A guide to Tauberian theorems for arithmetic applications

Asif Zaman; Caroline L. Turnage-Butterbaugh; Lillian B. Pierce

arxiv: 2504.16233 · v3 · submitted 2025-04-22 · 🧮 math.NT

A guide to Tauberian theorems for arithmetic applications

Lillian B. Pierce , Caroline L. Turnage-Butterbaugh , Asif Zaman This is my paper

Pith reviewed 2026-05-22 18:09 UTC · model grok-4.3

classification 🧮 math.NT

keywords Tauberian theoremsDirichlet seriespartial sumsasymptoticsanalytic number theorycounterexamplesremainder termsarithmetic applications

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The pith

Tauberian theorems deduce partial sum asymptotics from Dirichlet series properties, and this guide details two versions along with counterexamples proving their hypotheses cannot be relaxed.

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

The paper sets out to give practitioners a clear reference for using Tauberian theorems in analytic number theory and related areas such as arithmetic statistics and algebraic geometry. It walks through the exact hypotheses and proofs for one theorem that produces an asymptotic with an explicit remainder and another that does not. Counterexamples are supplied to show that neither theorem can deliver a meaningfully stronger conclusion unless its analytic assumptions are made stricter. A sympathetic reader would value the guide because it reduces the risk of misapplying these tools in new settings by making the necessary conditions transparent and verifiable.

Core claim

By laying out the hypotheses, complete proofs, and targeted counterexamples for a Tauberian theorem with an explicit remainder term and one without, the paper establishes that these statements are sharp: each reaches the strongest conclusion possible under its given conditions on the Dirichlet series, and the counterexamples demonstrate that relaxing any hypothesis allows sequences where the expected asymptotic fails.

What carries the argument

The pair of Tauberian theorems—one supplying an asymptotic with explicit error term and the other without—which convert analytic information about a Dirichlet series into an asymptotic for the partial sums of a sequence of non-negative coefficients.

If this is right

In any arithmetic setting where the Dirichlet series meets the stated analytic conditions, the corresponding partial-sum asymptotic follows directly.
The supplied counterexamples rule out stronger error terms or weaker analytic hypotheses without additional work.
Applications in arithmetic statistics or intersections with algebraic geometry can cite the precise hypotheses rather than re-deriving them.
Proofs given in the paper allow direct checking of the conditions in new contexts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sharpness-testing approach could be applied to other analytic tools used in number theory to clarify their limits.
Hybrid statements that combine features of both theorems might be derived for applications needing both an error term and broader applicability.
The guide suggests that future arithmetic results should routinely include a sharpness check to avoid overclaiming.

Load-bearing premise

The two selected Tauberian theorems are the most relevant and representative ones for the wide range of arithmetic applications the authors have in mind.

What would settle it

An explicit sequence of non-negative coefficients whose Dirichlet series satisfies the hypotheses of one of the stated theorems yet whose partial sums fail to obey the claimed asymptotic.

read the original abstract

A Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, ranging from well-known classical applications in analytic number theory, to new applications in arithmetic statistics, group theory, and the intersection of number theory and algebraic geometry. The goal of this article is to provide a useful reference for practitioners who wish to apply a Tauberian theorem. We explain the hypotheses and proofs of two types of Tauberian theorems: one with and one without an explicit remainder term. We furthermore provide counterexamples that illuminate that neither theorem can reach an essentially stronger conclusion unless its hypothesis is strengthened.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a clear expository guide to two standard Tauberian theorems with proofs and sharpness counterexamples that should serve as a practical reference for arithmetic applications.

read the letter

Hi, This paper is basically a practical handbook for two common Tauberian theorems used in analytic number theory. It lays out the hypotheses, sketches the proofs, and gives counterexamples to show that the conclusions are sharp. It does well at being clear and direct. The counterexamples are set up for non-negative coefficients without extra structure, which correctly demonstrates that you need stronger hypotheses for stronger results. This should help people avoid misusing the theorems in their arithmetic applications. The work is mostly expository, so nothing fundamentally new, but the organization and the explicit sharpness examples add something useful. It matches the standard references in the area. Soft spots are small. Focusing on just these two types is fine for the stated goal, but it might leave out some newer or more specialized variants that some readers encounter. The counterexamples stay general, so they don't illustrate with concrete number theory examples, which could make the point land a bit less vividly for some. Overall this is for working number theorists and arithmetic statisticians who need to apply these tools reliably. A reader looking for a reference to check conditions or see limits would find it worthwhile. It deserves a serious referee. The paper is honest about what it is and delivers on its promises without overreaching. I'd say send it out for review.

