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Let $n$ be a positive integer. Define the Gauss sum $$g(\ell,n):=\sum_{k=0}^{n-1}\,\omega_n^{k^2\ell}=\sum_{k=0}^{n-1}\,\exp\left(\frac{2\pi\text{i}k^2\ell}{n}\right)\,,$$ for every integer $\ell$. Here, $\omega_n$ is the primitive $n$-th root of unity $\exp\left(\frac{2\pi\text{i}}{n}\right)$, and $\text{i}$ is the imaginary unit $\sqrt{-1}$. From my observation (using Mathematica), for every integer $n>0$, we have $$\frac{g(1,n)}{\sqrt{n}}=\begin{cases} 1+\text{i}\,,&\mbox{if }n\equiv 0\pmod{4}\,, \\ 1\,,&\mbox{if }n\equiv 1\pmod{4}\,, \\ 0\,,&\mbox{if }n\equiv 2\pmod{4}\,, \\ \text{i}\,,&\mbox{if }n\equiv 3\pmod{4}\,. \end{cases}\tag{*}$$ The question is to compute $g(\ell,n)$ for general $\ell$ and $n$.

Note also that, for every $\ell\in\mathbb{Z}$, $g(\ell+n,n)=g(\ell,n)$ and $g(-\ell,n)=\overline{g(\ell,n)}$, where $\bar{z}$ is the complex conjugate of $z\in\mathbb{C}$. Thus, it suffices to evaluate $g(\ell,n)$ for integers $\ell$ with $0\leq \ell\leq \frac{n}{2}$. The value of $g(\ell,n)$ is of great interest for integers $\ell$ with $\gcd(\ell,n)=1$.

If the claim (*) is true, then I can find the multiplicities of the eigenvalues of the (inverse) discrete Fourier transform matrix $\mathbf{A}$ in my answer here with $\omega:=\omega_n$. That is, $+\sqrt{n}$ will have multiplicity $\left\lfloor\frac{n+4}{4}\right\rfloor$, $-\sqrt{n}$ will be of multiplicity $\left\lfloor\frac{n+2}{4}\right\rfloor$, $+\sqrt{n}\text{i}$ will be of multiplicity $\left\lfloor\frac{n+1}{4}\right\rfloor$, and $-\sqrt{n}\text{i}$ will be of multiplicity $\left\lfloor\frac{n-1}{4}\right\rfloor$. I am sure there are different approaches of getting these multiplicities, but I would like to see how the sum above is evaluated.

Notes:

  1. The only case in (*) I am able to prove is when $n\equiv 2\pmod{4}$. That is because $$\omega^{(k+n/2)^2}=-\omega^{k^2}$$ for any primitive $n$-th root of unity $\omega$ and for all $k=0,1,2,\ldots,n-1$. Consequently, if $n\equiv 2\pmod{4}$, then $g(\ell,n)=0$ for all $\ell\in\mathbb{Z}$ with $\gcd(\ell,n)=1$.

  2. i707107 has provided me with a great reference. Hence, I no longer need a proof for (*), except a different proof. However, I will still greatly appreciate if anybody can determine $g(\ell,n)$ for $\ell\not\equiv\pm1\pmod{n}$.

  3. As arthur's link shows, if $n:=p$ is an odd prime and $p\nmid \ell$, then $$g(\ell,p)=\left(\frac{\ell}{p}\right)\,g(1,p)\,,$$ where $\left(\frac{\ell}{p}\right)$ denotes the Legendre symbol of $\ell$ modulo $p$.

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Batominovski
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1 Answers1

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I found a very nice paper on the subject, using only elementary linear algebra to prove the statements you mention about the Gauss sum value and the multiplicities of the eigenvalues as you calculated them (although none of the explicit formula you mention are supplied in the document, it is easy to derive these from the information in the paper).

See https://mast.queensu.ca/~murty/quadratic2.pdf.

Note that the '(inverse) discrete Fourier transform matrix A' you mention is equivalent to the matrix used in the article (by changing the order of the base vectors).

Maestro13
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