An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields

Brandon Alberts

REVIEW 3 major objections 3 minor 20 references

An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves a Tauberian theorem whose error term is controlled by twisted-moment averages rather than by pointwise vertical bounds of the Dirichlet series, and uses it to obtain unconditional square-root-saving error bounds for the num

desk verdict The averaged-input Tauberian theorem is a real new tool and its proof looks sound; the applications, however, rest on a moment lemma that is false as stated, so the corollaries are conditional pending a fix. read the letter →

arxiv 2508.20814 v1 pith:U245ZNO4 submitted 2025-08-28 math.NT

Brandon Alberts This is my paper · ORCID

classification math.NT MSC 11M4511R4511M06

keywords TauberiantheoremDirichletseriestwistedmomentsabeliannumberfieldsdiscriminantcountingcyclicextensionsDedekindzetasquare-rootsavings

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

This paper establishes a new Tauberian theorem for Dirichlet series: to pass from coefficient averages to an asymptotic for a summatory function, one no longer needs a pointwise bound on |L(σ+it)| along vertical lines. Instead, a bound on twisted-moment averages of L is enough, and the resulting error term is tracked explicitly in terms of the average-growth exponent, the pole data, and contour values. The theorem is stated self-containedly so that other counting problems can use it directly. As a demonstration, the paper proves unconditional square-root-saving error bounds for counting cyclic extensions of Q by discriminant for C3, C4, C6, C8, C16, and C2p—families where such error terms were previously available only conditionally or not at all.

What carries the argument

The central object is the twisted-moment integral of the Mellin transform along a vertical line, together with an auxiliary average of the contour shift in Perron's formula. The method smooths N by repeated integration, represents the smooth sum as a contour integral, shifts the contour in a u-dependent way while averaging over u, and then unsmooths using finite-difference operators. Each shifted contour piece is bounded by the twisted-moment hypothesis and integration by parts, not by pointwise estimates on L. A single floating parameter T controls the trade-off between the residue terms, the averaged integral on the leftmost line, and the outer contour pieces, and is then optimized.

What would settle it

Evaluate I_3(5/6, ζ_{Q(ζ_6)}; T) = ∫_0^T |ζ_{Q(ζ_6)}(5/6+it)|^6 dt for growing T: Lemma 5.2 predicts growth at most T (log T)^18. If the measured log-power or growth exponent is larger, the moment input used for C6 fails and the unconditional error bound in Corollary 1.3 for C6 does not follow, even though the conditional theorem remains valid.

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Extended reading notes

Core claim

The central claim is Theorem 2.1: given a nondecreasing counting function N whose Mellin transform L(s,N) is meromorphic up to Re(s)=σ_a−δ and whose twisted averages satisfy a polynomial bound, the difference between N(X) and the sum of residues S_0(X) is bounded by explicit expressions involving only the moment exponents η, β, the constant Q, the residues, and the size of L(s)/s on a fixed contour. No pointwise bound of the form |L(σ+it)| ≪ (1+|t|)^ξ is used anywhere. Optimizing a floating contour parameter T yields the error X^{σ_a−δ}/max{η,1} (log X)^θ. For the applications, the twisted-moment hypothesis is verified by Hölder's inequality and integral-moment bounds for Dedekind zeta funct

Load-bearing premise

The load-bearing premise is that the twisted-moment bounds hold for the generating series used, which the paper verifies via Hölder and the Dedekind-zeta moment bound Lemma 5.2; that lemma is cited for the first moment only and assumed to extend to all higher integer moments.

Editorial extensions

If this is right

If the twisted-moment hypothesis holds, Tauberian error terms depend only on average growth of L, not on pointwise growth, as summarized in Theorem 1.1 and made explicit in Theorem 2.1.
Known pointwise bounds imply the new form by taking η=ξ+1+ε, recovering earlier errors of the shape O(X^{σ_a−δ}/(ξ+1+ε)).
Existing integral-moment bounds for L-functions can be plugged in 'out of the box' to prove new asymptotic expansions, for example in counting ideals in abelian number fields.
The counts of C_n-extensions of Q of bounded discriminant have unconditional square-root-saving error terms for n=3,4,6,8,16 and n=2p for odd primes p.
The method separates the contour-shifting part from the use of a functional equation, so future improvements in moment bounds can be converted directly into better counting errors.

Reading between the lines

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

The same template should convert any improved bound for twisted moments of L-functions into improved counting error bounds, so conjectural moment results would have immediate arithmetic consequences.
The restriction to C3, C4, C6, C8, C16, and C2p appears to reflect the availability of Dedekind-zeta moment bounds rather than a limitation of the theorem; verifying additional moment bounds would likely yield square-root savings for other abelian groups.
Because the error terms are explicit, one could first evaluate R_N(X) for a concrete Dirichlet series and then optimize T numerically, producing fully explicit asymptotic inequalities rather than order-of-magnitude statements.
The same averaged-input contour argument could be applied to lattice-point and divisor-sum problems whose Mellin transforms have known moment bounds but no usable pointwise vertical bounds.

