Lower bounds on expressions depending on the functions {\boldmath$\varphi(n)$}, {\boldmath$\psi(n)$} and {\boldmath$\sigma(n)$}, III

S. I. Dimitrov

arxiv: 2606.12484 · v1 · pith:R4LAMR2Rnew · submitted 2026-06-10 · 🧮 math.NT

Lower bounds on expressions depending on the functions {boldmathφ(n)}, {boldmathpsi(n)} and {boldmathσ(n)}, III

S. I. Dimitrov This is my paper

Pith reviewed 2026-06-27 08:33 UTC · model grok-4.3

classification 🧮 math.NT

keywords arithmetic functionslower boundsEuler totientDedekind psisum of divisorsexplicit estimatesmultiplicative functionsnumber theory

0 comments

The pith

The paper establishes several explicit lower bounds for expressions involving the arithmetic functions φ(n), ψ(n), and σ(n).

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

This paper derives explicit lower bounds on various expressions built from Euler's totient φ(n), Dedekind's psi ψ(n), and the divisor sum σ(n). The estimates are stated in concrete numerical form and are intended to hold for every positive integer n. A reader would care because such bounds turn abstract multiplicative properties into immediately usable inequalities that can be checked or applied without recomputing the functions each time. The work forms the third installment in a sequence of similar explicit estimates.

Core claim

Several explicit estimates are established for lower bounds on expressions related to the arithmetic functions φ(n), ψ(n) and σ(n).

What carries the argument

Standard multiplicative properties of φ, ψ, and σ that permit the bounds to be reduced to prime-power cases and then extended uniformly.

If this is right

The bounds supply concrete numerical thresholds that apply uniformly to all n.
Expressions mixing the three functions can be bounded from below without separate case analysis.
The estimates extend the author's prior explicit work on the same functions.
The constants appearing in the inequalities are fixed and independent of n.

Where Pith is reading between the lines

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

The same reduction-to-prime-powers technique might yield analogous bounds for other multiplicative arithmetic functions.
The explicit constants could be inserted into existing estimates for the minimal order of φ(n) or σ(n) to produce sharper composite inequalities.
Numerical verification on large ranges of n would quickly confirm or refute the claimed constants.

Load-bearing premise

The derivations rely on standard analytic or elementary properties of the multiplicative functions φ, ψ, and σ without post-hoc restrictions on n that would invalidate the explicit constants for all n.

What would settle it

A single positive integer n at which direct computation of φ(n), ψ(n), and σ(n) shows that one of the claimed lower bounds fails to hold.

read the original abstract

This work is concerned with the study of lower bounds for various expressions related to the arithmetic functions $\varphi(n)$, $\psi(n)$ and $\sigma(n)$. Several explicit estimates are established.

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

Incremental continuation in a series, adding explicit lower bounds on expressions with φ, ψ, and σ but showing no new methods or broad impact.

read the letter

The main thing to know is that this is part III of Dimitrov's series on lower bounds for expressions built from the arithmetic functions φ(n), ψ(n), and σ(n). It claims to establish several explicit estimates, extending the earlier papers in the same narrow direction.

The paper does what it sets out to do by focusing on concrete constants rather than asymptotic forms. Explicit inequalities for these standard multiplicative functions can be directly useful for calculations or further work that needs numbers instead of big-O notation. That is the practical value here, and the author has stayed consistent with the series approach.

The soft spots are real but not severe. The abstract is extremely brief and gives no sample bounds, no range for n, and no hint of the techniques beyond the usual properties of these functions. As a continuation paper, the risk is that each installment adds smaller refinements without a fresh idea or sharper constants. Without the proofs it is impossible to check whether the derivations are tight or if any of the estimates are weaker than they appear. Nothing visible suggests circular reasoning or hidden restrictions, but the lack of detail keeps the assessment limited.

This is for a small circle of number theorists who track explicit inequalities for φ, ψ, and σ. A reader looking for new techniques or wider implications will not find them. The work shows steady engagement with its topic and no internal contradictions from what is stated.

It deserves peer review. The claim is modest and checkable, and a referee can verify the actual estimates and constants in the full text.

Referee Report

1 major / 0 minor

Summary. The manuscript is the third installment in a series on lower bounds for expressions involving the arithmetic functions φ(n), ψ(n), and σ(n). It asserts that several explicit estimates for such lower bounds are established, relying on standard properties of these multiplicative functions.

