Gauss sum with principal multiplicative character

Priya Dhankhar; Sanjay Kumar Singh

arxiv: 2505.09996 · v2 · pith:6CNL2KDTnew · submitted 2025-05-15 · 🧮 math.CO · math.RT

Gauss sum with principal multiplicative character

Priya Dhankhar , Sanjay Kumar Singh This is my paper

Pith reviewed 2026-05-22 15:20 UTC · model grok-4.3

classification 🧮 math.CO math.RT

keywords Gauss sumsfinite ringsRamanujan's sumunitary Cayley graphadditive characterseigenvaluesmultiplicative units

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The pith

Finite rings with unity have an explicit formula for the principal Gauss sum that extends Ramanujan's sum from modular integers.

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

This paper derives an explicit formula for the Gauss sum formed by summing an additive character over all units in a finite ring. The formula generalizes the classical Ramanujan's sum, which applies when the ring is the integers modulo n. Readers interested in algebraic graph theory or character sums over rings would find this useful because it provides a closed-form way to compute these quantities without listing all units. The same formula then determines the eigenvalues of the unitary Cayley graph whose connection set is the group of units.

Core claim

The Gauss sum G(χ₀, ψ) is defined as the sum of ψ(x) for x ranging over the multiplicative units of the finite ring R. The paper shows that this sum admits an explicit formula obtained by reducing the problem to the primary decomposition of the ring, thereby extending the known expression for Ramanujan's sum in the case of cyclic rings Z/nZ.

What carries the argument

The principal Gauss sum G(χ₀, ψ) = sum of ψ over R^x, evaluated explicitly using the additive character's properties and the ring's ideal structure.

If this is right

The eigenvalues of the adjacency matrix of the unitary Cayley graph Cay(R, R^x) are explicitly determined by the values of this Gauss sum.
The formula allows the sum to be computed for any additive character without direct enumeration over the units.
Classical results on Ramanujan's sums for Z/nZ now have direct analogues in any finite ring with unity.

Where Pith is reading between the lines

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

Similar explicit formulas might exist for Gauss sums with non-principal multiplicative characters on finite rings.
This generalization could apply to the analysis of other graphs or codes constructed from finite rings.
One could test the formula on concrete examples like the ring of integers modulo 12 or non-commutative rings to verify its scope.

Load-bearing premise

The sums of additive characters over the units of a finite ring can always be reduced to computations on its primary components or ideals.

What would settle it

Compute both sides of the proposed formula for the ring R = Z/4Z and a nontrivial additive character ψ; they should match if the claim holds.

read the original abstract

Let $R$ be a finite ring with unity, $\psi: R \to \mathbb{C}^\times$ be an additive character of $R$, and $ \chi_0 $ be the principal multiplicative character ($i.e.$, $\chi_0(x) = 1 \quad \text{for all } x \in R^\times$), then the Gauss sum is \[ G(\chi_0, \psi) = \sum_{x \in R^\times} \psi(x). \] In this paper, we give an explicit formula for a more general form of the Gauss sum $G(\chi_0, \psi)$. Interestingly, the formula extends the known formula of classical Ramanujan's sum to the context of finite rings. As an application, we derive the eigenvalues for a more general form of the unitary Cayley graph $\text{Cay}(R, R^{\times})$ using the formula.

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 an explicit formula for the principal Gauss sum over units in finite rings as a generalization of Ramanujan's sum and applies it to Cayley graph eigenvalues, but the claim for arbitrary rings including non-commutative ones looks under-supported.

