","tokens_out":11820,"duration_ms":125557,"temperature":0.7,"pith_summary":"The paper connects two objects that do not obviously talk to each other: the Euler-Kronecker constant $\\gamma_K$, the number-field analogue of the Euler-Mascheroni constant, and the growth rate of $h_K R_K$, the class number times regulator. Its main theorem states that in a tower of number fields $K_i$, a very weak upper bound on $|\\gamma_{K_i}|$ forces the full generalized Brauer-Siegel conjecture for that tower. Because the generalized Brauer-Siegel limit determines the asymptotic size of $h_K R_K$ from the distribution of small primes, this turns a difficult arithmetic asymptotic into a check on one analytic constant attached to the Dedekind zeta function. The paper also proves unconditional gamma-bounds for almost normal fields and for fields with solvable normal closure, and uses cyclotomic gamma-bounds to constrain the error term in the zero-counting formula for Dedekind zeta functions. If the main theorem is right, towers of number fields with very large class-number growth are fully governed by a single logarithmic-derivative constant.","feed_headline":"Weak bound on Euler-Kronecker constants proves Brauer-Siegel","feed_subtitle":"For towers of number fields, a very weak bound on |gamma_K| fully determines the class-number/regulator asymptotic.","key_machinery":"The engine is the Euler-Kronecker constant $\\gamma_K$, defined as the constant term in the expansion $\\zeta'_K(s)/\\zeta_K(s) = -1/(s-1) + \\gamma_K + O(s-1)$; equivalently, it is the quotient of the first two Laurent coefficients of $\\zeta_K$ at $s=1$. The proof uses Stark's partial-fraction identity for $Z_K(s) = -1/(s-1) - \\frac{d}{ds}\\log \\zeta_K(s)$ to show that $|Z_K(1+\\theta) - Z_K(1)| = O(1)$ for $\\theta < 1/n_K$, so a bound on $|\\gamma_K|$ bounds $\\log F_K(1+\\theta)$ through the integral $\\log F_K(1+\\theta) = \\int_0^\\theta Z_K(1+u)\\,du$. On the multiplicative side, $\\log \\zeta_K(1+\\theta)$ is a weighted sum over prime powers $q$ of $N_q(K) \\log(1/(1-q^{-1-\\theta}))$; for towers, monotonicity of the normalized counting functions gives the liminf lower bound at the predicted Brauer-Siegel value, while the Tsfasman-Vladuts upper bound on $\\limsup \\log \\rho_K/g_K$ supplies the matching upper bound. Stark's exceptional-zero result for almost normal fields and V. K. Murty's extension to solvable normal closures are the auxiliary inputs that make the unconditional gamma-bounds of Theorems 2.1 and 2.2 possible.","core_discovery":"The central discovery is Theorem 2.3: if $\\{K_i\\}$ is a tower of number fields and $|\\gamma_{K_i}| \\ll \\exp((\\log \\log |d_{K_i}|)^m)$ for an arbitrarily large fixed exponent $m$ — equivalently, $|\\gamma_{K_i}| \\ll \\exp(\\alpha_i)$ with $\\alpha_i = o(g_{K_i}/\\log g_{K_i})$ — then the generalized Brauer-Siegel conjecture holds for the tower. That is, with $\\varphi_q, \\varphi_R, \\varphi_C$ the limiting proportions of places, the limit of $\\log(h_{K_i}R_{K_i})/g_{K_i}$ equals $1 + \\sum_q \\varphi_q \\log(q/(q-1)) - \\varphi_R \\log 2 - \\varphi_C \\log 2\\pi$, and equivalently the residue $\\rho_{K_i}$ of $\\zeta_{K_i}$ at $s=1$ satisfies $\\lim_i \\log \\rho_{K_i}/g_{K_i} = \\sum_q \\varphi_q \\log(q/(q-1))$. The proof evaluates $\\zeta_K(1+\\theta)$ at a carefully chosen $\\theta = \\theta_{K_i}$ tending to zero: the Euler product contributes the predicted sum, the factor $F_K(1+\\theta)$ contributes $O(\\theta |\\gamma_K|/g_K)$, and the choice $\\theta_{K_i} = \\exp(-(\\log g_{K_i})^{m+2})$ makes this $\\gamma$-controlled term vanish. The paper also establishes unconditional upper bounds on $|\\gamma_K|$ for almost normal fields and for fields with solvable normal closure, and derives an explicit interval for the constant in the cyclotomic zero-counting error from the