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Questions tagged [gauss-sums]

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

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Quadratic Fields and Gauss Sums

I am given the p-cyclotomic field as $K_p = \frac{\mathbb{Q}[x]}{(\Phi_p(x))}$ where $\Phi_p(x) = x^{p-1} + x^{p-2} + ... + x + 1 $ Then, a quadratic subfield L is defined such that $\mathbb{Q} \...
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Norm of Gauss Sum = p

I am given a non-trivial homomorphism $\chi : \left(\mathbb{Z} / p\mathbb{Z} \right)^\times \rightarrow \mathbb{C}^\times$, p is prime, and $\zeta$ is a primitive p-th root of unity. A generalized ...
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About sums similar to gauss sums

It is well known the case for sums like: $$ \sum_{i=0}^{p^n -1}\zeta^{-ai}, $$ where zeta is a primitive $p^n$-rooth of $1$. But, is there a standard formula for sums like: $$ \sum_{i=0}^{p^n -1}i^...
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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 ...
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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 ...
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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 ...
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$\lim_{n\to\infty}\frac{\sum_{i=1}^ni^{4\lambda}}{\Big(\sum_{i=1}^ni^{2\lambda}\Big)^2}=0$

Is it true that $\forall \lambda>0$ $$\lim_{n\to\infty}\frac{\sum_{i=1}^ni^{4\lambda}}{\Big(\sum_{i=1}^ni^{2\lambda}\Big)^2}=0$$ I cannot find a way to prove it, nor can I find a counterexample. ...
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On a $\mathbb C$-linear map from $M(p-1,\mathbb C)$ to $\mathbb C^\hat G$, where $p$ is an odd prime and $G=\mathbb Z/(p) ^\times$

Let $p$ be an odd prime and $G=(\mathbb Z/(p))^\times=\{1,2,...,p-1\}$ i.e. $G$ is a cyclic group of order $p-1$. Let $\hat G:=\{\chi:G \to \mathbb C^\times : \chi $ is a group homomorphism $\}$. For ...
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Gauss sum possible typo

Let $ψ: \mathbb{F_p} \to Z_p$ with the property $ψ(a+b)=ψ(a)ψ(b)$ where $Z_p$ denotes the p-adic integers. Assume further that $ψ$ is not trivial. I'm trying to follow my professor's work, but I ...
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Quadratic Gauss sum using $|Z|^2=Z \bar Z$

Given $Z=\sum_{k=1}^{n-1}\omega^{k^2}$ I'm asked to find $|Z|^2$, here's what I thought of: $$|Z|^2=Z \bar Z=\left( \sum_{k=1}^{n-1}\omega^{k^2} \right) \left( \sum_{k=1}^{n-1}\frac{1}{\omega^{k^2}} \...
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Why did people constructed Quadratic Gauss Sum? [duplicate]

I was studying number theory these days where Quadratic Gauss sum came up. https://en.wikipedia.org/wiki/Quadratic_Gauss_sum My question was that: What motivated them to construct Gauss Sum in the ...
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a special problem about numbers assigned on polygons

So the problem is stated as follow: We have $a$ numbers of regular $b$-sided polygons. We place them in a fashion such that the sides of polygons are parallel and the vertex of every polygon ...
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Gauss sum in regular 7-gon

The heptagon on the picture is a regular heptagon with side 1. What is the length of the dashed interval? This is a (kind of) 'geometric version of a quadratic Gauss sum for p=7' (this observation — ...
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Summation in Gaussian Variables

Let X and Y are Gaussian Variables. We know $Y=X+Z$. Let X and Z are Independent. How can I prove Y is a Gaussian Random Variable iff Z is a Random Variable? Can I use X, Z Orthogonal and Normal ...
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Gauss like sum evaluation / estimate

Let $\mathbb{F}_{p^2}$ be the finite field of cardinality $p^2$, $\chi$ be a (multiplicative) character of $\mathbb{F}_{p^2}$ and $e(.)=e^{2\pi i .}$. What is the evaluation of the following two sums, ...
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Show that $Z < \sqrt p$ if $\left( \frac{m - n}p \right) = 1, m \in \mathcal N, n \in \mathcal N, m \ne n$.

Would you please help me solve Exercise 4.2(b) on page 20 of the online document Characters. I repeat that exercise here: Let $p$ be a prime, $p \equiv 1$ (mod $4$), and let $\mathcal N$ be a set ...
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Multiplication of Gaussian pdfs

I have a sample mean given by: $$S_n=\frac{1}{n}\sum_{i=1}^nX_i$$ Where $X_i$ are i.i.d. Gaussian random variable, i.e., each of them has pdf: $$p(X_i=x_i)=\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x_i-\mu)...
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$ S^2= \left(\sum_{j=0}^{p-1} \epsilon^{j^2}\right)^2$ where $p$ are prime number, $\epsilon $ is primitive p-th root of unity.

