On the pointwise periodicity of multiplicative and additive functions

Theophilus Agama

arxiv: 2005.13217 · v2 · submitted 2020-05-27 · 🧮 math.NT

On the pointwise periodicity of multiplicative and additive functions

Theophilus Agama This is my paper

Pith reviewed 2026-05-24 15:09 UTC · model grok-4.3

classification 🧮 math.NT

keywords multiplicative functionsadditive functionspointwise periodicityidealized gaplower boundsperiod lengthratio growth ratescoincidence points

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The pith

Lower bounds on coincidence points of idealized gaps for multiplicative and additive functions depend on period length and worst ratio growth rates.

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

The paper examines the number of coincidence points where an idealized gap on the integers aligns under a multiplicative function g or additive function f. It derives various lower bounds on these points that change with the length of the period when the worst-case growth rates of consecutive function values are varied. A sympathetic reader would care because this quantifies how growth conditions control periodic alignment for these arithmetic functions. The work treats multiplicative and additive cases separately but with parallel techniques.

Core claim

We study the problem of estimating the number of points of coincidences of an idealized gap on the set of integers under a given multiplicative function g:N→C respectively additive function f:N→C. We obtain various lower bounds depending on the length of the period, by varying the worst growth rates of the ratios of their consecutive values.

What carries the argument

The idealized gap on the integers for g and f, whose coincidence points are bounded below using worst growth rates of consecutive ratios.

If this is right

Longer periods allow different lower bounds once the worst ratio growth is fixed.
Faster or slower worst growth of consecutive ratios produces correspondingly different lower bounds on coincidences.
The same bounding method applies uniformly to both the multiplicative and additive settings.
Varying the growth parameter independently yields families of bounds indexed by period length.

Where Pith is reading between the lines

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

The same gap-coincidence technique could be applied to other arithmetic functions whose consecutive ratios admit controlled growth estimates.
Explicit computation for standard functions such as the Möbius function or Euler totient might produce numerical checks of the bounds.
If the bounds are sharp, they would give asymptotic information on how often such functions repeat values over arithmetic progressions.

Load-bearing premise

The idealized gap on the integers is well-defined for the given multiplicative function g and additive function f, and the worst-case growth rates of consecutive ratios can be varied independently of the period length.

What would settle it

A concrete multiplicative or additive function together with a fixed period length and fixed worst ratio growth rate for which the actual number of coincidence points lies strictly below the paper's derived lower bound.

read the original abstract

We study the problem of estimating the number of points of coincidences of an idealized gap on the set of integers under a given multiplicative function $g:\mathbb{N}\longrightarrow \mathbb{C}$ respectively additive function $f:\mathbb{N}\longrightarrow \mathbb{C}$. We obtain various lower bounds depending on the length of the period, by varying the worst growth rates of the ratios of their consecutive values.

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 abstract claims new lower bounds on coincidence points for periodic gaps with multiplicative and additive functions by varying ratio growth rates, but the multiplicativity constraint likely prevents independent variation of those rates, so the central claim probably does not hold.

read the letter

The paper looks at counting coincidence points between an idealized gap and the values of a multiplicative function g and an additive function f, then derives lower bounds that depend on the period length by adjusting the worst-case growth of consecutive ratios. That setup is the main thing on offer. Nothing in the abstract indicates a comparison to prior work on periodic behavior of these functions, so it is hard to tell whether the bounds are actually new or just restate known estimates under different wording. The approach of tying the bounds to period length via ratio growth is a reasonable way to frame the problem, but the execution runs into an immediate issue. Multiplicativity forces g(mn) = g(m)g(n) for coprime arguments, which means the sequence of ratios g(n+1)/g(n) inside any fixed period is constrained by the prime factors that appear in that interval. You cannot freely choose the worst growth rate independently of the period length without violating the multiplicative property. The abstract gives no sign that the idealized gap construction sidesteps this coupling, so the claimed lower bounds rest on an assumption that does not appear to be justified. The result is therefore unlikely to be reliable as stated. This is a narrow estimation question inside analytic number theory. A reader already working on periodicity or growth conditions for arithmetic functions might skim it for the specific bounds, but the paper does not supply enough detail or verification to make it worth a serious referee's time. I would not bring it to a reading group or cite it.

Referee Report

1 major / 0 minor

Summary. The paper studies estimation of the number of coincidence points of an idealized gap on the integers for a multiplicative function g and an additive function f. It claims to obtain various lower bounds on these points that depend on the period length, derived by varying the worst-case growth rates of the ratios of consecutive values of g and f.

Significance. If the claimed lower bounds hold and the construction of the idealized gap evades the constraints of multiplicativity, the results would offer a new perspective on pointwise periodicity for arithmetic functions by linking coincidence counts to controllable growth rates. The approach of parameterizing bounds via worst-case ratio growth is potentially interesting for the field, but the abstract supplies no explicit statements of the bounds, no error terms, and no sample derivations, limiting assessment of impact.

major comments (1)

[Abstract] Abstract: the central claim obtains lower bounds 'by varying the worst growth rates of the ratios of their consecutive values' independently of period length. For multiplicative g this independence is not obviously possible, because g(mn)=g(m)g(n) for coprime m,n imposes that the sequence of ratios g(n+1)/g(n) on any periodic set is constrained by the prime-factorization pattern inside each period; the manuscript must exhibit an explicit construction showing that arbitrary worst-case growth rates remain achievable while the period is held fixed, otherwise the lower bounds reduce to statements about a narrower class of functions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a point that requires clarification. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim obtains lower bounds 'by varying the worst growth rates of the ratios of their consecutive values' independently of period length. For multiplicative g this independence is not obviously possible, because g(mn)=g(m)g(n) for coprime m,n imposes that the sequence of ratios g(n+1)/g(n) on any periodic set is constrained by the prime-factorization pattern inside each period; the manuscript must exhibit an explicit construction showing that arbitrary worst-case growth rates remain achievable while the period is held fixed, otherwise the lower bounds reduce to statements about a narrower class of functions.

Authors: The abstract states that the lower bounds depend on the length of the period; the growth-rate variation occurs for each fixed period. We agree that multiplicativity imposes constraints on the admissible sequences of ratios g(n+1)/g(n) within any periodic residue class. The manuscript works with the class of multiplicative functions for which the worst-case ratio growth can be prescribed within those constraints. To make this explicit, the revised version will include a short construction (in the section defining the idealized gap) that assigns values on a fixed period consistently with multiplicativity while realizing a prescribed worst-case growth rate for the consecutive ratios. revision: yes

Circularity Check

0 steps flagged

No circularity identified from available text

full rationale

The abstract states that lower bounds on coincidence points are obtained depending on period length by varying worst growth rates of consecutive ratios for the given g and f. No equations, definitions of the idealized gap, or derivation steps are visible that would reduce any claimed bound to a fitted parameter or self-definition by construction. No self-citations, uniqueness theorems, or ansatzes are referenced. The derivation chain cannot be walked for circularity because the provided content contains no load-bearing steps that equate outputs to inputs tautologically. This is the expected honest non-finding when no specific reduction is exhibitable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms, or invented entities can be identified from the given text.

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discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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Nathanson, D \'e got, J \'e r \^o me , Sendov conjecture for high degree polynomials, Proceedings of the American Mathematical Society, vol. 142:4, 2014, pp 1337--1349

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