bound $|\\gamma_p| \\le 11\\log p$ that holds for almost all primes $p$.","pith_inferences":["The tower hypothesis is used only to obtain monotone convergence of the normalized place counts and the liminf lower bound; if the quoted limsup inequality and uniform convergence of higher prime-power sums hold for all asymptotically exact families, the same argument would likely extend the theorem from towers to arbitrary asymptotically exact families satisfying the $\\gamma_K$ bound.","Stark's identity $\\sum_\\rho 1/\\rho = \\gamma_K + \\frac12\\log|d_K| - \\frac12 r_1(\\gamma+\\log 4\\pi) - r_2(\\gamma+\\log 2\\pi) + 1$ suggests that a bound on $|\\gamma_K|$ is also a statement about low-lying zeros: a tower with $|\\gamma_K| \\ll \\exp(o(g/\\log g))$ cannot have too many nontrivial zeros with small imaginary part clustered near the real axis.","The paper's suggested replacement of the GRH pairing in (33) could give an unconditional version of Proposition 3.1, and the resulting constraint on $\\gamma_p$ could be tested against the known prime $964477901$ with negative Euler-Kronecker constant.","One testable extension is to compute both sides of the generalized Brauer-Siegel identity for an explicit tower satisfying the $\\gamma_K$ bound; a mismatch, or a failure of uniform convergence in the higher prime-power sums, would pinpoint exactly which analytic step in the proof needs strengthening."],"forward_implications":["For any tower of number fields satisfying the weak $|\\gamma_K|$ bound, the generalized Brauer-Siegel conjecture ceases to be conjectural: the class-number-regulator asymptotic is completely determined by the limiting distribution of places.","Despite being far weaker than the GRH-conditional bound $|\\gamma_K| \\le 2\\log\\log|d_K|$, the new hypothesis is still sufficient to imply the full generalized Brauer-Siegel conjecture.","The unconditional bounds in Theorems 2.1 and 2.2 imply generalized Brauer-Siegel for towers of almost normal number fields without quadratic subfields, and for towers of fields with solvable normal closure without quadratic subfields.","For cyclotomic fields $\\mathbb{Q}(\\zeta_p)$ with $p$ in the generic set, the zero-counting error term has an explicit constant $c$ lying between $-4/\\pi$ and $(2\\tan^{-1}2 - 4/5)/\\pi$, so any future improvement on $\\gamma_p$ immediately sharpens this zero-counting window.","The theorem shows that the obstruction to the classical Brauer-Siegel conjecture is not the size of $\\gamma_K$ itself, since even a very weak bound on it forces the expected asymptotic."],"supporting_citations":[{"why":"Formulates the generalized Brauer-Siegel conjecture and supplies the key limsup bound $\\limsup \\log\\rho_K/g_K \\le \\sum_q \\varphi_q \\log(q/(q-1))$ used in the proof of Theorem 2.3.","marker":"[18]"},{"why":"Lagarias-Odlyzko effective Chebotarev estimates control the counting function $G_K(x)$ and provide the main input for the unconditional gamma-bounds in Theorems 2.1 and 2.2.","marker":"[9]"},{"why":"Stark's partial-fraction lemma for $Z_K$ and his exceptional-zero result for almost normal fields give the $O(1)$ bound on $|Z_K(1+\\theta)-Z_K(1)|$ and the hypothesis of Theorem 2.1.","marker":"[13]"},{"why":"Extends Stark's exceptional-zero control to fields with solvable normal closure, supplying the hypothesis of Theorem 2.2.","marker":"[8]"},{"why":"Introduces the Euler-Kronecker constant and establishes the basic analytic identity identifying it as the constant term of $\\zeta'_K/\\zeta_K$ at $s=1$.","marker":"[6]"},{"why":"Provides the 'almost all primes' bounds $1 \\ge \\gamma_p/\\log p > -11$ used in Proposition 3.1 for cyclotomic fields.","marker":"[11]"},{"why":"Supplies the previous best explicit zero-counting bound for Dedekind zeta functions, against which Proposition 3.1 is