Let $p$ are prime number, $\epsilon $ is primitive p-th root of unity. Calculate: $$ S^2= \left(\sum_{j=0}^{p-1} \epsilon^{j^2}\right)^2$$ $p=3;p=5$ the result are real number. $\epsilon^{j^2}=\...
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How to prove that below quantity is purely imaginary?

How do I prove that the following quantity is purely imaginary: $$\sum_{0\leq l_1<l_2<l_3<l_4\leq q-1} e^{-2\pi i \frac{(l_1^2-l_2^2+l_3^2-l_4^2)}{q} } $$ where $q$ is an odd number?
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Finding closed form of a summation $\sum_{i = \lfloor{\frac{n}{2}}\rfloor}^{n}i$

So i'm trying to find the clsoed form of $\sum_{i = \lfloor{\frac{n}{2}}\rfloor}^{n}i$ from what I know its probably going to be $? * (\lfloor{\frac{n}{2}}\rfloor + n)$ based on when I wrote out ...
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Infinite quadratic gauss sum.

I have a summation of the form: $$\sum_{m\in\mathbb{Z}}m\exp\left\{-\frac{B}{2}\left(z-\frac{2\pi iv}{B}-\frac{2\pi m}{BL_2}\right)^2\right\}$$ where $B, v, L_2$ are constants and $z$ is a variable. ...
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Gauss sum variation $\sum_{n=0}^{p-1}\left(\frac{a+bn}{p}\right)\zeta_p^{cn} = ?$

I'm having trouble evaluating this, for $a, b, p$ all pairwise coprime, $p$ an odd prime, $c$ any integer. $$\sum_{n=0}^{p-1}\left(\frac{a+bn}{p}\right)\zeta_p^{cn}$$ Any help/references would be ...
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Upper bound on the summation of roots of unity

Let $p$ be a prime number, and $\zeta = e^{(2\pi i/p)}$ be a $p$th root of unity. Given the gauss sum: $$g_t = \sum_{k = 0}^{p-1}\left(\frac{k}{p}\right)\zeta^{tk}$$ I'm trying to prove the upper ...
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Gauss sum variant: $ \sum_{ a, b \in \mathbb{Z} } e^{-i \pi (a^2+b^2)/p } \big( e^{\pi(a-b)} + e^{-\pi(a-b)} \big) $

Motivated by an example in Chern Simons theory, let $p \in \mathbb{Z}$ be prime, can anyone compute this sum: $$ \sum_{ a, b \in \mathbb{Z} } e^{-i \pi (a^2+b^2)/p } \big( e^{\pi(a-b)} + e^{-\pi(a-b)...
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Gauss Sum calculation

I want to calculate Gauss Sum of the field with 8 elements. I studied in Lidl & Niederreiter's book. $\chi(c)=e^{2 \pi i Tr(c)/p}$ for all $c\in F_q$, defines an additive character of $F_q$, ...
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A step from the proof of the Pólya-Vinogradov inequality

I am having trouble understanding one of the inequalities involved in the proof of the Pólya-Vinogradov inequality, more precisely $$ \left \vert{\frac{\sqrt p} p \sum_{a \mathop = 1}^{p-1} \frac{e^{...
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Gauss formula to add number of sequence for arbitrary range

Gauss formula to add numbers from $1-100$ is: $$ \frac{n(n+1)}{2}$$ How can this be made applicable for arbitrary range, lets say $3-30$? Is there an easy way of doing that rather then linearly ...
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How is distributed the sum of square of correlated z-score?

I have a $1\times k$ matrix representing $z$-scores and each element is correlated to each other according to a covariance matrix $\Sigma$. I would like to compute their sum of square and to know the ...
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Product of Gauss sums $\tau_p,\tau_q$

Let $p,q$ be different odd primes, and let $\tau_p = \sum\limits_{a=1}^{p}\left(\frac{a}{p}\right)e^{\frac{2\pi ia}{p}}$, $\tau_p = \sum\limits_{b=1}^{q}\left(\frac{b}{q}\right)e^{\frac{2\pi ib}{q}}$. ...
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How are Gauss sums the analogue of the gamma function for finite fields?

I've seen this statement on the internet in a few places and I don't really get the connection. Would anyone mind fleshing out the details? Thanks.
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Why is the absolute value of this Gauss sum obvious?