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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 core of this paper is a Tauberian theorem whose error term depends only on twisted-moment bounds, not pointwise bounds. That is genuinely new, and the proof via finite differencing with an auxiliary average is written out carefully. The explicit tracking of constants makes it a usable tool for arithmetic counting problems. I believe Theorem 2.1 is correct as proven.

The problem is the applications. Lemma 5.2, as stated, is false. For K=Q, m=1 it claims ∫|ζ(σ+it)|² dt ≪ T for every σ>0, but for 0<σ<1/2 the integral is of order T^{2−2σ}, which is larger than T. The author notes that [CN63] only proves the m=1 case and asserts the higher moments are “proven similarly,” but no approximate functional equation for ζ_K(s)^m is supplied. That is a real gap, not just a missing reference.

That said, the specific applications use the lemma at arguments ≥1/2, where the bounds are actually true: at σ=1/2 you get T log^{m²} T, and for σ>1/2 you get O(T). So Corollary 1.3 may well survive with a corrected lemma, and the error terms would even improve in some cases. But as written, the proof does not establish the stated bounds. The log exponents in Lemma 5.2 look wrong for general [K:Q], and the higher-m moment bounds are unsupported. The C3 case, for instance, uses m=1 for Q(ζ3) at real part 1/2; the true log power there is 1, not 2, so the paper's claimed error term is weaker than what a correct argument would give.

The sieve from étale algebras to fields is summarized rather than shown, but it follows an existing template, and the explicit constant computation in Proposition 5.5 is a nice addition. The author is candid about the CN63 caveat, which I appreciate.

I would send this to referees, with a clear request to fix Lemma 5.2 before publication. The Tauberian theorem itself is solid and publishable; the applications need revision to either prove the needed moment bounds or state them from a correct source.

Referee Report

3 major / 3 minor

Summary. The paper proves an explicit Tauberian theorem (Theorem 2.1, with the less explicit form Theorem 1.1) that bounds the difference between a summatory function and its residue expansion using only averaged/twisted moment bounds of the form ∫_T^{2T} L(σ+it)Z^{it} dt, rather than pointwise vertical bounds |L(σ+it)| ≪ (1+|t|)^ξ. The proof follows Landau's finite-differencing method: smoothing, contour shift with an auxiliary average, then unsmoothing. Sections 3–4 are detailed and internally coherent. The paper then applies the theorem to counting C_n-extensions of Q for n = 3, 4, 8, 16, and 2p, claiming unconditional square-root-saving error terms. The applications reduce the twisted-moment hypothesis to Hölder-type bounds for integral moments of Dedekind zeta functions, via Lemmas 5.2 and 5.3.

Significance. The Tauberian theorem itself is a genuine and useful contribution: it removes the pointwise-growth input that is standard in this circle of methods, tracks parameter dependence explicitly, and the contour argument with the auxiliary average is inventive. The paper also correctly identifies the connection to moments of L-functions. If the moment inputs were fully justified, the corollaries would be a significant improvement for the listed cyclic groups. However, the application section rests on moment assertions that are not established as stated, including a visibly false special case. The lasting value of the paper is therefore the general Tauberian framework; the numerical corollaries need correction or re-proof before they can be accepted as unconditional.

major comments (3)

[§5.2, Lemma 5.3] The lemma is false as stated. For K=Q and m=1 it asserts I_1(σ, ζ; T) = ∫_0^T |ζ(σ+it)|^2 dt ≪ T for every σ > 0. But for 0 < σ < 1/2 the functional equation gives |ζ(σ+it)|^2 ≍ t^{1−2σ}|ζ(1−σ+it)|^2, and since 1−σ > 1/2, the integral grows like T^{2−2σ} up to log factors, not O(T). The author's own caveat states that [CN63, Thm 4] is only for m=1; but even the m=1 statement is not a faithful restatement. Thus the assertion that the m≥2 cases are 'proven similarly' is unsupported, and the higher-m bounds used in the C4, C6, C8, C16, and C2p proofs are not established. This is load-bearing: those corollaries' error exponents depend directly on these moment bounds. A correct proof or reference for the needed higher moments, and a corrected threshold, are required.
[§5.2, Lemma 5.3] Lemma 5.3 asserts I_z(σ, ζ_K; T) ≪ T for every complex z and every σ ≥ 1, but I_z is not defined for negative z, and the applications use I_{-2} and I_{-4} at the boundary argument 4σ = 1 (e.g., in the C3, C4, C6, and C8 proofs). Negative powers require lower bounds on |ζ_K(σ+it)|, not just upper bounds. The quoted [BIR93, Theorem 3] and the remark about Brauer induction concern positive powers/Artin L-functions; they do not cover the reciprocal of ζ_K. Even for z=-1, K=Q, standard lower bounds give only |ζ(1+it)|^{-1} ≪ log t, which would produce a log factor in the integral. The exact β-values in Corollary 1.3 are therefore not justified by the text, even though the main power saving might survive with a larger logarithm exponent.
[§5.3, proof for C16] The C16 proof states that the 'first four factors' are bounded using Lemma 5.2. But one of those factors is ∫ |ζ_{Q(ζ16)}(15σ+15it)|^{-4} dt, which has a negative exponent and is outside the range m ≥ 0 of Lemma 5.2. It must rely on Lemma 5.3 (or another negative-moment input). This misattribution, together with the issues in Lemma 5.3, means the C16 error term as printed is not proven. The same pattern appears in the C2p proof. This is a local but consequential error in the application section.

