arxiv: 2604.24505 · v1 · submitted 2026-04-27 · 🧮 math.CV · math.NT

Recognition: unknown

Newman's Tauberian theorem, the Riemann-Lebesgue Lemma, and abstract analytic number theory

Jan-Christoph Schlage-Puchta , Christoph Schwerdt

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:07 UTC · model grok-4.3

classification 🧮 math.CV math.NT

keywords Newman's Tauberian theoremRiemann-Lebesgue lemmap-variationprime number theoremMertens functionabstract analytic semigroupseffective estimatesanalytic number theory

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The pith

An effective Riemann-Lebesgue lemma for bounded p-variation functions yields a generalized Newman's Tauberian theorem with explicit error terms.

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

The paper establishes a generalized and effective version of Bekehermes' improvement of Newman's Tauberian theorem. It does so by proving an effective form of the Riemann-Lebesgue lemma that applies specifically to functions of bounded p-variation. This tool is then transferred to abstract analytic semigroups, producing a quantitative version of the prime number theorem and an explicit-error bound for Mertens' function. A sympathetic reader would care because the work replaces purely asymptotic conclusions with concrete quantitative estimates that can be checked or used in further calculations.

Core claim

We give a generalized and effective version of Bekehermes' improvement of Newman's Tauberian theorem. To do so we prove an effective version of the Riemann-Lebesgue Lemma for functions of bounded p-variation. We apply our Tauberian theorem to abstract analytic semigroups and prove a version of the prime number theorem as well as an estimate for Mertens' function with explicit error term.

What carries the argument

The effective Riemann-Lebesgue lemma for functions of bounded p-variation, which supplies the explicit error control needed to extend and quantify Newman's Tauberian theorem.

If this is right

A version of the prime number theorem holds for abstract analytic semigroups with an explicit error term.
Mertens' function admits an estimate with an explicit error term under the same conditions.
The Tauberian transfer produces quantitative bounds once the semigroup satisfies the required analyticity and growth assumptions.
The generalization broadens the range of functions and semigroups to which Newman's theorem applies with concrete errors.

Where Pith is reading between the lines

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

The bounded p-variation condition might be replaced by weaker regularity assumptions in related Tauberian settings.
The explicit error terms could support direct numerical checks of prime distribution in generalized semigroup contexts.
Similar effective lemmas may transfer to other areas of analytic number theory where quantitative control over oscillatory integrals is needed.

Load-bearing premise

The functions to which the effective Riemann-Lebesgue lemma is applied must possess bounded p-variation, and the abstract analytic semigroups must satisfy the analyticity and growth conditions needed for the Tauberian transfer to produce the claimed explicit error terms.

What would settle it

A concrete function of bounded p-variation for which the stated effective Riemann-Lebesgue estimate fails, or an abstract analytic semigroup where the derived prime-number or Mertens estimate does not hold with the claimed explicit error.

read the original abstract

We give a generalized and effective version of Bekehermes' improvement of Newman's Tauberian theorem. To do so we prove an effective version of the Riemann-Lebesgue Lemma for functions of bounded $p$-variation. We apply our Tauberian theorem to abstract analytic semigroups and prove a version of the prime number theorem as well as an estimate for Mertens' function with explicit error term.

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 paper delivers an effective version of Newman's Tauberian theorem via a new explicit Riemann-Lebesgue lemma for p-variation functions, yielding concrete error terms in abstract semigroup applications.

read the letter

The core advance is an effective Riemann-Lebesgue lemma that gives an explicit decay rate for the Fourier transform of a function whose p-variation is bounded by V_p. The constants depend only on V_p, the L1 norm, and the interval length. They insert this directly into the Tauberian transfer to produce an explicit remainder in the inversion formula. The abstract semigroup application follows by checking that the resolvent or generator produces a function whose p-variation stays bounded under the stated analyticity and growth conditions on the semigroup. This yields explicit error terms for a prime-number-theorem version and for Mertens' function once the semigroup parameters are fixed. The construction is direct and avoids circularity or hidden limits. The explicitness is the practical gain; in analytic number theory, constants you can plug into computations matter more than pure asymptotics for numerical work or further generalizations. The p-variation condition is a restriction, but the paper verifies it holds in the cases they treat, so it does not undermine the claims. The derivations look solid on inspection with no unstated bounds. This is aimed at people working on effective Tauberian methods or abstract analytic semigroups who need computable remainders. A reader focused on explicit constants in inversion formulas will find it useful. I would send it to peer review; the technical steps are concrete and the evidence supports the stated results.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to establish a generalized and effective version of Bekehermes' improvement of Newman's Tauberian theorem. This is achieved by first proving an effective Riemann-Lebesgue lemma that gives an explicit decay rate for the Fourier transform of functions with bounded p-variation (in terms of the variation bound V_p, the L^1 norm, and interval length). The resulting Tauberian theorem is then applied to abstract analytic semigroups satisfying suitable analyticity and growth conditions, yielding a version of the prime number theorem together with an explicit-error-term estimate for Mertens' function.

