Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [gauss-sums]

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

1
vote
1answer
44 views

Sub-Sum of Roots of Unity

Let $\alpha$ be an algebraic integer of a cyclotomic field, and let $\theta_1, \theta_2, ..., \theta_n$ be roots of unity such that $$\sum_{i=1}^n \theta_i = 2\alpha.$$ Does there necessarily exists a ...
0
votes
0answers
31 views

Variance of sum of scaled Gaussians with zero mean

I'm interested in finding the Gaussian, that best approximates the sum of scaled Gaussians with zero mean. If we define a Gaussian with variance $\sigma^2$ and zero mean as $g(\sigma^2)$, i.e.: $g(\...
1
vote
0answers
43 views

Can a bound be given to $\sum_{a=1}^{p-1}\chi(a)\left(\frac{a^2-1}{p}\right)$, which is smaller than $p$, where $\chi$ is a dirichlet character?

Here $(\cdot)$ denotes the legendre symbol and $p$ be an odd prime number. All terms are of norm 1. So one bound is $p$. can one better bound be given to the sum?
0
votes
0answers
35 views

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} \...
3
votes
1answer
52 views

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 ...
1
vote
0answers
31 views

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^...
3
votes
1answer
116 views

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 ...
0
votes
1answer
30 views

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
1answer
30 views

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 \...
0
votes
1answer
21 views

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 ...
1
vote
0answers
10 views

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 ...
3
votes
0answers
38 views

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 ...
-1
votes
1answer
46 views

$\lim_{n\to\infty}\frac{\sum_{i=1}^ni^{4\lambda}}{\Big(\sum_{i=1}^ni^{2\lambda}\Big)^2}=0$

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. ...
3
votes
0answers
60 views

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 ...
1
vote
1answer
46 views

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 ...
0
votes
0answers
23 views

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}} \...
0
votes
1answer
84 views

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 ...
0
votes
1answer
37 views

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 ...
2
votes
1answer
173 views

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 — ...
1
vote
1answer
56 views

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 ...
1
vote
1answer
54 views

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, ...
0
votes
1answer
50 views

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 ...
1
vote
2answers
196 views

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)...
0
votes
2answers
56 views

$ 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}=\...
8
votes
0answers
179 views

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?
1
vote
2answers
60 views

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 ...
0
votes
1answer
103 views

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. ...
0
votes
1answer
48 views

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 ...
1
vote
0answers
61 views

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 ...
1
vote
0answers
25 views

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)...
0
votes
0answers
111 views

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 $F_q$, ...
0
votes
0answers
51 views

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^{...
2
votes
2answers
285 views

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 ...
1
vote
0answers
86 views

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 ...
1
vote
1answer
91 views

Product of Gauss sums $\tau_p,\tau_q$

Let $p,q$ be different odd primes, and let $\tau_p = \sum\limits_{a=1}^{p}\left(\frac{a}{p}\right)e^{\frac{2\pi ia}{p}}$, $\tau_p = \sum\limits_{b=1}^{q}\left(\frac{b}{q}\right)e^{\frac{2\pi ib}{q}}$. ...
2
votes
1answer
136 views

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.
0
votes
1answer
280 views

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?
1
vote
0answers
121 views

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 ...
0
votes
2answers
61 views

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{...
0
votes
1answer
92 views

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 ...
0
votes
1answer
15 views

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}...
2
votes
0answers
50 views

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 $...
1
vote
0answers
105 views

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 ...
1
vote
0answers
144 views

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^{...
2
votes
0answers
121 views

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} \...
4
votes
1answer
82 views

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 $...
0
votes
1answer
35 views

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{...
2
votes
1answer
123 views

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 $...
3
votes
0answers
178 views

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 ...
1
vote
1answer
68 views

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)...