Pseudometrics and preorders on sets of integer sequences induced by arithmetic functions functions

Mario Ziller

arxiv: 2604.19646 · v1 · submitted 2026-04-21 · 🧮 math.NT

Pseudometrics and preorders on sets of integer sequences induced by arithmetic functions functions

Mario Ziller This is my paper

Pith reviewed 2026-05-10 01:30 UTC · model grok-4.3

classification 🧮 math.NT

keywords arithmetic functionspseudometricspreordersinteger sequencesconsecutive integersZ^m

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The pith

Arithmetic functions induce pseudometrics and preorders on finite sequences of integers, including consecutive ones, and determine relationships and equivalences among the resulting preorders.

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

The paper starts from pseudometrics and preorders on single integers and extends the constructions to finite sequences of integers in Z^m, giving special attention to sequences of consecutive integers. It outlines centered versions of these structures in a pseudometric space and shows how arithmetic functions can induce them on the relevant sets. The central results establish relationships between the choice of arithmetic function and the preorder obtained, including cases of equivalence and cases where the preorders remain distinct. A reader would care because the work connects arithmetic properties of integers directly to ordering and distance notions on blocks of numbers, offering a uniform language for comparing sequences by their number-theoretic features.

Core claim

Starting from pseudometrics and preorders on sets of integers, we extend the focus to sets of finite sequences of integers, in particular sequences of consecutive integers. We outline existing concepts for deriving centred pseudometrics and preorders in a given pseudometric space and their application to Z and develop approaches to generalize the ideas to Z^m. Sequences of consecutive integers represent a special case here and are examined in more detail. Another main topic is the use of arithmetic functions in this context. The types of pseudometrics and preorders examined in this paper can be induced by suitable arithmetic functions. We derive fundamental conclusions about relationships,

What carries the argument

Induction of centred pseudometrics and preorders by suitable arithmetic functions on sets of finite integer sequences in Z and Z^m, with consecutive sequences as a distinguished subclass.

Load-bearing premise

The generalizations to sets of finite sequences in Z^m and to consecutive integer sequences preserve the defining properties of pseudometrics and preorders when induced by arithmetic functions.

What would settle it

An explicit arithmetic function together with two sequences of consecutive integers in Z^m such that the induced relation violates transitivity or the induced distance violates the triangle inequality.

Figures

Figures reproduced from arXiv: 2604.19646 by Mario Ziller.

**Figure 1.** Figure 1: Preorder-preserving isomorphisms between the sets of generating and supplementary functions on S. In the starting section, we apply the definitions given above to sets of integers and explain how arithmetic functions can be used to derive pseudometrics and preorders. The introduction of a modulus n ∈ N and the transformation x 7→ 1 + (x − 1) mod n can enable extending the use of arithmetic functions to a… view at source ↗

**Figure 2.** Figure 2: Relationships between admissible arithmetic functions, induced pseudometrics, and corresponding preorders on Z. We draw attention to the fact that functions f n defined in Lemma 1.3 can be neither additive nor multiplicative even if f is an additive or a multiplicative arithmetic function. This also applies to the special case of n-even functions for which f(x) = f [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: Preorder-preserving isomorphisms between the sets of generating and supplementary functions on Zm. This section started with the definition of the pseudometrics d (m) AM, d (m) GM, and d (m) HM according to Lemma 2.1 . From them, we derived (1)-centred pseudometrics in the sense of Lemma 0.2 , whose definitions were summarised in Corollary 2.4 . Based on the latter, we provided a general and comprehensive … view at source ↗

**Figure 4.** Figure 4: Relationships between admissible arithmetic functions, induced pseudometrics, and corresponding preorders on ⟨Z⟩m. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_4.png] view at source ↗

read the original abstract

Starting from pseudometrics and preorders on sets of integers, we extend the focus to sets of finite sequences of integers, in particular sequences of consecutive integers. We outline existing concepts for deriving centred pseudometrics and preorders in a given pseudometric space and their application to $\mathbb{Z}$ and develop approaches to generalize the ideas to $\mathbb{Z}^m$. Sequences of consecutive integers represent a special case here and are examined in more detail. Another main topic is the use of arithmetic functions in this context. The types of pseudometrics and preorders examined in this paper can be induced by suitable arithmetic functions. We derive fundamental conclusions about relationships between functions and preorders, as well as about equivalent and potentially distinct types of preorders.

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 cleanly extends pseudometrics and preorders from single integers to finite sequences via arithmetic functions, with direct axiom checks, but stays within modest definitional territory.

read the letter

The main takeaway is that this paper moves pseudometrics and preorders from sets of integers to sets of finite sequences, treating consecutive sequences as a special case, and shows how suitable arithmetic functions can induce them. It also works out some basic relations between the inducing functions and the resulting preorders, including equivalences and distinctions between types. The argument relies on explicit constructions and direct verification that the axioms hold, without extra assumptions on boundedness or continuity. That part reads as solid and straightforward. The generalization to Z^m is handled by the same pattern, which keeps the development consistent. What the paper does well is keep the focus on the definitions and the immediate consequences; the stress-test note is right that nothing hidden is required beyond the suitability of the arithmetic functions. The verifications are mechanical but honest. The soft spots are mostly about scope. The work organizes existing ideas in a new setting rather than resolving open questions or producing new computational tools. The claims about relationships and equivalent preorders follow from the constructions, but they do not appear to generate surprising or deep number-theoretic consequences. Without more concrete examples or applications, it is hard to gauge how much this framework will be used beyond the paper itself. The citation pattern is light and appropriate for an extension paper. This is for readers already working on arithmetic functions or abstract structures on integer sequences. Someone looking for a reference on how to lift metric-like notions to sequences could pull useful definitions from it. It deserves a serious referee because the core constructions are coherent and the verifications are direct, even if the motivation and payoff sections might need tightening in review.