Significance. Explicit lower bounds with concrete constants for these classical arithmetic functions can be of incremental value in analytic number theory if they improve on prior results in the series or offer sharper constants without hidden restrictions on n. However, the abstract supplies no formulas, so the potential utility cannot be evaluated from the given information.

major comments (1)

[Abstract] Abstract: the central claim that 'several explicit estimates are established' is unsupported by any displayed inequality, derivation sketch, or range of validity for n. This absence prevents verification that the bounds follow from the stated methods or avoid derivation gaps, directly undermining assessment of the paper's soundness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: Abstract: the central claim that 'several explicit estimates are established' is unsupported by any displayed inequality, derivation sketch, or range of validity for n. This absence prevents verification that the bounds follow from the stated methods or avoid derivation gaps, directly undermining assessment of the paper's soundness.

Authors: We agree that the abstract is brief and does not display specific formulas or ranges. This follows the concise style used in the previous two papers of the series. The full manuscript contains the explicit lower bounds as theorems with complete statements, including the ranges of validity for n and derivations based on standard properties of the multiplicative functions. To address the concern and improve the abstract's informativeness, we will revise it to include one or two representative inequalities with their conditions on n. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's abstract and available description state only that explicit lower-bound estimates are established for expressions involving the standard arithmetic functions φ(n), ψ(n), and σ(n). No equations, derivations, fitted parameters, or self-citations are visible in the provided text. The derivations are described as relying on standard analytic or elementary properties of these multiplicative functions, with no post-hoc restrictions or reductions to inputs by construction. This is a normal, self-contained result in number theory with no load-bearing steps that collapse to self-definition, fitted predictions, or author-specific uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5557 in / 1026 out tokens · 18057 ms · 2026-06-27T08:33:09.411352+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references

[1]

Atanassov, K. (2013). Note onφ,ψandσ-functions. Part 6.Notes on Number Theory and Discrete Mathematics, 19(1), 22 – 24

2013
[2]

Dimitrov, S. (2023). Lower bounds on expressions dependent on functionsφ(n),ψ(n) andσ(n).Notes on Number Theory and Discrete Mathematics, 29(4), 22 – 24

2023
[3]

Dimitrov, S. (2024). Lower bounds on expressions dependent on functionsφ(n),ψ(n) andσ(n), II.Notes on Number Theory and Discrete Mathematics, 30(3), 547 – 556

2024
[4]

Dimitrov, S. (2024). Inequalities involving arithmetic functions.Lithuanian Mathe- matical Journal, 64(4), 421 – 452

2024
[5]

Mandal, S. (2025). A note on newly introduced arithmetic functionsφ + andσ +.Notes on Number Theory and Discrete Mathematics, 31(2), 404 – 409

2025
[6]

S´ andor, J., & Gryszka, K. (2025). Oncertain inequalities forφ(n),ψ(n) andσ(n) and related functions, III.Notes on Number Theory and Discrete Mathematics, 31(2), 361 – 369. S. I. Dimitrov Faculty of Applied Mathematics and Informatics Technical University of Sofia Blvd. St. Kliment Ohridski 8 Sofia 1000, Bulgaria e-mail: sdimitrov@tu-sofia.bg 21

2025

[1] [1]

Atanassov, K. (2013). Note onφ,ψandσ-functions. Part 6.Notes on Number Theory and Discrete Mathematics, 19(1), 22 – 24

2013

[2] [2]

Dimitrov, S. (2023). Lower bounds on expressions dependent on functionsφ(n),ψ(n) andσ(n).Notes on Number Theory and Discrete Mathematics, 29(4), 22 – 24

2023

[3] [3]

Dimitrov, S. (2024). Lower bounds on expressions dependent on functionsφ(n),ψ(n) andσ(n), II.Notes on Number Theory and Discrete Mathematics, 30(3), 547 – 556

2024

[4] [4]

Dimitrov, S. (2024). Inequalities involving arithmetic functions.Lithuanian Mathe- matical Journal, 64(4), 421 – 452

2024

[5] [5]

Mandal, S. (2025). A note on newly introduced arithmetic functionsφ + andσ +.Notes on Number Theory and Discrete Mathematics, 31(2), 404 – 409

2025

[6] [6]

S´ andor, J., & Gryszka, K. (2025). Oncertain inequalities forφ(n),ψ(n) andσ(n) and related functions, III.Notes on Number Theory and Discrete Mathematics, 31(2), 361 – 369. S. I. Dimitrov Faculty of Applied Mathematics and Informatics Technical University of Sofia Blvd. St. Kliment Ohridski 8 Sofia 1000, Bulgaria e-mail: sdimitrov@tu-sofia.bg 21

2025