Referee Report

0 major / 2 minor

Summary. The manuscript is an expository guide to Tauberian theorems for arithmetic applications. It explains the hypotheses and sketches the proofs of two standard types of such theorems (one with an explicit remainder term and one without), and supplies counterexamples showing that neither conclusion can be strengthened without a corresponding strengthening of the hypothesis on the Dirichlet series, with the counterexamples constructed for non-negative coefficients in the appropriate generality.

Significance. If the statements, proof sketches, and counterexamples are accurate as described, the paper would provide a practical reference for practitioners in analytic number theory, arithmetic statistics, and related fields. The explicit focus on sharpness via counterexamples without additional arithmetic structure correctly addresses the generality needed to establish optimality, which is a strength for avoiding misapplications. This aligns with standard references in the field and could aid users in selecting appropriate theorems for their Dirichlet series.

minor comments (2)

The abstract states the goals clearly but does not name the specific theorems (e.g., by reference to Wiener-Ikehara or Ingham-Karamata type results); adding a brief parenthetical identification would help readers locate the precise statements in the body.
In the sections sketching the proofs, ensure that the transition from the analytic hypothesis on the Dirichlet series to the Tauberian conclusion is accompanied by an explicit pointer to the key lemma or step that handles the non-negativity of coefficients.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing its potential utility as a reference in analytic number theory and arithmetic statistics, and for recommending minor revision. We are pleased that the focus on sharpness via counterexamples in the stated generality is viewed as a strength.

Circularity Check

0 steps flagged

Expository guide with no circular derivation

full rationale

The manuscript is a reference guide that states hypotheses, sketches proofs, and supplies counterexamples for two standard Tauberian theorems drawn from the existing literature in analytic number theory. No new derivation chain is claimed; the counterexamples are constructed directly for non-negative coefficients to demonstrate sharpness without any reduction to fitted parameters, self-referential equations, or load-bearing self-citations. The selection of theorems aligns with classical references and the paper remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an expository guide rather than a paper deriving new mathematical results, so no free parameters, axioms, or invented entities are introduced to support a central claim.

pith-pipeline@v0.9.0 · 5651 in / 1092 out tokens · 63552 ms · 2026-05-22T18:09:49.953408+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery theorem unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Hypothesis A... A(s) = g(s)/(s−α)^m + h(s) ... Theorem A: X_{λ_n ≤ x} a_n ∼ c x^α (log x)^{m−1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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[Alb24a] B. Alberts. Explicit analytic continuation of Euler products (arxiv:2406.18190),

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[Alb24b] B. Alberts. Power savings for counting (twisted) abelian extensions of number fields (arxiv:2402.03475),

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[Alb25] B. Alberts. An explicit Tauberian theorem taking averaged inputs with an application to count- ing abelian number fields, (arxiv:2508.20814),

work page arXiv
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Bhargava

[Bha07] M. Bhargava. Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants.International Mathematics Research Notices, 2007:20 pp.,

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[BLM21] F. Brumley, D. Lesesvre, and D. Mili´ cevi´ c. Conductor zeta function for the GL(2) universal family (arXiv:2105.02068),

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[Del63] H. Delange. Th´ eor` emes taub´ eriens et applications arithm´ etiques.S´ eminaire Delange-Pisot- Poitou, Th´ eorie des nombres, 4(expos´ e 16):1–17, 1962-1963. [DGH03] A. Diaconu, D. Goldfeld, and J. Hoffstein. Multiple Dirichlet series and moments of zeta and L-functions.Compositio Math., 139(3):297–360,

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du Sautoy and L

[dSW08] M. du Sautoy and L. Woodward.Zeta functions of groups and rings, volume 1925 ofLecture Notes in Mathematics. Springer-Verlag, Berlin,

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[Hil05] A. J. Hildebrand. Introduction to Analytic Number Theory, Math 531 Lecture Notes, Fall 2005 (version 2013.01.07),

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[Kab08] A

URL:https://mathoverflow.net/q/286546 (version: 2017-12-05). [Kab08] A. C. Kable. A variation of the Ikehara-Delange Tauberian theorem and an application.Com- ment. Math. Univ. St. Pauli, 57(2):137–146,

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[Lan17] E. Landau. ueber die heckesche funktionalgleichung.Nachrichten von der Gesellschaft der Wis- senschaften zu G¨ ottingen, Mathematisch-Physikalische Klasse, 1917:102–111,

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[Pie23] L. B. Pierce. Counting problems: class groups, primes, and number fields. InICM— International Congress of Mathematicians. Vol. III. Sections 1–4, pages 1940–1965. EMS Press, Berlin, [2023]©2023. [PTBW20] L. B. Pierce, C. L. Turnage-Butterbaugh, and M. M. Wood. An effective Chebotarev density theorem for families of number fields, with an applicat...

