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Questions tagged [gauss-sums]

For questions on Gauss sums, a particular kind of finite sum of roots of unity.

In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically $$ G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r) $$ where the sum is over elements $r$ of some finite commutative ring $R$, $ψ(r)$ is a group homomorphism of the additive group $R^+$ into the unit circle, and $χ(r)$ is a group homomorphism of the unit group $R^×$ into the unit circle, extended to non-unit $r$ where it takes the value $0$. Gauss sums are the analogues for finite fields of the Gamma function.

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Trig identities analogous to $\tan(\pi/5)+4\sin(\pi/5)=\sqrt{5+2\sqrt{5}}$

The following trig identities have shown up in various questions on…
Franklin Pezzuti Dyer
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What is the Gauss sum equivalent of $\Gamma(s+1) = s\Gamma(s)$?

Gauss sums are analogous to the Gamma function: Fix a complex number $s$ with real part $>0$. Then we have a multiplicative character $\chi_s :\mathbb R^{\times}_{>0} \to \mathbb C^\times$ given by $x \mapsto x^s$. We also have an additive character…
Bruno Joyal
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Expected maximum of sub-Gaussian

I'm trying to answer the following question from the book high-dimensional probability: Let $X_1,X_2,\dots$ be a sequence of sub-gaussian random variables, which are not necessarily independent. Show that $E\bigg[ \max_i \frac{|X_i|}{\sqrt{1 + \log…
JohnSnowTheDeveloper
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How to prove that below quantity is purely imaginary?

How do I prove that the following quantity is purely imaginary: $$\sum_{0\leq l_1
SK1712
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Prime power Gauss sums are zero

Fix an odd prime $p$. Then for a positive integer $a$, I can look at the quadratic Legendre symbol Gauss sum $$ G_p(a) = \sum_{n \,\bmod\, p} \left( \frac{n}{p} \right) e^{2 \pi i a n / p}$$ where I use $\left( \frac{\cdot}{\cdot} \right)$ to be…
davidlowryduda
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Relation that holds for the Legendre symbol of an integer but not for the Jacobi symbol?

Let $p$ be a prime number and $\big(\frac{a}{p} \big)$ the Legendre symbol. Then we have the equality $$\sum_{a=1}^{p-1} \big(\frac{a}{p} \big) \zeta^a =\sum_{t=0}^{p-1} \zeta^{t^2},$$ where $\zeta$ is a primitive $p$th root of unity. This follows…
user65175
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How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are relatively prime integers. (Here is another question.…
user65175
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Gauss-type sums for cube roots

(Quadratic) Gauss sums express square root of any integer as a sum of roots of unity (or of cosines of rational multiples of $2\pi$, if you will) with rational coefficients. But Kronecker-Weber guarantees that any root of any integer can be…
Grigory M
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Showing $\sum_{m=1}^{2n-1} \sin \dfrac{\pi m^{2}}{2n}=\sqrt{n}$

$$\sum_{m=1}^{2n-1} \sin \dfrac{\pi m^{2}}{2n}=\sqrt{n}$$ I figured out this sum on my own by experimenting values on Wolfram Alpha but I am unsure how to prove it.I looked up on several sources about this and found "Gauss Sums" but it looks way…
匚ㄖㄥᗪ乇ᗪ
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Alternating quadratic Gauss sum

I would like to know if there exists an expression for the generalized alternating quadratic Gauss sum $$ g_N(a,b) = \sum_{k=0}^{N-1}(-1)^{k} e^{i2\pi\frac{ak(k+b)}{N}} $$ where $N$ is an odd prime and where $a,b \in \{1,2,\dots,N-1\}$. When $b=0$,…
dvdgrgrtt
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Gaussian periods

Let p be an odd prime number and $p-1=m d$ a decomposition into positive factors. Then there is a unique cyclic extension $K_d/\mathbb Q$ of degree d ramified only at p. It is contained in the cyclotomic field $\mathbb Q(\zeta)$, where $\zeta$ is a…
user75148
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A Gauss sum like summation

I would like to calculate the following sum. Let $\zeta$ be a primitive $n$ th root of unity for some integer $n$. Here $n$ is not necessarily prime. The sum is $$\sum_{j=1}^n (-1)^j \zeta^{\frac{j^2}{2}}.$$ I know how to find the value of the sum…
user65175
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Gauss Sum of a Field with Four Elements

I need to calculate a couple of Gauss sums to solve a problem I'm working on, but I keep getting the wrong answer because the absolute value of what I calculate is impossible for such a sum. Can anyone spot my mistake? Suppose $F$ is a field with…
D_S
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Determining the Value of a Gauss Sum.

Can we evaluate the exact form of $$g\left(k,n\right)=\sum_{r=0}^{n-1}\exp\left(2\pi i\frac{r^{2}k}{n}\right) $$ for general $k$ and $n$? For $k=1$, on MathWorld we have that $$g\left(1,n\right)=\left\{ \begin{array}{cc} (1+i)\sqrt{n} & \…
Eric Naslund
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Dirichlet characters as coboundaries of Gauss sums

Let $p$ be a prime number, and consider a Dirichlet character $\chi : (\mathbf Z/p\mathbf Z)^\times \to \mathbf C^\times$. Its image lands in the group $\mu_{p-1}$ of $(p-1)$-st roots of unity. The Gauss sum is defined as $\eta(\chi) := \sum_{a=1}^p…
Bruno Joyal
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