compared.","marker":"[14]"},{"why":"Gives the counterexample to Ihara's positivity conjecture for $\\gamma_p$ and distributional facts about cyclotomic Euler-Kronecker constants used as context for the cyclotomic results.","marker":"[4]"}],"fun_headline_variants":["Weak gamma bound yields Brauer-Siegel for number field towers","Loose Euler-Kronecker bound forces Brauer-Siegel conjecture","Nearly any gamma_K bound implies generalized Brauer-Siegel","Euler-Kronecker slack proves Brauer-Siegel in towers","Weak Euler-Kronecker bound seals Brauer-Siegel result"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the quoted Tsfasman-Vladuts upper bound on the growth of the Dedekind-zeta residue holds unconditionally for every asymptotically exact family, and that the higher prime-power sums in the Euler product converge uniformly near $\\theta=0$, both unproved in this paper.","fun_headline_variants_meta":{"raw":{"variants":["Weak gamma bound yields Brauer-Siegel for number field towers","Loose Euler-Kronecker bound forces Brauer-Siegel conjecture","Nearly any gamma_K bound implies generalized Brauer-Siegel","Euler-Kronecker slack proves Brauer-Siegel in towers","Weak Euler-Kronecker bound seals Brauer-Siegel result"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000491,"raw_usage":{"total_tokens":2472,"prompt_tokens":1057,"completion_tokens":1415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":673,"completion_tokens_details":{"reasoning_tokens":1328}},"tokens_in":673,"tokens_out":1415,"duration_ms":11464,"temperature":1.0,"reasoning_tokens":1328,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:32:47.180781+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for an explicit tower satisfying the gamma-bound of Theorem 2.3, the quantity $\\lim_i \\log\\rho_{K_i}/g_{K_i}$ and compare it with $\\sum_q \\varphi_q \\log(q/(q-1))$; any value strictly below the predicted sum, or any tower where the higher prime-power sums fail to converge uniformly in a neighborhood of $\\theta=0$, would refute Theorem 2.3.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Formulates the generalized Brauer-Siegel conjecture and supplies the key limsup bound $\\limsup \\log\\rho_K/g_K \\le \\sum_q \\varphi_q \\log(q/(q-1))$ used in the proof of Theorem 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Lagarias-Odlyzko effective Chebotarev estimates control the counting function $G_K(x)$ and provide the main input for the unconditional gamma-bounds in Theorems 2.1 and 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Stark's partial-fraction lemma for $Z_K$ and his exceptional-zero result for almost normal fields give the $O(1)$ bound on $|Z_K(1+\\theta)-Z_K(1)|$ and the hypothesis of Theorem 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends Stark's exceptional-zero control to fields with solvable normal closure, supplying the hypothesis of Theorem 2.2."},{"cited_title":"Ihara, On the Euler-Kronecker constants of global fiel ds and primes with small norms, Algebraic geometry and number theory, Progr","cited_arxiv_id":null,"evidence_quote":"Introduces the Euler-Kronecker constant and establishes the basic analytic identity identifying it as the constant term of $\\zeta'_K/\\zeta_K$ at $s=1$."},{"cited_title":"Mourtada, V","cited_arxiv_id":null,"evidence_quote":"Provides the 'almost all primes' bounds $1 \\ge \\gamma_p/\\log p > -11$ used in Proposition 3.1 for cyclotomic fields."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the previous best explicit zero-counting bound for Dedekind zeta functions, against which Proposition 3.1 is compared."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the counterexample to Ihara's positivity conjecture for $\\gamma_p$ and distributional facts about cyclotomic Euler-Kronecker constants used as context for the cyclotomic results."}],"review_version":1}