minor comments (3)

[§2.1] The integrals defining the polynomial bound for twisted moments are printed as ∫_T^{T0} L(σ+it,N) Z^{it} dt. With T0 fixed, this is empty for T > T0; presumably the intended range is ∫_{T0}^{T} or ∫_T^{2T}. The proofs in Section 4 consistently use intervals [T0,T] or [T,2T], so this is a typographical/notation issue, but it should be fixed.
[§5.3, C16 proof] The line 'asserting the number of C8-étale algebras is given by' should read 'C16-étale algebras' in the sentence just after applying Theorem 1.1.
[Throughout] The title in the header contains 'A VERAGED'; presumably 'AVERAGED'. There is also a recurring typo 'Dirichelt' for 'Dirichlet' in Section 5. These are harmless but should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Tauberian theorem is derived self-contained from its stated average-growth hypotheses; applications import external moment bounds, and the paper's self-citations are not used to define the main result.

full rationale

The core result, Theorem 2.1 / Theorem 1.1, is proved by Landau's finite-differencing method entirely from the stated hypotheses: meromorphic continuation, polar data, and the twisted-moment bounds (1.1). The auxiliary average N(X) = (1/T)∫_T^{2T} N(X) du is trivially an identity, and the contour-shifting argument uses integration by parts to convert (1.1) into bounds on the horizontal and outer vertical pieces. The unsmoothing step then recovers N(X) from its Riesz means via standard monotonicity inequalities. There is no step in which the conclusion is assumed or in which the error term is defined to be the input: the final bound is expressed in terms of η, β, Q and the residue data, but the residue data describe the main term, not the error, and the average bounds are external analytic inputs. The applications verify (1.1) by the trivial bound and Hölder, using moment bounds for Dedekind zeta functions from [CN63], [BIR93], and [Ten15]; these are independent external results, not consequences of the counting theorems being proved. The paper does cite the author's prior work [Alb24b] for the shape of the étale generating series (also reproved here in Proposition 5.1), for a nonvanishing criterion for constants, and for a prior power-saving count of C_p-extensions used in the C_{2p} subtraction. These self-citations are normal uses of prior results and are not the load-bearing justification for the Tauberian theorem itself; the cited results do not depend on the present paper. The most serious caveat is Lemma 5.2: the author admits [CN63, Theorem 4] is only stated for m=1, asserts the higher-m cases are 'proven similarly' without supplying the approximate functional equation for ζ_K(s)^m, and the lemma as stated is actually false for K=Q, m=1 when 0<σ<1/2. This is a correctness gap in the application chain, potentially invalidating the unconditional claims for C_4, C_6, C_8, C_16, and C_{2p}, but it is not a circularity: the moment bounds are inputs assumed for Theorem 2.1, and the theorem's derivation does not reduce to them by construction. Circularity score is therefore 0, with the correctness risk noted separately.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central theorem introduces no fitted constants: eta, beta, r_eta, Q, and delta are hypotheses supplied by the user and tracked explicitly, not derived from the target counting data. The applications rely on external moment theorems for Dedekind zeta functions (Lemmas 5.2-5.4) and on the author's earlier meromorphic continuation work [Alb24b]. The three domain_assumption entries above are the load-bearing inputs for Corollary 1.3; no new entities are postulated.

