Proof of Summability of the Negative Powers of the Fourier Weights of the Torus

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named. Natural numbers are read in $\mathbb{R}$ through the canonical map, as fixed in The Real Numbers: Standing Notation and Background (i); they are positive there by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and by claim 6 of that lemma the canonical map is strictly increasing, so that an inequality between natural numbers holds in $\mathbb{N}$ if and only if it holds in $\mathbb{R}$, the passage from $\mathbb{R}$ back to $\mathbb{N}$ using the trichotomy of claim 3 of Properties of the Order on the Natural Numbers. Write $\alpha=4\pi^{2}$, a positive real number as recorded in the statement. We use that $\le$ on $\mathbb{R}$ is a total order, so reflexive and transitive, and that $x\le y$ if and only if $0\le y-x$, by claim 3 of Elementary Arithmetic in an Ordered Field.

Claim 1 (The denominators). Let $k\in\mathbb{Z}^{n}$ and $i\in[n]$. Then $0\le\lVert k\rVert^{2}$ and $0\le k_{i}^{2}$ by claim 2 of Nonnegativity of Squares in an Ordered Field, so $0\le\alpha\lVert k\rVert^{2}$ and $0\le\alpha k_{i}^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field, applied with the nonnegative multiplier $\alpha$, together with $\alpha\cdot0=0$ from claim 1 of Zero Products and Elementary Identities in a Field. Hence

$$1\le\mu_{k},\qquad 1\le 1+\alpha k_{i}^{2},$$

and since $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field, the mixed transitivity of claim 2 of that lemma makes $\mu_{k}$ and $1+\alpha k_{i}^{2}$ positive; their inverses exist and are positive by claim 7 of that lemma. Moreover $\mu_{k}^{s}$ is positive: $0\le\mu_{k}$ gives $0\le\mu_{k}^{s}$ by claim 5 of Properties of Natural Number Powers in a Field, and $\mu_{k}^{s}\ne0$ by claim 4 of that lemma since $\mu_{k}\ne0$; so $0<\mu_{k}^{s}$, and its inverse exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field.

Claim 2 (Clause 1). Let $k\in\mathbb{Z}^{n}$; positivity was shown in claim 1. Let $i\in[n]$. By claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square one has $|k_{i}|\le\lVert k\rVert$, and both sides are nonnegative by claim 1 of Properties of the Absolute Value in an Ordered Field and claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square; so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|k_{i}|^{2}\le\lVert k\rVert^{2}$, and $|k_{i}|^{2}=k_{i}^{2}$ by claim 1 of Nonnegativity of Squares in an Ordered Field. Multiplying by the nonnegative $\alpha$ through claim 5 of Elementary Arithmetic in an Ordered Field and adding $1$ gives

$$1+\alpha k_{i}^{2}\le\mu_{k}\qquad\text{for every }i\in[n].$$

All these numbers are nonnegative by claim 1, so claim 5 of Properties of Finite Products, applied to the two families $i\mapsto 1+\alpha k_{i}^{2}$ and $i\mapsto\mu_{k}$ on $[n]$, gives

$$\prod_{i=1}^{n}\bigl(1+\alpha k_{i}^{2}\bigr)\le\prod_{i=1}^{n}\mu_{k}=\mu_{k}^{n},$$

the last equality being Natural Number Power of an Element of a Field. Since $1\le\mu_{k}$ and $n\le s$, claim 2 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities gives $\mu_{k}^{n}\le\mu_{k}^{s}$, these real powers agreeing with the natural powers by claim 1 of that lemma. The left-hand product is positive: its factors are positive by claim 1, so the product is nonnegative by claim 5 of Properties of Finite Products and nonzero by claim 4 of that lemma. So by transitivity and claim 1 of Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares (i),

$$\frac{1}{\mu_{k}^{s}}\le\frac{1}{\prod_{i=1}^{n}\bigl(1+\alpha k_{i}^{2}\bigr)} .$$

Finally, applying claim 2 of Properties of Finite Products to the families $i\mapsto 1+\alpha k_{i}^{2}$ and $i\mapsto(1+\alpha k_{i}^{2})^{-1}$, whose pointwise product is the constant family $1$ with $\prod_{i=1}^{n}1=1^{n}=1$ by Natural Number Power of an Element of a Field and claim 2 of Properties of Natural Number Powers in a Field, shows that $\prod_{i=1}^{n}(1+\alpha k_{i}^{2})^{-1}$ is the inverse of $\prod_{i=1}^{n}(1+\alpha k_{i}^{2})$. This proves clause 1.

