On curious properties of the general size-biased distribution

Alexander E. Patkowski

arxiv: 2412.19347 · v3 · submitted 2024-12-26 · 🧮 math.NT

On curious properties of the general size-biased distribution

Alexander E. Patkowski This is my paper

Pith reviewed 2026-05-23 07:11 UTC · model grok-4.3

classification 🧮 math.NT

keywords size-biased distributionRiemann xi-functionexpected valuefunctional equationsRiemann hypothesis

0 comments

The pith

A general size-biased distribution from the Riemann xi-function has an expected value obeying special functional equations tied to the Riemann hypothesis.

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

The paper builds on an earlier construction of a general size-biased distribution derived from the Riemann xi-function via Ferrar's work. It identifies properties of the expected value of this distribution that arise through particular functional equations. These properties are then related to recent developments on the Riemann hypothesis. A reader would care if the construction supplies a probabilistic object whose moments encode information about the xi-function.

Core claim

The general size-biased distribution related to the Riemann xi-function possesses an expected value that satisfies special functional equations, and these relations connect to recent developments concerning the Riemann hypothesis.

What carries the argument

The general size-biased distribution related to the Riemann xi-function, defined using Ferrar's work, whose expected value produces the functional equations under study.

If this is right

The expected value of the distribution satisfies functional equations linked to the xi-function.
These equations provide relations that connect the distribution to recent Riemann hypothesis investigations.
The analytic setup allows further properties of the distribution to be derived from its expected value.

Where Pith is reading between the lines

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

The size-biasing approach might supply a probabilistic sampling method for studying the zeros of the xi-function if the functional equations are confirmed.
Analogous size-biased constructions could be tested on other L-functions to see whether similar expected-value equations appear.

Load-bearing premise

The general size-biased distribution related to the Riemann xi-function is well-defined and has the analytic properties needed for the expected-value calculations and functional equations to hold.

What would settle it

A direct calculation or numerical check showing that the expected value fails to satisfy the stated functional equations would disprove the claimed properties.

read the original abstract

We offer further results on a general size-biased distribution related to the Riemann xi-function we presented in [9] using the work of Ferrar. Curious properties associated with its expected value are presented, which are related to special functional equations. We also relate our observations to some recent developments related to the Riemann hypothesis.

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 an incremental extension of the author's own prior work on a size-biased distribution for the xi-function, adding expected-value properties but without new foundational results.

read the letter

The main takeaway is that Patkowski is extending his own earlier construction from [9] of a general size-biased distribution tied to the Riemann xi-function via Ferrar's results. The additions here are some properties of the expected value that connect to functional equations, plus a relation of those observations to recent developments around the Riemann hypothesis. The paper does a straightforward job of carrying the probabilistic framing forward and noting those links. If the expected-value calculations hold, they give one more angle on how size-biasing might interact with the analytic properties of the xi-function. The soft spots are straightforward and central. Everything rests on the definition and analytic features (integrability, positivity, continuation) established in [9], and this manuscript does not re-derive or independently check them. That makes the new claims vulnerable to any gaps in the base construction, exactly as the stress-test note flags. The connection to recent RH work also reads as observational rather than a substantive new link or test. This paper is only for the small set of people already tracking this specific thread in analytic number theory. A broader reader in number theory or probability will not find enough standalone value. I would not bring it to a reading group or cite it in the next year. It shows clear thinking on its own terms but does not rise to the level that justifies referee time.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the author's prior work [9] by defining a general size-biased distribution tied to the Riemann xi-function via Ferrar's construction. It reports curious properties of the expected value of this distribution, associates them with special functional equations, and links the observations to recent developments on the Riemann hypothesis.

Significance. If the functional equations and expected-value properties are rigorously established, the work could supply a probabilistic lens on the analytic continuation and zero-distribution features of the xi-function, offering a potential bridge to RH-related results. The approach is novel in its use of size-biasing, but its impact hinges on independent verification of the underlying analytic properties.

major comments (2)

[Introduction and §2] The central claims concerning expected-value properties and functional equations rest entirely on the analytic features (positivity, integrability, continuation) of the size-biased distribution as constructed in [9]; the present manuscript supplies no re-derivation, error bounds, or independent checks of these features, rendering the new results non-self-contained.
[§3] No explicit statement is given of the measure or density used for the expectation, nor of the domain on which the functional equations are asserted to hold; without these, it is impossible to assess whether the claimed relations follow from the definition or require additional assumptions.

minor comments (2)

[Abstract] The abstract and introduction should clarify the precise sense in which the distribution is 'general' and distinguish the new functional equations from those already appearing in [9].
[Introduction] References to 'recent developments related to the Riemann hypothesis' should be expanded with specific citations rather than left at the level of a general allusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to improve clarity and self-containment where appropriate.

read point-by-point responses

Referee: [Introduction and §2] The central claims concerning expected-value properties and functional equations rest entirely on the analytic features (positivity, integrability, continuation) of the size-biased distribution as constructed in [9]; the present manuscript supplies no re-derivation, error bounds, or independent checks of these features, rendering the new results non-self-contained.

Authors: We agree that the new results build directly on the analytic properties (positivity, integrability, and continuation) established in our prior work [9]. To address the concern about self-containment, the revised manuscript will include a concise summary of the key relevant theorems from [9] in the introduction, with explicit citations. A full re-derivation or new error bounds would duplicate the content of [9] and fall outside the scope of this short note on further properties. We maintain that the claims follow from the definitions and results in [9] without additional assumptions, but we accept that better exposition is needed. revision: partial
Referee: [§3] No explicit statement is given of the measure or density used for the expectation, nor of the domain on which the functional equations are asserted to hold; without these, it is impossible to assess whether the claimed relations follow from the definition or require additional assumptions.

