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

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6
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1answer
273 views

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 $\zeta$ ...
6
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0answers
121 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 ...
5
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1answer
127 views

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 ...
5
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1answer
189 views

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 ...
5
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1answer
215 views

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 ...
4
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1answer
281 views

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 ...
4
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1answer
239 views

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 ...
4
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0answers
143 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 ...
3
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0answers
98 views

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 ...
2
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2answers
43 views

$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$

$$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$$ where $c \in \mathbb Z_9$, $w=e^{2\pi i/9}$ and $\mathbb Z_9$ is the ring of integers modulo 9.
2
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1answer
49 views

Dirichlet character modulo p

How can I prove that if $\chi$ is a non-principal character modulo $p$ prime, then $\chi (-1) = \overline{\chi} (-1)= \pm 1$ and $\sum_{x=1}^p \chi (x) e^{2\pi i x}=0$? For the first question, I just ...
2
votes
1answer
303 views

A Trigonometric Sum Related to Gauss Sums

This is a problem given to me by fractals on Art of Problem Solving. I couldn't solve it so I'm posting it here for some thoughts on it. Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} ...
2
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1answer
83 views

Number of solutions of $N(y^{2}+x^{3}=1)=p+2ReJ(\chi,\rho)$

This is similar to a question I recently asked about. It is from Ireland's Number theory book, ch.8, ex.27 b,c. I think I can do the first part of this question, but I think there might be a trick ...
2
votes
1answer
52 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 ...
2
votes
1answer
374 views

Quadratic Gauss Sums

Let $p$ be an odd prime and $\zeta \not = 1$ be a $p^{th}$ root of unity. Let $R$ denote the set of all quadratic residues in $\mathbb{F}_p^*$. If $\alpha=\sum_{r\in R} \zeta^r$, prove that $$\alpha ...
2
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1answer
50 views

Multi-dimensional MLE Guassian

I wonder that what is the mu and sigma formula MLE(maximum likelihood estimates) for a 3 dimension guassian ? It is the same form as 1 and 2 dimension (+ 1 mu and sigma for the new vector) ?
2
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0answers
32 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
33 views

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 ...
1
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1answer
84 views

Gamma function and Gauss sums

In this Wikipedia article appears this : "Gauss sums are the analogues for finite fields of the Gamma function." What was the relation between gamma functions and non-finite fields?
1
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1answer
88 views

Definition: Gauss Sum - Where is the error?

In my algebraic number theory lecture we defined Gauss sums as follows. However, I am quite unsure whether this definition is correct. My intuition says "there is a mistake somewhere". I tried ...
1
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1answer
159 views

Fractions in limits of a summation

What if on the sum there is a fraction in the limit? $\sum_{m=k/12}^{k}$ or $\sum_{m=0}^{k/12+1}$ thank you very much! what type of sequence is used for summing this type of interval?
1
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1answer
75 views

Absolute value of a cubic Gauss sum (over Field $\mathbb{F}_p$ )

I'm interested in the quantity $|\sum_{x \in \mathbb{F}_p} \omega^{ax^3+bx^2+cx}|$ where $a \in \mathbb{F}_p^*,b,c \in \mathbb{F}_p$ and $\omega$ is a primitive $p$-th root of unity i.e. ...
1
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1answer
204 views

Gauss Newton minimization of 2D linear function

Given the input-output relation: $ \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} =p_1 \begin{pmatrix} p_2 & p_3 \\ p_4 & p_4 ...
1
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1answer
32 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: ...
1
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1answer
79 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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0answers
105 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 ...
1
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1answer
384 views

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 ...
0
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1answer
49 views

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) ...
0
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1answer
45 views

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 ...
0
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1answer
21 views

Gaussian sums values

I have the following problem: Denoting $S(q,a,\chi ) = \sum_{x=1}^q \chi (x) e(ax/q)$, where $\chi $ is an arbitrary character modulo $q$, I have to prove $$\sum_{a=1}^q \vert S(q,a,\chi ) \vert ^2 = ...
0
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1answer
35 views

Help Engineering maths

I am really stuck on this, I've done the first part need help on the second part. The potential, V, between two hollow, concentric metallic spheres is given by the solution of the following equation ...
0
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1answer
56 views

Properties of a Jacobi sum for $p=1\bmod 4$

I'm struggling with Ireland and Rosen, chapter 8, exercise 7. Suppose that $p=1\bmod 4$ and that $\chi$ is a character of order 4. Write $\chi^2=\rho$ a character of order 2. Show that ...
0
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1answer
108 views

Equivalent definitions of the quadratic gauss sum

In Ireland and Rosen, the quadratic Gauss sum of $a$, $g_a$, is defined by $g_a=\sum_{t=0}^{p-1}(\frac tp)\zeta^{at}$ with $\zeta$ a $p$th root of unity, $p$ an odd prime and $(\frac\cdot\cdot)$ the ...
0
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1answer
42 views

Gauß sum and primitive character

I am working with Daniel Marcus "Number Field" Book. And I have a question to the following Lemma: $$\tau_k(\chi)=\left\{\begin{array}{ll} \bar\chi(k)\tau(\chi), & \textrm{if }(k,m)=1 \\ ...
0
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1answer
111 views

Gaussian elimination with partial pivoting doubts

I have the following doubts about Gauss algorithm with partial pivoting: Say that I sum to the second row the first row multiplied by $k$. In the $L$ matrix, should I sum to the second row the first ...
0
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1answer
16 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}]} ...
0
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0answers
23 views

Explanation of Gauss sums' preference for the North-East?

Define the Gauss sums algebraically as follows: $$\mathcal{G}_q = \sum_{a=0}^{q-1} e^{2 \pi i a^2 / q}$$ Then the result ends up being $\sqrt{q}, 0, i \sqrt{q}, (1+i)\sqrt{q}$ depending on $q \equiv ...
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1answer
166 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 ...