On coprime residue averaging and spectral properties of the circle Laplacian

Abstract We examine the standard Laplacian $\widehat{H} = -\pi\partial_\theta^2$ on the circle $S^1$ through the lens of averaging operators $P_n$ associated to the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^\times$ of coprime residues. These operators act diagonally in the Fourier basis via Ramanujan sums $c_n(k)/\varphi(n)$, and we show that their limiting behaviour provides an arithmetic characterisation of the integer Fourier modes. The heat kernel of $\widehat{H}$ equals Jacobi's theta function $\theta(t) = \sum_{k\in\mathbb{Z}} e^{-\pi k^2 t}$, whose Mellin transform yields standard connections to zeta functions. The work presents a purely operator-theoretic perspective on how coprime residue structure interacts with spectral theory.

1. Introduction

1.1 Mathematical context

The spectral analysis of differential operators with arithmetic constraints has long been of interest in mathematical analysis and number theory. A classic example is the Laplacian on the circle $S^1$, whose eigenfunctions $e^{ik\theta}$ ($k\in\mathbb{Z}$) form the Fourier basis of $L^2(S^1)$. The eigenvalues $\pi k^2$ appear in the heat kernel $\theta(t) = \sum_{k\in\mathbb{Z}} e^{-\pi k^2 t}$, which via Mellin transform connects to the Riemann zeta function.

1.2 Objective and approach

This note explores how arithmetic structure—specifically the coprime residue groups $(\mathbb{Z}/n\mathbb{Z})^\times$—interacts with the spectral theory of the circle Laplacian. Rather than constructing new operators, we examine the standard operator $\widehat{H} = -\pi\partial_\theta^2$ on its usual domain $H^2_{\text{per}}(S^1)$, but introduce a family of averaging operators $P_n$ that encode the $\varphi(n)$-coprime structure.

The key observation is that these $P_n$ operators:

Act diagonally on the Fourier basis via Ramanujan sums
Converge strongly to projection onto constants as $n\to\infty$
Provide an arithmetic characterisation of which Fourier modes are compatible with coprime averaging

All analytic results used are classical; the contribution is in packaging them around this arithmetic averaging perspective and making explicit the role of $\varphi(n)$ and Ramanujan sums.

1.3 Structure

Section 2 defines the averaging operators $P_n$ and establishes their basic properties. Section 3 reviews the standard circle Laplacian and its spectral theory. Section 4 examines how the $P_n$ operators interact with this spectral structure. Section 5 connects to heat kernels and theta functions, while Section 6 recalls the standard Mellin transform connections to zeta functions. Section 7 offers concluding remarks.

2. The $\varphi(n)$-averaging operators

2.1 Algebraic preliminaries

For $n\geq 2$, the set of coprime residues modulo $n$:

$$ (\mathbb{Z}/n\mathbb{Z})^\times = \{ r \in \{1,\dots,n-1\} : \gcd(r,n) = 1 \} $$

forms a multiplicative group of order $\varphi(n)$, Euler's totient function. This is a purely algebraic fact with no empirical component.

2.2 Definition and basic properties

Let $S^1 = \mathbb{R}/(2\pi\mathbb{Z})$ with angular coordinate $\theta$.

Definition 2.1.

For each $n\geq 2$, define the averaging operator $P_n: L^2(S^1) \to L^2(S^1)$ by $$ (P_n f)(\theta) = \frac{1}{\varphi(n)} \sum_{r \in (\mathbb{Z}/n\mathbb{Z})^\times} f\left(\theta + \frac{2\pi r}{n}\right). $$

Each $P_n$ is bounded ($\|P_n\| \leq 1$) and self-adjoint, being an average of unitary translation operators.

2.3 Fourier basis action

For the Fourier basis functions $e_k(\theta) = e^{ik\theta}$, $k\in\mathbb{Z}$:

Lemma 2.2.

For $f(\theta) = \sum_{k\in\mathbb{Z}} a_k e^{ik\theta}$, $$ (P_n f)(\theta) = \sum_{k\in\mathbb{Z}} a_k \frac{c_n(k)}{\varphi(n)} e^{ik\theta}, $$ where $c_n(k) = \sum_{r\in(\mathbb{Z}/n\mathbb{Z})^\times} e^{2\pi i k r/n}$ is the Ramanujan sum.

Proof. Direct computation: $$ \begin{align*} (P_n f)(\theta) &= \frac{1}{\varphi(n)} \sum_{r\in(\mathbb{Z}/n\mathbb{Z})^\times} \sum_{k\in\mathbb{Z}} a_k e^{ik(\theta + 2\pi r/n)} \\ &= \sum_{k\in\mathbb{Z}} a_k e^{ik\theta} \left( \frac{1}{\varphi(n)} \sum_{r\in(\mathbb{Z}/n\mathbb{Z})^\times} e^{2\pi i k r/n} \right) \\ &= \sum_{k\in\mathbb{Z}} a_k \frac{c_n(k)}{\varphi(n)} e^{ik\theta}. \end{align*} $$ ◼

Thus each $P_n$ acts diagonally on the Fourier basis, with eigenvalues $c_n(k)/\varphi(n)$.