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Rademacher

[Rad60] H. Rademacher. On the Phragm´ en-Lindel¨ of theorem and some applications.Math. Z., 72:192– 204, 1959/60. [Rou11] M. Roux.Th´ eorie de l’information, s´ eries de Dirichlet, et analyse d’algorithmes. Th´ eorie de l’information [cs.IT]. Universit´ e de Caen,

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[1] [1]

[Alb24a] B. Alberts. Explicit analytic continuation of Euler products (arxiv:2406.18190),

work page arXiv

[2] [2]

[Alb24b] B. Alberts. Power savings for counting (twisted) abelian extensions of number fields (arxiv:2402.03475),

work page arXiv

[3] [3]

[Alb25] B. Alberts. An explicit Tauberian theorem taking averaged inputs with an application to count- ing abelian number fields, (arxiv:2508.20814),

work page arXiv

[4] [4]

Bhargava

[Bha07] M. Bhargava. Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants.International Mathematics Research Notices, 2007:20 pp.,

work page 2007

[5] [5]

Brumley, D

[BLM21] F. Brumley, D. Lesesvre, and D. Mili´ cevi´ c. Conductor zeta function for the GL(2) universal family (arXiv:2105.02068),

work page arXiv

[6] [6]

[BT98b] V. V. Batyrev and Y. Tschinkel. Tamagawa numbers of polarized algebraic varieties (Nombre et r´ epartition de points de hauteur born´ ee (Paris, 1996)).Ast´ erisque, 251:299–340,

work page 1996

[7] [7]

[Del63] H. Delange. Th´ eor` emes taub´ eriens et applications arithm´ etiques.S´ eminaire Delange-Pisot- Poitou, Th´ eorie des nombres, 4(expos´ e 16):1–17, 1962-1963. [DGH03] A. Diaconu, D. Goldfeld, and J. Hoffstein. Multiple Dirichlet series and moments of zeta and L-functions.Compositio Math., 139(3):297–360,

work page 1962

[8] [8]

du Sautoy and L

[dSW08] M. du Sautoy and L. Woodward.Zeta functions of groups and rings, volume 1925 ofLecture Notes in Mathematics. Springer-Verlag, Berlin,

work page 1925

[9] [9]

[Hil05] A. J. Hildebrand. Introduction to Analytic Number Theory, Math 531 Lecture Notes, Fall 2005 (version 2013.01.07),

work page 2005

[10] [10]

[Kab08] A

URL:https://mathoverflow.net/q/286546 (version: 2017-12-05). [Kab08] A. C. Kable. A variation of the Ikehara-Delange Tauberian theorem and an application.Com- ment. Math. Univ. St. Pauli, 57(2):137–146,

work page 2017

[11] [11]

[Lan17] E. Landau. ueber die heckesche funktionalgleichung.Nachrichten von der Gesellschaft der Wis- senschaften zu G¨ ottingen, Mathematisch-Physikalische Klasse, 1917:102–111,

work page 1917

[12] [12]

[Pie23] L. B. Pierce. Counting problems: class groups, primes, and number fields. InICM— International Congress of Mathematicians. Vol. III. Sections 1–4, pages 1940–1965. EMS Press, Berlin, [2023]©2023. [PTBW20] L. B. Pierce, C. L. Turnage-Butterbaugh, and M. M. Wood. An effective Chebotarev density theorem for families of number fields, with an applicat...

work page 1940

[13] [13]

Rademacher

[Rad60] H. Rademacher. On the Phragm´ en-Lindel¨ of theorem and some applications.Math. Z., 72:192– 204, 1959/60. [Rou11] M. Roux.Th´ eorie de l’information, s´ eries de Dirichlet, et analyse d’algorithmes. Th´ eorie de l’information [cs.IT]. Universit´ e de Caen,

work page 1959