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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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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{...
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1answer
389 views

Decompose a sum of Gaussian curves

I have a data set with 2 different curves in a .csv file. Both curves are a sum of Gaussian curves and I'd like to be able to decompose these curves into their substituent addends: ...
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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 ...
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1answer
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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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542 views

Is there a way to directly compute maximum of a sum of several Gaussian functions?

I have a problem which goes as follows. I am trying to predict the value of a variable $x$. I also have a set of measurements (the actual context is an image) $x^i$. I know from some training ...
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0answers
250 views

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 $\chi_s :\mathbf R^{\times}_{>0} \to \mathbf C^\times$ given ...
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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?
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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 ...
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416 views

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 ...
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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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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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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 ...
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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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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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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?
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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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0answers
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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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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 ...
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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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0answers
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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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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 ...
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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 ...
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143 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^{...
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0answers
145 views

A Gauss sum over a field.

Let $K$ be a field (not necessarily $\mathbb C$) and let $\zeta=\zeta_n$ be a primitive $n$th root of unity in $\bar K$. I would like to know if there is a formula calculating $$ \sum_{k=1}^n \zeta^{-...
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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(\...
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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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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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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 $F_q$, ...
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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^{...