Some Generalizations of Totient Function with Elementary Symmetric Sums

N. Uday Kiran; Udvas Acharjee

arxiv: 2511.19502 · v2 · pith:4TFPZOEEnew · submitted 2025-11-23 · 🧮 math.NT

Some Generalizations of Totient Function with Elementary Symmetric Sums

Udvas Acharjee , N. Uday Kiran This is my paper

Pith reviewed 2026-05-21 18:49 UTC · model grok-4.3

classification 🧮 math.NT

keywords totient functionelementary symmetric sumsproduct formsfinite fieldslinear congruencesquadratic formsnumber theory

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The pith

Generalized totient functions with elementary symmetric sums equate to counting polynomial zeros over finite fields and solving restricted linear congruences

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

The paper generalizes the totient function by incorporating elementary symmetric polynomials into its definition. It derives explicit product forms for the case using the second elementary symmetric sum and offers observations on the arbitrary jth case. The central result establishes an equivalence among deriving these product forms, counting zeros of certain polynomials over finite fields, and solving a broad class of restricted linear congruence problems that include a greatest common divisor constraint on a quadratic form. This equivalence unifies distinct problems in number theory and is illustrated through a concrete example of the congruence problem. A sympathetic reader would care because methods developed in one of these areas could transfer directly to the others.

Core claim

The authors demonstrate the equivalence between obtaining product forms for generalized totient functions, counting zeros of specific polynomials over finite fields, and resolving a broad class of restricted linear congruence problems with a greatest common divisor constraint on a quadratic form, following from earlier generalizations using the first and kth elementary symmetric polynomials.

What carries the argument

The generalized totient function defined by replacing the standard sum with the jth elementary symmetric polynomial evaluated on the prime power factors of the divisors.

Load-bearing premise

The generalizations of the totient function via the jth elementary symmetric polynomial admit explicit product forms and satisfy the stated equivalences to zero-counting and congruence problems for the broad class claimed.

What would settle it

A concrete generalized totient function lacking an explicit product form, or a specific restricted linear congruence with quadratic-form gcd constraint that cannot be solved via the equivalence, would disprove the central claim.

Figures

Figures reproduced from arXiv: 2511.19502 by N. Uday Kiran, Udvas Acharjee.

read the original abstract

We generalize certain totient functions using elementary symmetric polynomials and derive explicit product forms for the totient functions involving the second elementary symmetric sum. This work follows from the work of Toth [The Ramanujan Journal, 2022] where the totient function was generalized using the first and the kth elementary symmetric polynomial. We also provide some observations on the behavior of the totient function with an arbitrary jth elementary symmetric polynomial. We then outline a method for solving a certain the restricted linear congruence problem with a greatest common divisor constraint on a quadratic form, illustrated by a concrete example. Most importantly, we demonstrate the equivalence between obtaining product forms for generalized totient functions, counting zeros of specific polynomials over finite fields, and resolving a broad class of restricted linear congruence problems .

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

Extends Toth with explicit product formulas for the second symmetric sum but the broad equivalence claim for congruences rests on a single example.

read the letter

The punchline is that this extends the Toth 2022 work with explicit product forms for the totient using the second elementary symmetric sum and claims equivalences to finite-field zero counting and restricted congruences, but the broad-class part of the equivalence rests on a single concrete example. They do provide the product forms and some observations for arbitrary j, which is the concrete new material. The example for the restricted linear congruence with the quadratic gcd constraint gives a sense of how these might apply in practice. The derivations seem to be the strength here, assuming they are carried through properly in the full text. It is good that they cite the prior work directly and try to connect to other problems. The soft spot is the generality of the equivalence claim. The stress test points out that it is illustrated by one quadratic example, and without a uniform reduction or parametrization for the broad class, the assertion that it resolves a broad class looks overstated. That is the part that needs the most scrutiny. Overall the paper stays within standard number theory techniques and does not show signs of circular reasoning or invented entities. The citation pattern is minimal but appropriate. This paper is for people already working on generalizations of arithmetic functions like the totient and their links to other areas such as finite fields or congruence equations. A specialist in that corner might pick up the new formulas, but it is not something that changes the landscape. I recommend sending it for peer review so that the proofs can be verified and the scope of the equivalence clarified.

Referee Report

2 major / 0 minor

Summary. The manuscript generalizes certain totient functions by replacing the usual sum with the j-th elementary symmetric polynomial, following Toth (2022) for the first and k-th cases. It derives explicit product forms specifically for the second elementary symmetric sum, offers observations on the general-j behavior, outlines a method for solving restricted linear congruences subject to a gcd constraint on a quadratic form (illustrated by one concrete example), and asserts a three-way equivalence among (i) obtaining such product forms, (ii) counting zeros of certain polynomials over finite fields, and (iii) resolving a broad class of the restricted congruences.