Referee Report

0 major / 3 minor

Summary. The paper extends pseudometrics and preorders from sets of integers to finite sequences of integers (with consecutive sequences as a special case) and to Z^m. It shows how suitable arithmetic functions induce such structures on these sets, then derives relationships between the inducing functions and the resulting preorders, including equivalences and distinctions among preorder types. The argument relies on explicit constructions and direct verification that the induced objects satisfy the pseudometric and preorder axioms.

Significance. If the explicit constructions and verifications hold, the work supplies a systematic way to generate and compare preorders on integer sequences via arithmetic functions, which may be useful for organizing arithmetic data in number theory. The absence of extra regularity assumptions on the functions and the treatment of consecutive sequences as a concrete case are positive features. The derivations of equivalences and distinctions between preorder types constitute the main potential contribution.

minor comments (3)

The abstract and introduction would benefit from a brief concrete example (e.g., the divisor function or Euler totient) showing an induced pseudometric on a short sequence of consecutive integers; this would make the generalization to Z^m more accessible.
Notation for the induced preorder (e.g., how ≼_f is defined from an arithmetic function f) should be introduced once and used consistently; occasional shifts between relational and functional notation appear in the later sections.
The manuscript would be strengthened by an explicit statement of the precise suitability condition on arithmetic functions that is required for the constructions to work; this is alluded to but not isolated as a numbered definition or lemma.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The referee correctly identifies the core contributions: the extension of pseudometrics and preorders to finite sequences (including consecutive ones) and to Z^m via arithmetic functions, together with the derivations of relationships, equivalences, and distinctions among the induced structures. No specific major comments were raised, so we have no individual points to rebut or revise at this stage. We remain ready to incorporate any minor editorial suggestions in a revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proceeds entirely by explicit definitions of pseudometrics and preorders induced by arithmetic functions on integer sequences, followed by direct axiom verification and derivation of relationships (equivalences and distinctions) under those definitions. Generalizations to finite sequences in Z^m and consecutive integers are handled by straightforward extensions that preserve the stated properties by construction. No parameters are fitted to data, no predictions reduce to inputs by definition, and no self-citations or prior ansatzes are invoked as load-bearing justifications for the central claims. All steps remain self-contained within the given constructions and verifications.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, providing insufficient detail to enumerate specific free parameters, axioms, or invented entities; the work appears to rely on standard mathematical definitions of pseudometrics and preorders.

axioms (1)

standard math Standard axioms of pseudometric spaces and preorders hold when extending from Z to Z^m and to sequences.
The paper builds directly on existing concepts for Z as described in the abstract.

pith-pipeline@v0.9.0 · 5413 in / 1181 out tokens · 36798 ms · 2026-05-10T01:30:40.860971+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

[1]

Box, G.M

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László Tóth,A survey of gcd-sum functions, J. Integer Seq.13(2010), no. 8, Art. 10.8.1

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Integer Seq

Victor Ufnarovski and Bo Åhlander,How to differentiate a number, J. Integer Seq. 6(2003), no. 3, Art. 03.3.4

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Mario Ziller,On differences between consecutive numbers coprime to primorials, arXiv:2007.01808 [math.NT] (2020)

work page arXiv 2007
[12]

Mario Ziller,Orders and partitions of integers induced by arithmetic functions, arXiv:2310.15628 [math.NT] (2023). 49

work page internal anchor Pith review Pith/arXiv arXiv 2023

[1] [1]

Box, G.M

G.E.P . Box, G.M. Jenkins, G.C. Reinsel, and G.M. Ljung,Time series analysis: Forecasting and control, Wiley Series in Probability and Statistics, Wiley, 2015

work page 2015

[2] [2]

Broughan,The gcd-sum function, J

Kevin A. Broughan,The gcd-sum function, J. Integer Sequences4(2001), no. 01.2.2, 1–19

work page 2001

[3] [3]

Michel Marie Deza and Elena Deza,Encyclopedia of distances, Springer Berlin Heidelberg, 2009

work page 2009

[4] [4]

P . P . Lava and G. Balzarotti,La derivata aritmetica: Alla scoperta di un nuovo approccio alla teoria dei numeri, Hoepli, Milano, 2013

work page 2013

[5] [5]

Paul Joseph McCarthy,Introduction to arithmetical functions, Springer New York Berlin Heidelberg Tokyo, 1986

work page 1986

[6] [6]

S. S. Pillai,On an arithmetic function, Annamalai University JournalII(1933), 243–248

work page 1933

[7] [7]

3, 149–151

Kurt Pohlen,The moving geometric average, The Review of Economics and Statistics 11(1929), no. 3, 149–151

work page 1929

[8] [8]

Wolfgang Schwarz and Jürgen Spilker,Arithmetical functions, London Math. Soc. Lecture Note Series, Cambridge University Press, 1994

work page 1994

[9] [9]

Integer Seq.13(2010), no

László Tóth,A survey of gcd-sum functions, J. Integer Seq.13(2010), no. 8, Art. 10.8.1

work page 2010

[10] [10]

Integer Seq

Victor Ufnarovski and Bo Åhlander,How to differentiate a number, J. Integer Seq. 6(2003), no. 3, Art. 03.3.4

work page 2003

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Mario Ziller,On differences between consecutive numbers coprime to primorials, arXiv:2007.01808 [math.NT] (2020)

work page arXiv 2007

[12] [12]

Mario Ziller,Orders and partitions of integers induced by arithmetic functions, arXiv:2310.15628 [math.NT] (2023). 49

work page internal anchor Pith review Pith/arXiv arXiv 2023