assumptions (4)

domain assumption Moments of Dedekind zeta functions satisfy the bounds of Lemma 5.2 for all integers m >= 0, extending [CN63, Theorem 4] beyond its stated m=1 case.
Used in Section 5.3 to bound the first moment of the generating Dirichlet series via Hoelder's inequality. The author asserts the higher m values are proven similarly, but no proof is supplied in the paper.
domain assumption Lemma 5.3 ([BIR93, Theorem 3]) applies with complex and negative z, in particular to inverse powers of zeta functions at sigma >= 1.
Used in Section 5.3 to bound integrals of inverse zeta powers such as |zeta(4 sigma + i t)|^{-4}; the paper does not demonstrate that the cited theorem covers negative exponents.
domain assumption The generating Dirichlet series D_et^Q(G; s) admits a meromorphic continuation to Re(s) >= 1/(2 a(G)), as stated in Proposition 5.1, building on [Alb24b, Cor 3.3] and Wright's work.
Proposition 5.1 is proved in the paper, but its starting point is the author's prior result [Alb24b]. This continuation is the analytic precondition for applying Theorem 1.1 to the counting problems.
domain assumption Tenenbaum's inverse-zeta bound |zeta(sigma + i t)|^{-1} << log|t| at sigma >= 1 - c/log|t| (Lemma 5.4).
Cited from [Ten15] and used in the C16 and C2p cases to handle the factor zeta(2 a(G) s)^{-1} at the edge of the region. No proof is included, as it is a standard cited result.

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Pith. "Pith review of An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields." pith.science (2026). https://pith.science/paper/U245ZNO4

@misc{pith2026250820814,
  author       = {Pith},
  title        = {Pith review of: An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U245ZNO4}},
  note         = {Machine review of arXiv:2508.20814}
}

abstract

Given a Dirichlet series $L(s) = \sum a_n n^{-s}$, the asymptotic growth rate of $\sum_{n\le X} a_n$ can be determined by a Tauberian theorem. Bounds on the error term are typically controlled by the size of $|L(\sigma+it)|$ for fixed real part $\sigma$. We modify this approach to prove new Tauberian theorems with error terms depending only on the average size of $L(\sigma+it)$ as $t$ varies, and we take care to track explicit dependence on various parameters. This often leads to stronger error bounds, and introduces strong connections between asymptotic counting problems and moments of $L$-functions. We provide self-contained statements of Tauberian theorems in anticipation that these results can be used ``out of the box'' to prove new asymptotic expansions. We demonstrate this by proving square root saving error bounds for the number of $C_n$-extensions of $\mathbb{Q}$ of bounded discriminant when $n=3$, $4$, $8$, $16$, or $2p$ for $p$ an odd prime.

Figures

Figures reproduced from arXiv: 2508.20814 by the authors.

**Figure 1.** The integrals over C and γ are independent of u so that lim TÑ8 1 T ż 2T T ˆż C ` ż γ ˙ “ ż C ` ż γ , as they appear in the statement. It now suffices to evaluate the limit for the remaining contour integrals. 3.0.1. The outer vertical integrals. In this section we evaluate lim TÑ8 1 2πiT ż 2T T ż c˘i8 c˘iu Lps, Nq Γpsq Γps ` k ` 1q X s`k dsdu. Restricting to k ě 2 we bound [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Explicit analytic continuation of euler products

Brandon Alberts. Explicit analytic continuation of euler products. arXiv preprint arXiv:2406.18190 , 2024. Preprint available at https://arxiv.org/abs/2406.18190

arXiv 2024

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arXiv 2024

[3] [3]

An application of the H ooley- H uxley contour

R Balasubramanian, A Ivi \'c , and K Ramachandra. An application of the H ooley- H uxley contour. Acta Arithmetica , 65(1):45--51, 1993

work page 1993

[4] [4]

Functional equations with multiple gamma factors and the average order of arithmetical functions

K Chandrasekharan and Raghavan Narasimhan. Functional equations with multiple gamma factors and the average order of arithmetical functions. Annals of Mathematics , 76(1):93--136, 1962

work page 1962

[5] [5]

The approximate functional equation for a class of zeta-functions

K Chandrasekharan and Raghavan Narasimhan. The approximate functional equation for a class of zeta-functions. Mathematische Annalen , 152(1):30--64, 1963

work page 1963

[6] [6]

L-functions and random matrices

J Brian Conrey. L-functions and random matrices. In Mathematics unlimited—2001 and Beyond , pages 331--352. Springer, 2001

work page 2001

[7] [7]

The H asse norm principle for abelian extensions

Christopher Frei, Daniel Loughran, and Rachel Newton. The H asse norm principle for abelian extensions. American Journal of Mathematics , 140(6):1639–1685, 2018

work page 2018

[8] [8]

Sharp conditional bounds for moments of the riemann zeta function

Adam J Harper. Sharp conditional bounds for moments of the riemann zeta function. arXiv preprint arXiv:1305.4618 , 2013. Preprint available at https://arxiv.org/abs/1305.4618

arXiv 2013