# Questions tagged [gauss-sums]

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

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### Quadratic Gauss sums and Fourier Analysis

Lately I have been reading this very interesting book Mathematics++: Selected Topics Beyond the Basic Courses by Kantor, Matoušek and Šámal, especially Chapter 3 on Fourier analysis. In Section 5: ...
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### Tate gamma factor as a principal value integral

Let $F$ be a local field, $\chi$ a multiplicative character of $F^{\ast}$, and $\psi$ an additive character of $F$. The gamma factor $\gamma(s,\chi,\psi)$ is defined by means of the local functional ...
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### Sum of the Multiplicative Legendre's symbol

Let $p$ a be odd prime show that $\displaystyle\sum_{a=1}^{p-2}\left(\frac{a(a+1)}{p}\right)=-1$. Nota Bene : The $(\frac{a}{p})$ is $\textbf{Legendre Symbol}$
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### Prove a formula to evaluate Legendre Symbol $(\frac{-3}{p})$ using quadratic Gauss Sums (i.e. without using quadratic reciprocity)

I've been trying to find a question similar to this, but have failed to find any. The full statement is let $\zeta = e^{\frac{2\pi i}{3}}$, and use the fact that $(2 \zeta + 1)^2 = -3$ and quadratic ...
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### Why does the following summation equivalence hold true?

$$\sum_{i=1}^{n} \sum_{j=i}^{n}(j-i)=\sum_{i=1}^{n} \sum_{j^{\prime}=0}^{n-i} j^{\prime}=\sum_{i=1}^{n} \sum_{j^{\prime}=1}^{n-i} j^{\prime}$$ Can someone explain to me how we got from the first to ... 393 views

### Prove that $\sum _{x=0}^{p-1}e^{\frac {2\pi ix^{2}}{p}}={\sqrt {p}}$ , $p \equiv 1{\pmod {4}}$

Prove that : $$\sum _{x=0}^{p-1}e^{\frac {2\pi ix^{2}}{p}}={\begin{cases}{\sqrt {p}}&{\text{if}}\ p\equiv 1{\pmod {4}}\\i{\sqrt {p}}&{\text{if}}\ p\equiv 3{\pmod {4}}\end{cases}}$$ where $p$ ...
1 vote
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### Gauss sums and Dirichlet characters

Currently, I'm attending an Analytic Number Theory course, and in the lecture notes I've come across the following statement: Does anyone know how to prove this, or at least can give me a reference? ...
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### Distribution of Gauss sums on a circle in the Argand plane

It is a fairly well-known fact that the Gauss sum of a non-trivial character $\chi$ modulo a prime $p$ is always a complex number with an absolute value of $\sqrt{p}$. In other words, when the Gauss ...
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### A special type of Gauss sum

In the work of my thesis I came up with a problem that is elementary, but I can't figure out its proof. Let $p$ be an odd prime, let $(\mathbb{Z}/p^n\mathbb{Z})^\times$ denote the multiplicative ...
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### Multiple sum variable notation

I have a question regarding multiple sum variables. Is it correct to rewrite this sum like this ? $$\sum_{1\leq i<j\leq n}^{}\frac{i}{j} = \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{i}{j}$$ and how ...
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### How can one compute power sums with negative exponent?

Is there a general method to find the sum of powers with negative exponent? For example: $\sum_{i = 1}^{N} i^a$ with $a \in \mathbb{Z} - \mathbb{N}$
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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 ...
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### What is the reason for taking $\omega$ to be a primitive $q$-th root unity rather than taking any $q$-th root of unity?

Let $p$ and $q$ be two distinct odd primes. Let $\omega$ be a primitive $q$-th root of unity. Consider the sum $$S = \sum_{x \in \Bbb {F_q}^*} \left ( \frac x q \right ) {\omega}^x.$$ Prove ...
1 vote
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### Show that $S^p = \sum\limits_{x \in \Bbb {F_q}^*} \left ( \frac x q \right ) {\omega}^{xp}.$

Let $a \in \Bbb Z$ and $p,q$ be primes. Define $\left (\frac a p \right )$ as follows $:$ \left(\frac{a}{p}\right) = \begin{cases}\;\;\,0&\text{ if }p \text { divides } a\\+1&\text{ if } a \...
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### quadratic gauss sum calculation in sage

I tried to calculate quadratic gauss sum in SAGE but it works just for primes 3 and 5 which are $i\sqrt{3}$ and $\sqrt{5}$ respectively. p=3 print sum((legendre_symbol(x,p))*(e^(2*piIx/p)) for x in ...
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### Jacobi sums Gaussian Sum. Show $J(\rho^{'},\chi^{'}) = (- J(\rho,\chi))^s$
I want to show that $J(\rho^{'},\chi^{'}) = (- J(\rho,\chi))^s$, where $\rho^{'},\chi^{'}$ are characters of a finite field $F_{p^s}$ and $\chi,\rho$ are characters a finite field $F_p$. My work: I ...
### Computing number of solutions for equations in $F_{m^s}$ (finite field with $m^s$ elements)
First I want to give you some context. Then I will ask my questions. Context Consider the equations $a_1x_1^{l_1} + \dots + a_rx_r^{l_r} = b$ with $a_1, \dots , a_r \in F_m^{*}$, where $F_m$ is a ...