Authors: We thank the referee for this observation. In the revised version, we will explicitly state the measure and density used to define the expectation in §3 and specify the precise domain on which the functional equations are asserted to hold. This will allow direct verification that the relations follow from the given definitions. revision: yes

Circularity Check

1 steps flagged

Central claims rest on self-cited prior definition of the size-biased distribution in [9]

specific steps

self citation load bearing [Abstract]
"We offer further results on a general size-biased distribution related to the Riemann xi-function we presented in [9] using the work of Ferrar. Curious properties associated with its expected value are presented, which are related to special functional equations. We also relate our observations to some recent developments related to the Riemann hypothesis."

The functional equations and expected-value properties are asserted for the distribution whose well-definedness and analytic features (required for the calculations) are imported wholesale from the author's prior paper [9]; the new claims therefore inherit any gaps in that construction without independent support in the present text.

full rationale

The paper's derivation begins with the general size-biased distribution tied to the Riemann xi-function, which is taken directly from the author's own prior work [9] (via Ferrar) without re-derivation or independent verification of its analytic properties (positivity, integrability, continuation) in this manuscript. The expected-value functional equations and RH links are then presented as further results on that object. This matches self_citation_load_bearing: the load-bearing premise reduces to the self-citation, and no external benchmark (e.g., machine-checked or externally falsifiable construction) is supplied here to break the dependence.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit list of free parameters or invented entities. The work rests on the prior definition of the size-biased distribution in [9] and standard properties of the Riemann xi-function.

pith-pipeline@v0.9.0 · 5558 in / 1149 out tokens · 28419 ms · 2026-05-23T07:11:22.282999+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J(x) = J(1/x)) echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

E(Xk f(Xk)) = E(f(1/Xk)) for the general size-biased distribution x^{-1} vk(x) with vk from Mellin of ξ^k

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.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
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

11 extracted references · 11 canonical work pages

[1]

Biane, J

P. Biane, J. Pitman and M. Yor, Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. AMS, 4 (2001), 435–465

work page 2001
[2]

Borovkov

A.A. Borovkov. Probability Theory. Universitext. Springer, London, 2013

work page 2013
[3]

K.L. Chung. Excursions in Brownian motion. Arkiv fur Mat ematik, 14:155–177, 1976

work page 1976
[4]

de Bruijn, The roots of trigonometric integrals, Duke Math

N.G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950) 197–226

work page 1950
[5]

H. M. Edwards. Riemann ’s Zeta Function, 1974. Dover Publications

work page 1974
[6]

W. L. Ferrar, Some solutions of the equation F (t) = F (t−1), J. London Math. Soc. 11 (1936), 99–103

work page 1936
[7]

On the de Bruijn–Newman constant, Adv

Ki, H., Kim, Y.-O., Lee, J. On the de Bruijn–Newman constant, Adv. Math. 222, 281–306 (2009)

work page 2009
[8]

A. E. Patkowski, On a solution to a functional equation, Journal of Applied Analysis, vol. 28, no. 1, 2022, pp. 91-93

work page 2022
[9]

A. E. Patkowski, A general size-biased distribution, Journal of Applied Analysis, vol. 29, no. 2, 2023, pp. 347-351

work page 2023
[10]

R. B. Paris, D. Kaminski, Asymptotics and Mellin–Barnes Integrals. Cambridge University Press. (2001)

work page 2001
[11]

E. C. Titchmarsh, The theory of the Riemann zeta function, Oxford University Press, 2nd edition, 1986. 1390 Bumps River Rd. Centerville, MA 02632 USA E-mail: alexpatk@hotmail.com, alexepatkowski@gmail.com

work page 1986

[1] [1]

Biane, J

P. Biane, J. Pitman and M. Yor, Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. AMS, 4 (2001), 435–465

work page 2001

[2] [2]

Borovkov

A.A. Borovkov. Probability Theory. Universitext. Springer, London, 2013

work page 2013

[3] [3]

K.L. Chung. Excursions in Brownian motion. Arkiv fur Mat ematik, 14:155–177, 1976

work page 1976

[4] [4]

de Bruijn, The roots of trigonometric integrals, Duke Math

N.G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950) 197–226

work page 1950

[5] [5]

H. M. Edwards. Riemann ’s Zeta Function, 1974. Dover Publications

work page 1974

[6] [6]

W. L. Ferrar, Some solutions of the equation F (t) = F (t−1), J. London Math. Soc. 11 (1936), 99–103

work page 1936

[7] [7]

On the de Bruijn–Newman constant, Adv

Ki, H., Kim, Y.-O., Lee, J. On the de Bruijn–Newman constant, Adv. Math. 222, 281–306 (2009)

work page 2009

[8] [8]

A. E. Patkowski, On a solution to a functional equation, Journal of Applied Analysis, vol. 28, no. 1, 2022, pp. 91-93

work page 2022

[9] [9]

A. E. Patkowski, A general size-biased distribution, Journal of Applied Analysis, vol. 29, no. 2, 2023, pp. 347-351

work page 2023

[10] [10]

R. B. Paris, D. Kaminski, Asymptotics and Mellin–Barnes Integrals. Cambridge University Press. (2001)

work page 2001

[11] [11]

E. C. Titchmarsh, The theory of the Riemann zeta function, Oxford University Press, 2nd edition, 1986. 1390 Bumps River Rd. Centerville, MA 02632 USA E-mail: alexpatk@hotmail.com, alexepatkowski@gmail.com

work page 1986