Picking up the partial sums of the M\"{o}bius function problem with probabilistic number theory

Maxie Dion Schmidt

arxiv: 2604.23517 · v1 · submitted 2026-04-26 · 🧮 math.NT · math.PR

Picking up the partial sums of the M\"{o}bius function problem with probabilistic number theory

Maxie Dion Schmidt This is my paper

Pith reviewed 2026-05-08 05:20 UTC · model grok-4.3

classification 🧮 math.NT math.PR

keywords Möbius functionMertens functionprobabilistic number theorypartial sumsΩ(n)μ²(n)asymptotic growthDirichlet generating functions

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The pith

Probabilistic assumptions on Ω(n) and μ²(n) independence prove the asymptotic growth of |M(x)| / √x

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

The paper seeks to establish that probabilistic assumptions regarding the independence of the additive function Ω(n) and the indicator μ²(n) for n ≤ x allow one to recover the limiting asymptotic growth of |M(x)| / √x. This is achieved by summing identities derived from hybrid functions that link the Möbius partial sums to prime counting functions and the Liouville lambda function. A sympathetic reader would care because these assumptions are presented as substantially easier to make rigorous than those depending on the Riemann hypothesis or the linear independence of zeta zeros, potentially advancing unconditional proofs in analytic number theory. The approach builds directly on prior work relating these sums through Dirichlet generating functions involving the prime zeta function.

Core claim

By summing the identities involving the functions g(n), |g(n)|, and C_Ω(n) under the probabilistic independence assumptions for Ω(n) and μ²(n), the limiting asymptotic growth of |M(x)| / √x is recovered without relying on the Riemann hypothesis.

What carries the argument

Hybrid multiplicative-to-additive functions g(n) defined via the Dirichlet generating function ζ(s)^{-1}(1+P(s))^{-1} where P(s) is the prime zeta function, along with the variants |g(n)| and C_Ω(n), which connect partial sums of μ(n) to those involving π(x) and λ(n) through ω(n) and Ω(n).

If this is right

The growth of the Mertens function partial sums can be analyzed using probabilistic number theory.
Results on |M(x)| follow from assumptions on the independence of certain arithmetic functions.
Similar techniques may extend to other partial sum problems in number theory.
The need for deep analytic assumptions like zero independence is bypassed.

Where Pith is reading between the lines

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

If the independence can be established for a density of n, the method might apply more broadly to random models of multiplicative functions.
Computational checks of the joint distribution of Ω(n) and μ²(n) could provide empirical support or refutation.
Connections to other hybrid sum identities in the literature could be explored using the same probabilistic lens.

Load-bearing premise

The probabilistic assumptions about the independence of the values of Ω(n) and μ²(n) for n ≤ x at large x.

What would settle it

A calculation showing that the correlation between Ω(n) and μ²(n) does not vanish as x grows large, or that |M(x)| / √x fails to exhibit the predicted limiting growth in explicit computations.

read the original abstract

We revisit several hybrid multiplicative-to-additive type functions from a recent preprint article. These functions, $g(n)$ with Dirichlet generating function (DGF) $\zeta(s)^{-1} (1+P(s))^{-1}$ for $\Re(s) > 1$ where $P(s) = \sum_p p^{-s}$ is the prime zeta function, $|g(n)| = \lambda(n) g(n)$ with DGF $\zeta(2s)^{-1}(1-P(s))^{-1}$, and $C_{\Omega}(n)$ with DGF $(1-P(s))^{-1}$. Each of these function variants are defined in terms of the additive (respectively, strongly additive) functions $\omega(n)$ and $\Omega(n)$. These two auxiliary functions are used in the prior manuscript to relate partial sums of the classical M\"{o}bius function, $\mu(n)$, to signed partial sums involving the prime counting function, $\pi(x)$, and the Liouville lambda function, $\lambda(n) := (-1)^{\Omega(n)}$. In this article, we explore summing the identities from the first manuscript using several probabilistic assumptions about the independence of the values of $\Omega(n)$ and $\mu^2(n)$ for $n \leq x$ at large $x$. We recover proofs of the limiting asymptotic growth of $|M(x)| / \sqrt{x}$ whose hypotheses promise to be substantially more attainable to make rigorous than past results from other authors relying on the Riemann Hypothesis or assumption of the linear independence of the simple, non-trivial zeros of $\zeta(s)$.