Claim 3 (A cube containing finitely many lattice points). Let $\kappa$ be as in clause 2 and let $N\in\mathbb{N}$. Then there is $M\in\mathbb{N}$ such that

$$-M\le\kappa(j)_{i}\le M\qquad\text{for every }j\in[N]\text{ and every }i\in[n].$$

Indeed, put $t_{j}=\sum_{i=1}^{n}|\kappa(j)_{i}|$ for $j\in[N]$ and $T=\sum_{j=1}^{N}t_{j}$. All the summands are nonnegative by claim 1 of Properties of the Absolute Value in an Ordered Field, hence so is each $t_{j}$ by claim 5 of Properties of Finite Sums; by claim 6 of that lemma $|\kappa(j)_{i}|\le t_{j}$ and $t_{j}\le T$, so $|\kappa(j)_{i}|\le T$ by transitivity, and $0\le T$ by claim 5 again. By claim 4 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities the number $M=\lfloor T\rfloor+1$ is a natural number with $T<M$. Hence $|\kappa(j)_{i}|\le M$ by the mixed transitivity of claim 2 of Elementary Order Arithmetic in an Ordered Field, and claim 6 of Properties of the Absolute Value in an Ordered Field turns this into the two-sided bound asserted.

Claim 4 (Carrying the lattice points into a cube of tuples). Let $\kappa$, $N$ and $M$ be as in claim 3, let $Q=M+M+1$, and let $[Q]^{n}$ be the set of $n$-tuples in $[Q]$, which is nonempty and finite by claim 3 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets. Put $F=\{\kappa(j):j\in[N]\}$.

For $k\in F$ and $i\in[n]$ the real number $k_{i}+M+1$ is an integer, by claim 2 of Arithmetic, Order, Discreteness and Intervals of the Integers together with $k_{i}\in\mathbb{Z}$ from Lattice-Periodic Functions and the Periodic Function Classes (i) and $1\in\mathbb{Z}$, $M\in\mathbb{Z}$ from The Integers as a Subset of the Real Numbers. From $-M\le k_{i}\le M$ we get $1\le k_{i}+M+1\le Q$. Being an integer that is at least $1$, and so positive, $k_{i}+M+1$ is the image of a natural number: by The Integers as a Subset of the Real Numbers it is $0$, or $\iota(a)$, or $-\iota(a)$ for some $a\in\mathbb{N}$, and the first and third are excluded because $\iota(a)$ is positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field and hence $-\iota(a)$ is negative by claim 4 of Elementary Order Arithmetic in an Ordered Field. That natural number lies in $[Q]$, since the inequality $k_{i}+M+1\le Q$ may be read in $\mathbb{N}$, as recorded at the start of this proof.

Let $\beta:F\to[Q]^{n}$ send $k$ to the map on $[n]$ whose value at $i$ is the natural number just described. It is injective: if $\beta(k)=\beta(k')$ then $k_{i}+M+1=k'_{i}+M+1$, hence $k_{i}=k'_{i}$, for every $i\in[n]$, so $k=k'$ by claim 1 of Euclidean Points as Tuples of Real Numbers.

Claim 5 (Clause 2). Let $\kappa$ be as in clause 2. By claim 1 the terms $\tfrac{1}{\mu_{\kappa(j)}^{s}}$ are positive, in particular nonnegative. Let $N\in\mathbb{N}$, and let $M$, $Q$, $F$ and $\beta$ be as in claims 3 and 4.

