Lattice Sums of Powers of the Fourier Weights in Dimension at Most Two: Power Sums, the Growth of the Wick Constants, and the Quartic Convolution Sums

In the setting of The Real Numbers: Standing Notation and Background, let $n\in\mathbb{N}$ with $n\le2$, and let $\mu_{k}=1+4\pi^{2}\lVert k\rVert^{2}$ ($k\in\mathbb{Z}^{n}$) be the Fourier weights of Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families, where $\pi$ is the number of The Number Pi §pi and $\lVert\cdot\rVert$ is the Euclidean norm; thus $1\le\mu_{k}$. Powers $s^{r}$ of a positive real number $s$ with a real exponent $r$ are those of Real Power of a Positive Real Number. For $l\in\mathbb{N}$, cube sums and cube-summable families on $\mathbb{Z}^{l}$, and their lattice sums $\sum_{k\in\mathbb{Z}^{l}}$, are those of Cube Sums of Families on the Integer Lattice, with the cubes $\Gamma_{N}$ ($N\in\mathbb{N}$) of The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points; families on $\mathbb{Z}^{3n}$ are written as functions of $(k_{1},k_{2},k_{3})\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}^{n}$.

1. (Power sums)¶ For every real number $s$ with $n<2s$, the family $k\mapsto\mu_{k}^{-s}$ on $\mathbb{Z}^{n}$ is cube-summable.

2. (Growth of the cube sums)¶ For every positive real number $\varepsilon$ and every $N\in\mathbb{N}$,

$$\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}\ \le\ \bigl(1+4\pi^{2}nN^{2}\bigr)^{\varepsilon}\sum_{k\in\mathbb{Z}^{n}}\mu_{k}^{-(1+\varepsilon)},$$

the lattice sum on the right being defined by claim 1, since $n\le2<2+2\varepsilon$.

3. (Quartic convolution sums)¶ Let $\delta\in\mathbb{R}$ satisfy $0\le\delta<1/4$, and for $(k_{1},k_{2},k_{3})\in\mathbb{Z}^{3n}$ write $k_{4}=-(k_{1}+k_{2}+k_{3})$. For every $i\in\{1,2,3,4\}$ the family

$$(k_{1},k_{2},k_{3})\ \mapsto\ \mu_{k_{i}}^{\delta}\,\mu_{k_{1}}^{-1}\mu_{k_{2}}^{-1}\mu_{k_{3}}^{-1}\mu_{k_{4}}^{-1}$$

on $\mathbb{Z}^{3n}$ is cube-summable.