Higher Euler-Kronecker Constants of Number fields

Samprit Ghosh

REVIEW 2 major objections 6 minor 1 cited by

Higher Euler-Kronecker Constants of Number fields

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.17946 v1 pith:LVZ7IF54 submitted 2024-11-26 math.NT

Samprit Ghosh This is my paper · ORCID

classification math.NT MSC 11R4211M0611M3811M2011M36

keywords Euler-KroneckerconstantsDedekindzetafunctionLaurentseriescoefficientsvonMangoldtexplicitformulaLi'scriteriongeneralizedRiemannhypothesisStieltjes

open problems The Riemann Hypothesis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $\zeta_K' / \zeta_K$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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).

There 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.

On 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.

My 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.

Referee Report

2 major / 6 minor

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.

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 (2)

[Section 2, Proposition 2.2; Section 3, proof of Theorem 1.4] 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 2, Eq. (12); Section 3, Corollary 3.1 and Remark 3.2] 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.

minor comments (6)

[Section 1.2, Theorem 1.5] 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 1.2, Theorem 1.4] 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 2, Proposition 2.1] 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 5, Lemmas 5.1 and 5.4] 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 5, Lemma 5.5] 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 1.2, Theorem 1.5 and Remark 1.6] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: both main formulas are derived from the defining Laurent expansion via partial summation and explicit formulas, with no fitted input or load-bearing self-citation.

full rationale

The paper defines gamma_{K,r} as Laurent coefficients of zeta'_K/zeta_K at s=1 and then proves arithmetic formulas for them. Theorem 1.4's formula (4) is obtained by applying Proposition 2.2, which is a partial-summation identity: C_r is expressed as the limit of sum b_n log^r n / n minus C/(r+1) log^{r+1} x. The right-hand side is not used to define gamma_{K,r}; it is a consequence of the Laurent expansion and the identity b_n = Lambda_K(n). The proof is self-contained apart from standard analytic number theory (prime ideal theorem error terms, Serre's GRH bound, Lagarias-Odlyzko zero-free region, Stark's exceptional-zero lemma). Theorem 1.8's function f(r,x) is computed exactly in Lemma 5.4 as the contribution of the poles at s=0 and s=1; it is not fitted to the claimed limit. Lemma 5.5 and Lemma 5.6 evaluate the zero and Gamma contributions by contour integration. The resulting identity (7) is equivalent to the explicit formula (19), not to a restatement of the definition. The cited results are external (Ihara, Serre, Lagarias-Odlyzko, Stark, Dixit-Murty) and the paper contains no load-bearing self-citation. A caveat is that Proposition 2.2 is stated under E(x)=O(x^b) with b<1, which is stronger than the known unconditional error term for the prime ideal theorem; this is a proof gap in the unconditional formulation, not circularity. The conditional GRH bounds are conditional on an external hypothesis and are not circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters were fitted to data; the choice of the auxiliary variable x in the proof of Theorem 1.5 is a proof device, not a parameter of the final statement. The central claims rest on standard analytic number theory: continuation and explicit formulas for Dedekind zeta functions, prime ideal theorem error bounds, GRH for the conditional estimates, zero-free regions and zero-counting for the contour argument, and Minkowski's discriminant bound. The higher Euler-Kronecker constants are definitions from an existing Laurent expansion, not independently posited entities.

assumptions (5)

standard math Dedekind zeta function zeta_K has analytic continuation with only a simple pole at s=1 and a Hadamard product/explicit formula.
Invoked throughout; needed for the Laurent expansion (3), for the explicit formula (16), and for the contour residue computations in Section 5.
domain assumption The prime ideal theorem error term E(x)=B(x)-x is O(x^b) for some 0 at most b less than 1.
Proposition 2.1 requires this growth to justify partial summation and taking the limit x to infinity in Theorem 1.4; for Dedekind zeta functions it is a standard consequence of the prime ideal theorem.
domain assumption GRH for zeta_K: every nontrivial zero has real part 1/2.
Explicitly assumed in Theorem 1.5 and used at (14) through Serre's bound for Delta_K(x).
standard math Lagarias-Odlyzko zero-free region and Iwaniec-Kowalski zero-counting estimate (Lemma 4.1).
Used in Lemma 5.5 to justify the limit of the zero contribution and in Theorem 4.3 to bound sums over zeros.
standard math Minkowski's discriminant bound, giving n_K/log|d_K| bounded.
Used in the proof of Theorem 1.5 to absorb the n_K log x term inside the GRH error estimate.

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Cite this review

Pith. "Pith review of Higher Euler-Kronecker Constants of Number fields." pith.science (2026). https://pith.science/paper/LVZ7IF54

@misc{pith2026241117946,
  author       = {Pith},
  title        = {Pith review of: Higher Euler-Kronecker Constants of Number fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LVZ7IF54}},
  note         = {Machine review of arXiv:2411.17946}
}

abstract

The higher Euler-Kronecker constants of a number field $K$ are the coefficients in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function about $s=1$. These coefficients are mysterious and seem to contain a lot of arithmetic information. In this article, we study these coefficients. We prove arithmetic formulas satisfied by them and prove bounds. We generalize certain results of Ihara.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Moments of Higher Logarithmic Derivatives and Exceptional Zeros in Cyclotomic Fields
math.NT 2025-09 reject novelty 5.0 of 10

Higher logarithmic-derivative moments at s=1 over Dirichlet characters of prime modulus converge to an explicit series, but the advertised exceptional-zero application is missing from the body.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

[1]

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Sumaia Saad Eddin, The signs of the Stieltjes constants associated with the Dedekind zeta function, Proc. Japan Acad., 94, Ser. A, No. 10 (2018)

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Bombieri and J

E. Bombieri and J. C. Lagarias, Complements to Li’s criterion for the Riemann hypothesis., Journal of Number Theory , 77 (1999), 274–287

work page 1999

[2] [2]

W . E. Briggs and R. G. Buschman, The power series coefﬁcients of functions deﬁned Dirichlet series, Illinois Journal of Mathematics 5 (1961), no. 1, 43 – 44

work page 1961

[3] [3]

F . C. Brown, Li’s criterion and zero-free regions of L-functions., Journal of Number Theory , 111 (2005), 1–32

work page 2005

[4] [4]

Cassels and A

J.W .S. Cassels and A. Fr ¨ olich,Algebraic Number Theory , Proceedings of an Instructional Conference by London Mathematical Society and International Mathematic al Union, Academic press., 1976

work page 1976

[5] [5]

Davenport, Multiplicative Number Theory , Third edition, Springer V erlag, New Y ork, 2000

H. Davenport, Multiplicative Number Theory , Third edition, Springer V erlag, New Y ork, 2000

work page 2000

[6] [6]

Murty , On Ihara’s conjectures for Euler-Kronecker constants , Acta Arithmetica 210 (2023), 95–123

Anup Dixit and Ram M. Murty , On Ihara’s conjectures for Euler-Kronecker constants , Acta Arithmetica 210 (2023), 95–123

work page 2023

[7] [7]

Japan Acad., 94, Ser

Sumaia Saad Eddin, The signs of the Stieltjes constants associated with the Dedekind zeta function, Proc. Japan Acad., 94, Ser. A, No. 10 (2018)

work page 2018

[8] [8]

Hermite and T

C. Hermite and T. J. Stieltjes, Correspondance d’Hermite et de Stieltjes, I and II, , edited by B. Baillaud and H. Bourget, Gauthier-Villars, Paris (1905)

work page 1905