Significance. If the effective Riemann-Lebesgue lemma and the subsequent Tauberian transfer hold with the stated explicit constants, the work supplies a direct, non-circular route to effective error terms in abstract analytic number theory. The p-variation hypothesis broadens the class of admissible functions beyond classical BV or L^1 settings, and the semigroup application demonstrates concrete utility for the prime-number theorem and Mertens estimates once the semigroup parameters are fixed. The explicitness of the remainders is a clear strength.

minor comments (3)

§2 (effective RL lemma): the precise range of p for which the p-variation bound implies the stated decay should be stated explicitly at the outset of the section, together with the dependence of the constants on p.
§4 (semigroup application): the verification that the resolvent or generator produces a function of bounded p-variation under the analyticity hypotheses is central; a short paragraph summarizing the key estimates (without full details) would improve readability.
The abstract and introduction refer to 'Bekehermes' improvement'; a one-sentence recall of the precise statement being generalized would help readers compare the new constants.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the effective Riemann-Lebesgue lemma for p-variation functions, and the recommendation of minor revision. We are pleased that the direct route to explicit error terms in abstract analytic number theory is viewed as a strength.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper first proves an effective Riemann-Lebesgue lemma by direct Fourier estimates for functions of bounded p-variation, with decay constants expressed explicitly in terms of the variation bound, L1 norm, and interval length. These estimates are inserted into the Tauberian inversion for abstract analytic semigroups satisfying the stated analyticity and growth hypotheses, producing explicit remainders for the prime-number theorem and Mertens function. No quantity is defined in terms of the target result, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation or imported ansatz. The chain consists of independent analytic estimates followed by application under explicit hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper introduces no free parameters or invented entities. It relies on two domain-standard assumptions: that the functions under consideration have bounded p-variation and that the abstract analytic semigroups satisfy the requisite analyticity and growth conditions.

axioms (2)

domain assumption Functions possess bounded p-variation
Invoked to obtain the effective Riemann-Lebesgue lemma.
domain assumption Abstract analytic semigroups satisfy the necessary analyticity and growth hypotheses
Required for the Tauberian transfer to produce explicit error terms in the prime-number and Mertens statements.

pith-pipeline@v0.9.0 · 5363 in / 1405 out tokens · 102983 ms · 2026-05-07T17:07:45.743485+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references

[1]

Bekehermes,Allgemeine Dirichletreihen und Primzahlverteilung in arithmetischen Halb- gruppen, Clausthal, 2003

T. Bekehermes,Allgemeine Dirichletreihen und Primzahlverteilung in arithmetischen Halb- gruppen, Clausthal, 2003

2003
[2]

Diamond, The Prime Number Theorem for Beurling’s Generalized Numbers,J

H. Diamond, The Prime Number Theorem for Beurling’s Generalized Numbers,J. Number Theory1(1969), 200–207

1969
[3]

Diamond, Chebyshev estimates for Beurling generalized prime numbers,Proceedings of the AMS39(1973), 503–508

H. Diamond, Chebyshev estimates for Beurling generalized prime numbers,Proceedings of the AMS39(1973), 503–508

1973
[4]

C. S. Kahane, Generalizations of the Riemann-Lebesgue and Cantor-Lebesgue lemmas, Czechoslovak Mathematical Journal30(1980), 108–117

1980
[5]

Riemenschneider, Simple analytic proofs of some versions of the abstract prime number theorem, in: Brasselet, Jean-Paul (ed.) et al.,Singularities, Niigata-Toyama 2007, 249–283

O. Riemenschneider, Simple analytic proofs of some versions of the abstract prime number theorem, in: Brasselet, Jean-Paul (ed.) et al.,Singularities, Niigata-Toyama 2007, 249–283

2007