I came across the Gauss sum discussed in the following post in a problem from my Galois theory course: https://mathoverflow.net/a/71282. Why exactly is the square of its norm obvious?
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Upper bound for $\frac{\|x\|_1}{\|x\|_2}$ if each entry of $x\in R^d$ is i.i.d. sampled from Gaussian distribution $N(0,1)$

In the question, $\|x\|_1=\sum_{i=1}^d|x_i|$ with $|\cdot|$ being the absolute value, and $\|x\|_2=\sqrt{\sum_{i=1}^d x_i^2}$. In general, $\frac{\|x\|_1}{\|x\|_2}\leq \sqrt{d}$ always holds for ...
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How to come up with Gauss arithmetic progression solution in this sum

I need to solve this: $\sum\limits_{k=0}^n k\\$ Using this specific method: $n + \sum\limits_{k=0}^{n-1} k\ = 0 + \sum\limits_{k=1}^n k\\$ Now this has to evaluate to: $\ n+\frac{n(n-1)}{2} = \frac{...
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Upper bound for the multidimensional gaussian integral

How to prove that, for any integer $d \geq 1$ and any $\alpha > 1/2$, $$\int_{n^{\alpha}}^{\infty} r^{d-1} e^{-\frac{r^2 d}{n}} dr$$ goes to $0$ with $n$? This can be interpreted (up to a ...
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Summation of gaussians

Suppose we have given constants $A_i, x_i (i=1..N)$ Is it possible to approximately calculate the sum of N gaussians in less than N iterations for any x? (may be with some preprocessing) $$\sum_{i=1}...
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Module isomorphisms and coordinates modulo $p^n$

Let $p$ be a prime and $n \in \mathbb{N}$ is such that $p^n > 2$. We let $\alpha \in \mathbb{N}$ be such that $0 < \alpha < n$. Let $R := \mathbb{Z}_{p^n}$ denote the ring of integers modulo $...
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Cross-correlation of Gaussian and Jacobian sums

I recently came upon the following kind of sum and I'm wondering if anyone has seen it before, or could point out something interesting about them. Let $F$ be a finite field with $q > 2$ elements ...
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Gauss sum of a multiplication of two multiplicative characters of a finite field

Let $F$ be a finite field with $q$ elements and characteristic $p$. Let $E$ be a proper extension over $F$ of degree $n$. Let $\psi$ be the canonical additive character of $E$ defined by $\psi(x) = e^{...
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Gaussian likelihood - test two observations for same parent population

If I have an observation $x$ with a Gaussian distributed observational error of standard deviation $\sigma$ then the sum of likelihoods of that observation having the error free values $x_1^{\prime} \...
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Gauss sums and module endomorphisms

Let $p$ be an odd prime and $n \in \mathbb{N}$. Let $a,b,c$ be arbitrary integers such that $ab \neq 0$. We write $p^{\alpha}A = a$ and $p^{\beta}B = B$ for some $\alpha, \beta \in \mathbb{N}_0$ and $...
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Complex inequality question

I am trying to understand why the following holds: \begin{align*} \Re((1-\imath)(A+B)) \geq \Re((1-\imath)A) - \sqrt{2}|B|, \end{align*} where, \begin{align*} A:= \sum_{x=1}^{[\sqrt{k}]} e\left(\frac{...
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An identity involving Gauss sums and convolution

For a Dirichlet character $\chi$ modulo $N$, the Gauss sum attached to $\chi$ is given by $$G_\chi(m) = \sum_{k \in \mathbb{Z}_N} \chi(k) e^{2\pi i mk/N}.$$ Suppose one has an $N$-periodic function $...
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Determination of quartic Gauss sums

Typically, the Gauss sum over $\mathbb{F}_p$ of order $k$ means the quantity $$\sum_{n=0}^{p-1} e^{2\pi i n^k/p}.$$ In the book Gauss and Jacobi Sums of Berndt, Evans and Williams, a more general sum ...
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Gauss sum of character $\psi \neq 1$

I am trying to solve Let $1 \neq \psi$ be a charachter of $\mathbb{F}_p$ and define $$G(\psi) = \sum_{x\in \mathbb{F}_p} \psi(x^2) $$ Proof that $|G(\psi)|^2 = p$. What I tried so far: $$|G(\psi)...
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Summation Sequence Question

I need to find the summation of $ab^{-k}$ from $k=5$ to $n$ using Gauss' Law. Here's what I have so far: $$\begin{align}S_n&=(ab^{-5}+ab^{-6}+ab^{-7}+\cdots+ab^{-n}+ab^{-n}+ab^{-(n-1) }+ab^{-(n-2)...
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Summation Sequence

I'm supposed to use Gauss' law to find the summation of $6k$ from $k=5$ to $n$. Here is my work: $$6(5)+6(6)+6(7)+⋯+6(n)\\+6(n)+6(n-1)+6(n-2)+...+6(5)$$ When these are added together I get $2S=(30+...
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Writing a Gauss sum as a sum over divisors

Let $\chi$ be a Dirichlet character modulo $q$ induced by a primitive character $\chi^*$ modulo $d$ for some divisor $d$ of $q$. Let $n$ be a positive integer, and consider the generalised Gauss sum \[...