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

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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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Reversing the $T(n) = \frac{n(n+1)}{2}$ formula

Can someone reverse this formula? Sorry if it's too basic, i'm an old guy and my math is rusty :-D $s =\frac{n(n+1)}{2}$ If it ain't clear, by reverse I mean obtaining the value of ...
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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: http://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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61 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 $...
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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(\...
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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 \[...
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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 = ...
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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 ...
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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. $\omega=e^{\...
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114 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?
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148 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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37 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 $$...
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137 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 $J(\chi,\...
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Equivalent definitions of the quadratic gauss sum

In Ireland and Rosen, the quadratic Gauss sum of $a$, $g_a$, is defined by $$g_a:=\sum_{t=0}^{p-1}\left(\frac tp\right)\zeta_p^{at}$$ with $\zeta_p$ a $p$th root of unity, $p$ an odd prime and $(\...
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49 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 \\ ...
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$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,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.
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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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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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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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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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110 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 double-...
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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 ...
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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 (-...
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379 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} \sin\dfrac{2\pi\...
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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 $\zeta$ ...
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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 ...
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238 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 \zeta^{\frac{...
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120 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 ...
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140 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 ...
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55 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) ?
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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 \end{...
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273 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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212 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?
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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 ...
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305 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 ...
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271 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 $$g\left(1,n\right)=\...