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

The paper gives a probabilistic re-derivation of the known |M(x)|/√x growth using independence of Ω(n) and μ²(n), but the assumptions are not shown to be easier than RH or zero independence.

read the letter

The main takeaway is that this extends the author's prior preprint on hybrid functions to recover the standard asymptotic growth for the Mertens function M(x) under a probabilistic independence model for Ω(n) and square-freeness up to x. It does not establish a new fact about M(x), but rearranges existing connections via Dirichlet generating functions for g(n), the signed version with λ(n), and C_Ω(n) to sum identities and reach the target bound on |M(x)|/√x. The setup links the Möbius partial sums back to the prime zeta function and additive functions in a consistent way, which is a clean organizational step. The probabilistic averaging is applied directly to those summed identities. The soft spot is the independence assumption itself. It is chosen because it produces the expected growth rate, yet the abstract offers no reduction or external benchmark showing why proving independence of these two functions is substantially simpler than proving linear independence of zeta zeros or the Riemann hypothesis. Both Ω(n) and μ²(n) are controlled by the same prime factorization statistics that sit behind the analytic conjectures, so the circularity risk remains. No explicit error terms or verification steps appear in the summary, which leaves the attainability claim as a promise rather than a demonstrated improvement. This is for readers already working on probabilistic number theory or alternative routes around the Riemann hypothesis for arithmetic sums. Someone tracking the Mertens problem and its connections to additive functions will see the angle clearly, but the work will not displace the classical literature. I would send it to referees to check the details of the summation and whether the independence hypotheses can be compared rigorously to existing ones.

Referee Report

2 major / 2 minor

Summary. The manuscript revisits hybrid multiplicative-to-additive functions g(n), |g(n)|, and C_Ω(n) whose Dirichlet generating functions involve ζ(s)^{-1} and the prime zeta function P(s). These are used to relate partial sums of the Möbius function μ(n) to signed sums involving π(x) and the Liouville function λ(n) = (-1)^{Ω(n)}. The authors then sum the resulting identities under probabilistic independence assumptions on the values of Ω(n) and μ²(n) for n ≤ x at large x, recovering a proof of the limiting asymptotic growth of |M(x)|/√x. The central claim is that these probabilistic hypotheses are substantially more attainable than the Riemann Hypothesis or linear independence of the non-trivial zeros of ζ(s).

Significance. If the independence assumptions can be made rigorous with controlled error terms and shown to be strictly weaker than standard analytic hypotheses, the work would supply a probabilistic-number-theoretic route to the square-root growth of M(x), a longstanding problem in analytic number theory. The approach of combining prior multiplicative-to-additive identities with independence statements on additive functions is conceptually interesting and could, in principle, avoid heavy zero-density or zero-independence machinery.

major comments (2)

[Abstract] Abstract and the paragraph following the statement of the probabilistic assumptions: the claim that the independence hypotheses 'promise to be substantially more attainable' is not supported by any explicit reduction, benchmark comparison, or proof sketch showing why independence of Ω(n) and μ²(n) is easier to establish than linear independence of zeta zeros; both ultimately rest on questions of prime distribution.
[Main derivation section] The derivation that sums the prior identities under the independence assumption: no explicit error estimates, variance calculations, or verification steps are supplied to confirm that the assumed independence controls the summed remainder terms sufficiently to yield the claimed growth rate of |M(x)|/√x; the selection of the independence statement appears guided by the target asymptotic rather than by independent justification.

minor comments (2)

The definitions of the auxiliary functions g(n), |g(n)|, and C_Ω(n) are given only via their DGFs; explicit arithmetic definitions in terms of ω(n) and Ω(n) should be restated in the main text for readability.
The manuscript refers to 'a recent preprint article' without a full bibliographic citation; the reference list should include the precise arXiv identifier or title of the prior work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address the two major comments point by point below, indicating the revisions we will make.

read point-by-point responses

Referee: [Abstract] Abstract and the paragraph following the statement of the probabilistic assumptions: the claim that the independence hypotheses 'promise to be substantially more attainable' is not supported by any explicit reduction, benchmark comparison, or proof sketch showing why independence of Ω(n) and μ²(n) is easier to establish than linear independence of zeta zeros; both ultimately rest on questions of prime distribution.

Authors: We acknowledge that the manuscript provides no explicit reduction, benchmark, or proof sketch establishing that the independence of Ω(n) and μ²(n) is easier to rigorize than linear independence of zeta zeros. The phrasing reflects a heuristic view drawn from the fact that probabilistic models for additive functions have been successfully analyzed via moment methods and sieve techniques under assumptions such as Bombieri–Vinogradov, whereas zero independence requires stronger arithmetic control. Both indeed connect to prime distribution. We will revise the abstract and the subsequent paragraph to qualify the claim as a heuristic expectation rather than a supported assertion, and add a short discussion referencing known results on the distribution of Ω(n). revision: partial
Referee: [Main derivation section] The derivation that sums the prior identities under the independence assumption: no explicit error estimates, variance calculations, or verification steps are supplied to confirm that the assumed independence controls the summed remainder terms sufficiently to yield the claimed growth rate of |M(x)|/√x; the selection of the independence statement appears guided by the target asymptotic rather than by independent justification.

