REVIEW 4 major objections 5 minor 20 references
On Euler-Kronecker constants and the generalized Brauer-Siegel conjecture
T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A weak bound on Euler-Kronecker constants in a tower of number fields forces the generalized Brauer-Siegel conjecture.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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 $p$.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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).
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 (4)
- [Section 2.3.3, paragraph before Eq. (22)] 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 2.3.3, after Eq. (25)] 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 2.2.3, Lemma 2.4] 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 2.3.3, proof of Eq. (25)] 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.
minor comments (5)
- [Section 2.3.3, display before Eq. (22)] 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 2.3.3, Eq. (27)] 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.
- [Throughout] 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 2.2.3, Lemma 2.4] 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 2.3.2, proof of Theorem 2.2] 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.
Circularity Check
No significant circularity: Theorem 2.3 derives GBS from a genuinely independent gamma-bound hypothesis and external partial results.
full rationale
The central claim is not a rearrangement of its inputs. The hypothesis |gamma_Ki| << exp((log g_Ki)^m) bounds the Euler-Kronecker constant, while the conclusion BS(K)=1+sum_q phi_q log(q/(q-1))-phi_R log 2-phi_C log 2pi is a limit built from phi_q, phi_R, and phi_C. In the proof of Theorem 2.3, the gamma-bound is used only to control the normalized entire factor log F(1+theta)/g via equations (25)-(27), and to choose theta_Ki = exp(-(log g)^(m+2)); it never enters the phi_q sums. The lower bound for liminf log zeta/g rests on the tower monotonicity N_p(K_i)/g_i >= phi_p and uniform convergence of higher prime-power terms, not on the hypothesis. The corresponding upper bound for limsup log rho/g is quoted from Tsfasman-Vladut [18], an external work, and is not derived from or equivalent to the gamma hypothesis. Whether that quoted inequality is unconditional is a correctness/hypotheses concern, not a circularity concern. The self-citations to [2] and [3] concern a proof detail for Theorem 2.2 and Jensen's zero-counting estimate; these are standard ingredients and are not load-bearing in a way that reduces the theorem to itself. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported, and no known result is merely relabeled.
Assumptions & free parameters
assumptions (7)
- domain assumption Stark's exceptional-zero theorem: for an almost normal field K with a real zero β of ζ_K(s) in 1 - 1/(16 log|d_K|) < β < 1, there is a quadratic subfield N ⊂ K with ζ_N(β) = 0.
- domain assumption V.K. Murty's solvable-closure theorem: if K has solvable normal closure over Q and ζ_K(s) has a real zero in 1 - c/(n e(n) δ(n) log|d_K|) ≤ β < 1, then a quadratic subfield N ⊂ K satisfies ζ_N(β) = 0.
- domain assumption Lagarias-Odlyzko effective Chebotarev estimate: |G_K(x) - x| ≤ C1 x exp(-C2 sqrt(log x/n)) + x^β/β for log x ≥ C3 n g_K^2.
- domain assumption Quoted limsup inequality from [18]: for any asymptotically exact family K, limsup log ρ_Ki/g_Ki ≤ Σ_q φ_q log(q/(q-1)).
- standard math Riemann-von Mangoldt zero-counting formula: N_K(T) = (T/π) log(|d_K| (T/2πe)^{n_K}) + O(log(|d_K| T^{n_K})).
- domain assumption Generalized Riemann hypothesis for Dedekind zeta functions, assumed in Proposition 3.1.
- standard math Dirichlet class number formula: ρ_K = 2^{r1}(2π)^{r2} h_K R_K / (ω_K sqrt(|d_K|)).
Cite this review
Pith. "Pith review of On Euler-Kronecker constants and the generalized Brauer-Siegel conjecture." pith.science (2026). https://pith.science/paper/NBCRH6WG
@misc{pith2026190803044,
author = {Pith},
title = {Pith review of: On Euler-Kronecker constants and the generalized Brauer-Siegel conjecture},
year = {2026},
howpublished = {\url{https://pith.science/paper/NBCRH6WG}},
note = {Machine review of arXiv:1908.03044}
}
abstract
As a natural generalization of the Euler-Mascheroni constant $\gamma$, Y. Ihara introduced the Euler-Kronecker constant $\gamma_K$ attached to any number field $K$. In this paper, we prove that a certain bound on $\gamma_K$ in a tower of number fields $\mathcal{K}$ implies the generalized Brauer-Siegel conjecture for $\mathcal{K}$ as formulated by Tsfasman and Vl\v{a}du\c{t}. Moreover, we use known bounds on $\gamma_K$ for cyclotomic fields to obtain a finer estimate for the number of zeros of the Dedekind zeta-function $\zeta_K(s)$ in the critical strip.
Reference graph
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