2.4 Limiting behaviour

A key property is how these eigenvalues behave as $n\to\infty$:

Lemma 2.3 (Hardy and Wright, Theorem 272).

For fixed $k\in\mathbb{Z}$, $$ \frac{c_n(k)}{\varphi(n)} \to \begin{cases} 1 & \text{if } k = 0, \\ 0 & \text{if } k \neq 0. \end{cases} $$

Proof. The identity $c_n(k) = \mu(n/\gcd(n,k)) \varphi(n)/\varphi(n/\gcd(n,k))$ yields $$ \frac{c_n(k)}{\varphi(n)} = \frac{\mu(n/\gcd(n,k))}{\varphi(n/\gcd(n,k))}. $$ For $k=0$, $\gcd(n,0)=n$ so $\mu(1)/\varphi(1)=1$. For $k\neq 0$, the function $n\mapsto \mu(n/\gcd(n,k))$ is bounded (by 1 in absolute value) while $\varphi(n/\gcd(n,k)) \to \infty$ as $n\to\infty$ through values with $\gcd(n,k)$ fixed. ◼

An immediate consequence:

Corollary 2.4.

$P_n$ converges strongly to the orthogonal projection $P_0$ onto constants: $$ P_0 f = \frac{1}{2\pi}\int_0^{2\pi} f(\theta)\,d\theta. $$

Proof. For trigonometric polynomials $f$, Lemma 2.3 gives pointwise convergence of Fourier coefficients. For general $f\in L^2(S^1)$, use density and the uniform bound $\|P_n\|\leq 1$. ◼

3. The circle Laplacian

3.1 Standard construction

Consider the Hilbert space $H = L^2(S^1)$ with normalised Lebesgue measure $d\theta/(2\pi)$.

Definition 3.1.

The circle Laplacian is $\widehat{H} = -\pi\frac{d^2}{d\theta^2}$ with domain $$ D(\widehat{H}) = H^2_{\text{per}}(S^1) = \{ f\in L^2(S^1) : f,f',f''\in L^2(S^1),\ f^{(j)}(0)=f^{(j)}(2\pi),\ j=0,1 \}. $$

This is the standard self-adjoint realisation of the Laplacian on $S^1$.

3.2 Spectral theory

Theorem 3.2.

The operator $\widehat{H}$ has pure point spectrum $\sigma(\widehat{H}) = \{\pi k^2 : k\in\mathbb{Z}\}$, with eigenfunctions $e_k(\theta)=e^{ik\theta}$ forming a complete orthonormal basis of $L^2(S^1)$.

Proof. Direct computation gives $\widehat{H}e_k = \pi k^2 e_k$. Completeness of $\{e_k\}_{k\in\mathbb{Z}}$ in $L^2(S^1)$ is the standard Fourier series result. ◼

3.3 Commutation with averaging

Proposition 3.3.

For all $n\geq 2$, $[\widehat{H}, P_n] = 0$.

Proof. Both operators act diagonally on the Fourier basis: $\widehat{H}$ multiplies by $\pi k^2$, while $P_n$ multiplies by $c_n(k)/\varphi(n)$. Hence they commute. ◼

This commutation expresses compatibility between the geometric operator $\widehat{H}$ and the arithmetic operators $P_n$.

4. Arithmetic characterisation of eigenfunctions

4.1 Strong averaging condition

One might consider imposing a strong invariance condition:

Definition 4.1.

A function $f\in L^2(S^1)$ satisfies the strong $\varphi(n)$-averaging condition if $$ \lim_{n\to\infty} \|f - P_n f\| = 0. $$

Lemma 4.2.

$f$ satisfies the strong condition if and only if $f$ is constant.

Proof. By Corollary 2.4, $P_n f\to P_0 f$ (projection onto constants). Hence $\|f-P_n f\|\to 0$ implies $f = P_0 f$, i.e., $f$ is constant. ◼

Thus the strong condition is too restrictive for non-trivial spectral analysis.

4.2 Compatibility with eigenfunctions

A more fruitful approach is to examine how eigenfunctions of $\widehat{H}$ interact with the averaging:

Theorem 4.3.

For an eigenfunction $e_k(\theta)=e^{ik\theta}$ of $\widehat{H}$, $$ P_n e_k = \frac{c_n(k)}{\varphi(n)} e_k. $$ Moreover, $P_n e_k \to 0$ weakly for $k\neq 0$, and $P_n e_0 = e_0$ for all $n$.

Proof. The first statement is Lemma 2.2. For $k\neq 0$, Lemma 2.3 gives $c_n(k)/\varphi(n)\to 0$, so $P_n e_k\to 0$ in norm (hence weakly). For $k=0$, $c_n(0)=\varphi(n)$ so $P_n e_0 = e_0$. ◼

This shows that among eigenfunctions, only the constant one ($k=0$) is preserved by the averaging process in the limit $n\to\infty$.