Authors: Under the stated independence assumptions the formal summation eliminates cross terms, so that the variance of the summed quantity is the sum of the individual variances and the claimed growth follows directly. We agree that the absence of explicit error estimates and variance calculations leaves the control of remainder terms unverified. The independence statement is motivated by the Erdős–Kac theorem and the known asymptotic independence of Ω(n) for distinct n in probabilistic models, rather than being chosen solely to obtain the target asymptotic. We will add a new subsection containing variance computations under the independence hypothesis, showing that the remainders are of strictly lower order, together with references to related work on moments of additive functions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses stated probabilistic assumptions to recover known asymptotic

full rationale

The paper sums identities from a prior preprint under explicitly stated probabilistic independence assumptions on Ω(n) and μ²(n) for n ≤ x. These assumptions are presented as external inputs chosen for their potential attainability, not derived from or equivalent to the target |M(x)|/√x asymptotic by construction. No equations reduce the result to a fit, self-definition, or self-citation chain that forces the outcome. The prior work provides the identities, but the new assumptions add independent content. The claim of more attainable hypotheses is a correctness question, not a circularity reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests entirely on an unproven probabilistic independence assumption between Ω(n) and μ²(n); no free parameters, invented entities, or additional axioms are introduced beyond this modeling choice and standard properties of the Möbius and Liouville functions.

axioms (1)

domain assumption Values of Ω(n) and μ²(n) are independent for n ≤ x at large x
Invoked to sum the identities from the prior manuscript and obtain the asymptotic for M(x)

pith-pipeline@v0.9.0 · 5590 in / 1475 out tokens · 102072 ms · 2026-05-08T05:20:55.088376+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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T. M. Apostol.Introduction to Analytic Number Theory. Springer–Verlag, 1976

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Arratia, A

R. Arratia, A. D. Barbour, and Simon Tavaré.Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society Publishing House, 2003

work page 2003

[3] [3]

Billingsley

P. Billingsley. On the central limit theorem for the prime divisor function.Amer. Math. Monthly, 76(2):132–139, 1969

work page 1969

[4] [4]

Billingsley.Probability and measure

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work page 1994

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P. D. T. A. Elliott.Probabilistic Number Theory I: Mean-Value Theorems. Springer New York, 1979

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P. D. T. A. Elliott.Probabilistic Number Theory II: Central Limit Theorems. Springer New York, 1979

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Erdős and M

P. Erdős and M. Kac. The Gaussian errors in the theory of additive arithmetic functions.American Journal of Mathematics, 62(1):738–742, 1940

work page 1940

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G. H. Hardy and E. M. Wright.An Introduction to the Theory of Numbers. Oxford University Press, 2008 (Sixth Edition)

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G. Hurst. Computations of the Mertens function and improved bounds on the Mertens conjecture.Math. Comp., 87:1013–1028, 2018

work page 2018

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Hwang and S

H. Hwang and S. Janson. A central limit theorem for random ordered factorizations of integers.Electron. J. Probab., 16(12):347–361, 2011

work page 2011

[11] [11]

A.E. Ingham. On two conjectures in the theory of numbers. Amer. J. Math., 64:313–319, 1942

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Kotnik and H

T. Kotnik and H. té Riele. The Mertens conjecture revisited.Algorithmic Number Theory,7th Interna- tional Symposium, 2006

work page 2006

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Kotnik and J

T. Kotnik and J. van de Lune. On the order of the Mertens function.Exp. Math., 2004

work page 2004

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H. L. Montgomery and R. C. Vaughan.Multiplicative Number Theory: I, Classical Theory. Cambridge, 2006

work page 2006

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N. Ng. The distribution of the summatory function of the Móbius function.Proc. London Math. Soc., 89(3):361–389, 2004

work page 2004

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A. M. Odlyzko and H. J. J. té Riele. Disproof of the Mertens conjecture.J. Reine Angew. Math, 1985

work page 1985

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F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors.NIST Handbook of Mathematical Functions. Cambridge University Press, 2010

work page 2010

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Ribenboim.The new book of prime number records

P. Ribenboim.The new book of prime number records. Springer, 1996

work page 1996

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J. B. Rosser and L. Schoenfeld. Approximate formulas for some functions of prime numbers.Illinois J. Math., 6:64–94, 1962

work page 1962

[20] [20]

M. D. Schmidt. Exact formulas for partial sums of the Möbius function expressed by partial sums weighted by the Liouville lambda function. Preprint available online (2022):https://arxiv.org/abs/ 2102.05842. 9

work page arXiv 2022

[21] [21]

N. J. A. Sloane. The Online Encyclopedia of Integer Sequences, 2021.http://oeis.org

work page 2021

[22] [22]

Soundararajan

K. Soundararajan. The Liouville function in short intervals (after Matomäki and Radziwiłł).S einaire Boubake, 2015-2016, no. 1119(390):453–472, 2017

work page 2015

[23] [23]

Soundararajan

K. Soundararajan. Partial sums of the Möbius function.Annals of Mathematics, 2009

work page 2009

[24] [24]

E. C. Titchmarsh.The theory of the Riemann zeta function. Clarendon Press, 1951

work page 1951

[25] [25]

Tenenbaum.Introduction to Analytic and Probabilistic Number Theory

G. Tenenbaum.Introduction to Analytic and Probabilistic Number Theory. American Mathematical Society, 2015. 10

work page 2015