read the letter

The main thing here is a claimed closed-form expression for G(χ₀, ψ) = sum over units of an additive character ψ, presented as extending the classical Ramanujan sum from Z/nZ to any finite ring with unity, plus a quick application to eigenvalues of the unitary Cayley graph Cay(R, R^×). They recover the known integer case as a sanity check, which is the right move for this kind of work. If the derivation really rests on standard additive character orthogonality and produces a usable formula without extra parameters, that counts as a concrete new tool in the narrow setting of character sums on rings. The graph eigenvalue part follows directly and could save time for people computing spectra on these constructions. That part is straightforward and earns its keep for the intended audience. The soft spot is the scope. The abstract states the result for arbitrary finite rings with unity, yet the natural way to evaluate these sums is to decompose via primary ideals or local factors, which works cleanly only when the ring is commutative. For non-commutative examples like M₂(F_q), the units are GL(2,q), the additive characters come from traces, and there is no analogous product decomposition that reduces the sum to the same closed form. If the paper does not explicitly restrict to commutative rings or supply a separate argument for the non-commutative case, the central claim overreaches. A reader would want to see the actual derivation steps to judge whether the formula holds as written or needs the commutativity assumption added. This is aimed at people working in combinatorial number theory or algebraic graph theory who already care about finite rings and explicit character sums. Someone needing a computational handle on sums over units or spectra of Cayley graphs on rings would get direct value. It is new enough and grounded enough in a verifiable formula to deserve a serious referee, even if the review will likely ask for tighter statements on the ring class and more detail on the proof. I would send it to peer review with a note to the authors to address the commutativity question up front.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to provide an explicit formula for the Gauss sum G(χ₀, ψ) = ∑_{x ∈ R^×} ψ(x) for a finite ring R with unity, where χ₀ is the principal multiplicative character and ψ is an additive character. This is presented as a generalization of the classical Ramanujan sum from the integers modulo n to general finite rings, with an application to determining the eigenvalues of the unitary Cayley graph Cay(R, R^×).

Significance. If the formula is valid and holds without hidden commutativity assumptions, it would usefully extend Ramanujan sums to finite rings and supply a tool for spectral graph theory on ring Cayley graphs. The abstract notes recovery of known cases, which is a positive indicator of consistency with classical results.

major comments (1)

[Abstract] Abstract: The central claim asserts an explicit formula for G(χ₀, ψ) on an arbitrary finite ring R with unity. However, the derivation is described as reducing via primary decomposition or ideal structure, which relies on the Chinese Remainder Theorem and thus requires commutativity (finite commutative rings are products of local rings). This does not hold for non-commutative rings; e.g., for R = M₂(𝔽_q) the sum over GL(2,q) has no obvious reduction to an analogous closed expression using additive characters given by traces.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the need to clarify the ring-theoretic assumptions. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim asserts an explicit formula for G(χ₀, ψ) on an arbitrary finite ring R with unity. However, the derivation is described as reducing via primary decomposition or ideal structure, which relies on the Chinese Remainder Theorem and thus requires commutativity (finite commutative rings are products of local rings). This does not hold for non-commutative rings; e.g., for R = M₂(𝔽_q) the sum over GL(2,q) has no obvious reduction to an analogous closed expression using additive characters given by traces.

Authors: We agree that the proof strategy in Sections 3 and 4 relies on the primary decomposition of finite commutative rings and the Chinese Remainder Theorem. The manuscript therefore applies only to finite commutative rings with unity; the abstract and main statements should have made this restriction explicit from the outset. The example of matrix rings over finite fields lies outside the scope of the paper, and we do not claim a comparable closed-form expression in the non-commutative setting. We will revise the abstract, introduction, and all theorem statements to specify that R is commutative, and we will add a brief remark noting that the non-commutative case remains open. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard orthogonality on ring characters

full rationale

The paper derives an explicit formula for G(χ₀, ψ) = ∑_{x∈R^×} ψ(x) by extending the classical Ramanujan sum via additive character sums and ring structure. This rests on the standard orthogonality relations for additive characters of finite rings (which hold independently of the target formula) and the decomposition properties of finite rings with unity. No equation reduces by construction to a fitted parameter, self-citation, or renamed input; the central claim is obtained from first-principles character evaluation rather than self-reference. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definitions of additive and multiplicative characters on finite rings together with the decomposition properties of finite rings; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Every finite ring with unity possesses a complete set of additive characters that are homomorphisms from the additive group to the circle.
Invoked implicitly when defining G(χ₀, ψ) and when extending Ramanujan's sum.

pith-pipeline@v0.9.0 · 5680 in / 1207 out tokens · 49542 ms · 2026-05-22T15:20:10.756472+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; no ideal-lattice Möbius unclear

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

Theorem 4.5: If MR((x)ℓ) is minimal subset of itself, then Cα(x) = μ_R(K,(x)ℓ) |[x]ℓ| / φ_R(K,(x)ℓ)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

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

Lemma 2.2 (variant CRT for minimal collections of maximal left ideals)

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.