4.3 Interpretation

The $P_n$ operators provide an arithmetic filter on the Fourier basis:

Constant function ($k=0$): preserved exactly by all $P_n$
Non-constant integer frequencies ($k\neq 0$): attenuated, converging to zero
Non-integer frequencies: not eigenfunctions of $\widehat{H}$ on $L^2(S^1)$ due to periodicity requirement

Thus the integer frequency condition—which geometrically comes from $2\pi$-periodicity—is also arithmetically natural from the coprime averaging perspective.

5. Heat kernel and theta function

5.1 Heat kernel computation

Theorem 5.1.

The heat kernel of $\widehat{H}$ is $$ K(t,\theta,\theta') = \sum_{k\in\mathbb{Z}} e^{-\pi k^2 t} e^{ik(\theta-\theta')}, $$ and its trace is $$ \theta_{\widehat{H}}(t) = \operatorname{Tr}(e^{-t\widehat{H}}) = \sum_{k\in\mathbb{Z}} e^{-\pi k^2 t}. $$

Proof. By the spectral theorem and Theorem 3.2, $$ e^{-t\widehat{H}} e_k = e^{-\pi k^2 t} e_k. $$ Thus $K(t,\theta,\theta') = \sum_k e^{-\pi k^2 t} e_k(\theta)\overline{e_k(\theta')}$. Setting $\theta=\theta'$ and integrating gives the trace. ◼

5.2 Jacobi theta function

The trace $\theta_{\widehat{H}}(t)$ is Jacobi's theta function $\theta(t)$. Its functional equation follows from Poisson summation:

Theorem 5.2 (Poisson summation).

For $t > 0$, $$ \theta(t) = t^{-1/2} \theta(1/t). $$

Proof. Apply Poisson summation to the Gaussian $f(x)=e^{-\pi t x^2}$: $$ \sum_{k\in\mathbb{Z}} e^{-\pi t k^2} = \sum_{k\in\mathbb{Z}} \widehat{f}(k) = \sum_{k\in\mathbb{Z}} t^{-1/2} e^{-\pi k^2/t} = t^{-1/2} \theta(1/t). $$ ◼

6. Connection to zeta functions

6.1 Mellin transform representation

Theorem 6.1.

For $\operatorname{Re}(s) > 1$, $$ \pi^{-s} \Gamma(s) \zeta(2s) = \frac{1}{2} \int_0^\infty (\theta(t)-1) t^{s/2} \frac{dt}{t}. $$

Proof. For $k\geq 1$, $$ \int_0^\infty e^{-\pi k^2 t} t^{s/2} \frac{dt}{t} = \pi^{-s} k^{-2s} \Gamma(s). $$ Summing over $k\geq 1$ yields the result. ◼

6.2 Completed zeta function

Define Riemann's completed zeta function as

$$ \xi(s) = \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s). $$

Theorem 6.2 (Riemann, 1859).

$\xi(s) = \xi(1-s)$.

Proof. Using Theorem 6.1 with $s$ replaced by $s/2$, and the functional equation for $\theta(t)$ (Theorem 5.2), yields the result by standard manipulation (see Titchmarsh §2.1). ◼

6.3 Comparison with Epstein zeta functions

Epstein zeta functions arise from Laplacians on higher-dimensional tori, summing over lattice points $\mathbb{Z}^d$ rather than $\mathbb{Z}$. In those cases, the spectrum involves quadratic forms in several variables. Our one-dimensional setting is simpler, with spectrum $\{\pi k^2\}$. The $\varphi(n)$-averaging perspective developed here could potentially generalise to such higher-dimensional settings, but we do not pursue this here.

7. Concluding remarks

We have examined the standard circle Laplacian $\widehat{H} = -\pi\partial_\theta^2$ from the perspective of coprime residue averaging operators $P_n$. Key observations include:

The $P_n$ act diagonally on the Fourier basis via Ramanujan sums $c_n(k)/\varphi(n)$
They converge strongly to projection onto constants: $P_n \to P_0$
Only the constant eigenfunction ($k=0$) is preserved in the limit; others are attenuated
This provides an arithmetic characterisation of the integer frequency condition that is geometrically imposed by $2\pi$-periodicity
The heat kernel equals Jacobi's theta function, connecting via Mellin transform to standard zeta function theory

The work illustrates how arithmetic structure—encoded through the groups $(\mathbb{Z}/n\mathbb{Z})^\times$—interacts with spectral theory. All analytic results used are classical; the novelty lies in the packaging around the $\varphi(n)$-averaging operators and the explicit connection to Ramanujan sums.

Potential extensions include:

Generalisation to Dirichlet characters and $L$-functions
Higher-dimensional analogues on tori
Connections to Hecke operators and modular forms

These directions remain for future investigation.