The map $j\mapsto\kappa(j)$ from $[N]$ to $F$ is a bijection: it is onto $F$ by the definition of $F$, and the value at a point of $F$ is attained at only one $j$ because $\kappa$ is injective. Hence $F$ is nonempty and finite, by claim 4 of Basic Properties of Finite Sets applied to $[N]$, which has $N$ elements by claim 1 of that lemma. Writing $g(k)=\tfrac{1}{\mu_{k}^{s}}$ for $k\in\mathbb{Z}^{n}$, claim 1 of Properties of a Sum over a Finite Index Set and then claim 2 of that lemma, applied to the bijection just described, give

$$\sum_{j=1}^{N}\frac{1}{\mu_{\kappa(j)}^{s}}=\sum_{j\in[N]}g\bigl(\kappa(j)\bigr)=\sum_{k\in F}g(k).$$

Let $\Phi:[Q]^{n}\to\mathbb{R}$ be the map with

$$\Phi(f)=\prod_{i=1}^{n}\frac{1}{1+\alpha\,(f(i)-M-1)^{2}},$$

each factor being defined and positive as in claim 4 of Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares (iv); hence $0\le \Phi(f)$ by claim 5 of Properties of Finite Products. For $k\in F$ and $i\in[n]$ the $i$-th component of $\beta(k)$ satisfies $\beta(k)(i)-M-1=k_{i}$, so $\Phi(\beta(k))=\prod_{i=1}^{n}(1+\alpha k_{i}^{2})^{-1}$, which dominates $g(k)$ by clause 1. Therefore, by clause 1 of Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits (i) applied on the index set $F$,

$$\sum_{k\in F}g(k)\le\sum_{k\in F}\Phi\bigl(\beta(k)\bigr).$$

Let $E=\{\beta(k):k\in F\}\subseteq[Q]^{n}$, a nonempty set on which $\beta$ is a bijection from $F$, by its injectivity. By claim 2 of Properties of a Sum over a Finite Index Set and then clause 3 of Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits (iii), applied to the nonnegative $\Phi$ on $[Q]^{n}$ and its nonempty subset $E$,

$$\sum_{k\in F}\Phi\bigl(\beta(k)\bigr)=\sum_{f\in E}\Phi(f)\le\sum_{f\in[Q]^{n}}\Phi(f).$$

Let $c$ be the family on $[n]$ of families on $[Q]$ with $c_{im}=\bigl(1+\alpha(m-M-1)^{2}\bigr)^{-1}$, independent of $i$. Then $\prod_{i=1}^{n}c_{i\,f(i)}=\Phi(f)$ for every $f\in[Q]^{n}$, so Generalized Distributivity: Expanding a Product of Finite Sums gives

$$\sum_{f\in[Q]^{n}}\Phi(f)=\prod_{i=1}^{n}\Bigl(\sum_{m=1}^{Q}c_{im}\Bigr).$$

Each of these $n$ sums equals $\sum_{m=1}^{2M+1}\bigl(1+\alpha(m-M-1)^{2}\bigr)^{-1}$ and is therefore at most $1+\tfrac{4}{\alpha}$ by clause 4 of Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares (iv); each is also nonnegative by claim 5 of Properties of Finite Sums, its summands $c_{im}$ being positive as recorded above. Hence claim 5 of Properties of Finite Products gives

$$\prod_{i=1}^{n}\Bigl(\sum_{m=1}^{Q}c_{im}\Bigr)\le\prod_{i=1}^{n}\Bigl(1+\frac{4}{\alpha}\Bigr)=\Bigl(1+\frac{4}{\alpha}\Bigr)^{n},$$

by Natural Number Power of an Element of a Field. Since $\alpha=4\pi^{2}$ and $\pi^{2}$ is nonzero, field arithmetic gives $\tfrac{4}{\alpha}=\tfrac{1}{\pi^{2}}$.

Combining the displays, every partial sum satisfies

$$\sum_{j=1}^{N}\frac{1}{\mu_{\kappa(j)}^{s}}\le\Bigl(1+\frac{1}{\pi^{2}}\Bigr)^{n},$$

so the set of partial sums is bounded above by that number. By claim 1 of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series the series converges with sum the least upper bound of the partial sums, and that least upper bound is at most $(1+\tfrac{1}{\pi^{2}})^{n}$, since the latter is an upper bound. This proves clause 2.