Significance. If the product formulas and the claimed equivalences are rigorously established with general proofs, the work would usefully extend the literature on generalized arithmetic functions and supply a concrete bridge between totient-type counting, finite-field algebraic geometry, and Diophantine congruence problems. The explicit product forms, if parameter-free and verifiable, would be a computational asset; the equivalence, if shown uniformly rather than by example, could yield new solution techniques for a wider family of congruences.

major comments (2)

[Abstract / congruence-method section] Abstract and the section outlining the restricted-linear-congruence method: the central claim of equivalence for a 'broad class' of restricted linear congruences with quadratic-form gcd constraint is supported only by a single concrete example. No general parametrization of the class or uniform reduction mapping arbitrary members to the totient-product and zero-counting statements is supplied; this leaves the generality of the three-way equivalence unproven and load-bearing for the paper's main assertion.
[Abstract] Abstract: the derivations of explicit product forms for the generalized totient functions (especially the second elementary symmetric sum case) are asserted without any displayed equations, intermediate steps, or verification against the Toth (2022) baseline; the full manuscript must contain these derivations for the claims to be checkable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to each major comment below and indicate the revisions planned to strengthen the presentation.

read point-by-point responses

Referee: [Abstract / congruence-method section] Abstract and the section outlining the restricted-linear-congruence method: the central claim of equivalence for a 'broad class' of restricted linear congruences with quadratic-form gcd constraint is supported only by a single concrete example. No general parametrization of the class or uniform reduction mapping arbitrary members to the totient-product and zero-counting statements is supplied; this leaves the generality of the three-way equivalence unproven and load-bearing for the paper's main assertion.

Authors: We agree that the equivalence is currently demonstrated via an outline of the method together with a concrete example. In the revision we will add an explicit general parametrization of the class of restricted linear congruences subject to a gcd constraint on a quadratic form, together with a uniform reduction that maps arbitrary members of this class to the corresponding finite-field zero-counting statements and the product-form expressions. This will render the three-way equivalence rigorous for the full class rather than illustrative. revision: yes
Referee: [Abstract] Abstract: the derivations of explicit product forms for the generalized totient functions (especially the second elementary symmetric sum case) are asserted without any displayed equations, intermediate steps, or verification against the Toth (2022) baseline; the full manuscript must contain these derivations for the claims to be checkable.

Authors: The abstract is a concise summary and therefore omits equations and intermediate steps, as is conventional. The full manuscript already contains the explicit derivations of the product forms for the second elementary symmetric sum case, including the intermediate algebraic steps and direct comparisons with the baseline results of Toth (2022). To improve checkability we will insert forward references from the abstract to the precise sections containing these derivations. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations and equivalences are independent of inputs

full rationale

The paper cites Toth 2022 as external prior work for the initial generalization of totient functions via elementary symmetric polynomials and then derives new explicit product forms for the case of the second symmetric sum, provides observations for arbitrary j, and outlines a congruence-solving method illustrated by one concrete quadratic-form example. The three-way equivalence is presented as a demonstrated result connecting these elements rather than assumed or fitted by construction. No equation reduces to a prior input by definition, no parameter is fitted to a subset and relabeled as a prediction, and no load-bearing step relies on a self-citation chain. The central claims therefore retain independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard properties of elementary symmetric polynomials and totient functions from prior literature; no new free parameters, invented entities, or ad-hoc axioms are stated.

axioms (1)

standard math Standard algebraic properties of elementary symmetric polynomials and the classical totient function hold and extend in the expected way to the jth symmetric sum.
Implicitly used when generalizing the totient and deriving product forms.

pith-pipeline@v0.9.0 · 5658 in / 1276 out tokens · 62527 ms · 2026-05-21T18:49:04.993027+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/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat (recovery theorem) unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We generalize certain totient functions using elementary symmetric polynomials and derive explicit product forms for the totient functions involving the second elementary symmetric sum... demonstrate the equivalence between obtaining product forms for generalized totient functions, counting zeros of specific polynomials over finite fields, and resolving a broad class of restricted linear congruence problems.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using the above result, we obtain a formula for the totient function φ_{e2}(n)... Nk(e2,p) = pk−1 + (p−1)p(k−1)/2 η(...) (quadratic-form zero counts)

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

1 extracted references · 1 canonical work pages

[1]

E1460.The American Mathe- matical Monthly, 68(9):932–933, 1961

[AGC61] Masao Arai, Jiyu Gakuen, and Leonard Carlitz. E1460.The American Mathe- matical Monthly, 68(9):932–933, 1961. [BKS+17] Khodakhast Bibak, Bruce M Kapron, Venkatesh Srinivasan, Roberto Tauraso, and L´ aszl´ o T´ oth. Restricted linear congruences.Journal of Number Theory, 171:128–144, 2017. [Bra26] A Brauer. L¨ osung der aufgabe 30, jber. deutsch. m...

work page arXiv 1961

[1] [1]

E1460.The American Mathe- matical Monthly, 68(9):932–933, 1961

[AGC61] Masao Arai, Jiyu Gakuen, and Leonard Carlitz. E1460.The American Mathe- matical Monthly, 68(9):932–933, 1961. [BKS+17] Khodakhast Bibak, Bruce M Kapron, Venkatesh Srinivasan, Roberto Tauraso, and L´ aszl´ o T´ oth. Restricted linear congruences.Journal of Number Theory, 171:128–144, 2017. [Bra26] A Brauer. L¨ osung der aufgabe 30, jber. deutsch. m...

work page arXiv 1961