Questions tagged [gauss-sums]

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

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1answer
41 views

Gaussian like sum with binomial coefficients

Is there any way to calculate expressions of the form $$ f(q)=\sum_{n=0}^N\binom{n}{k}\exp({2\pi i\tfrac{n}{N}\cdot q}) $$ where $q\in\mathbb{Q}$? It reminds me quite alot of Gauss sums, but I don't ...
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72 views

Understanding the proofs in “Elementary aspects of the Verlinde formula and of the Harder–Narasimhan–Atiyah–Bott formula” from Don Zagier

As the proofs of Theorem 1 in "elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula" are very terse I could not understand all proofs. The paper can be ...
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0answers
26 views

Square of quadratic gauss sum

I know that if $g$ is the quadratic Gauss sum for $r$th root of unity, then $g^2=\pm r$. I know, how to prove it using $g=\sum_{i=0}^{r-1}\left(\frac ir\right)\zeta^i$, but can we prove that using the ...
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1answer
19 views

Quadratic gauss sum equivalence of definition

Let $r$ be a prime number and $\zeta_r$ be the $r$th rot of unity. I know quadratic gauss sum can be expressed in two ways, $g=\sum_{i=0}^{r-1}\zeta_r^{i^2}$ and $g=\sum_{i=0}^{r-1}(\frac{i}{r})\...
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1answer
49 views

Quadratic form Gauss sum and quadratic residue

Let $Q(n_1,n_2,\ldots,n_r)$ be a quadratic form of several variables with integral coefficients. Let $q$ be a prime and let $0<a<q$. We are interested in the associated Gauss sum $$S(a,q)=\sum_{...
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1answer
50 views

Prove a formula to evaluate Legendre Symbol $(\frac{-3}{p})$ using quadratic Gauss Sums (i.e. without using quadratic reciprocity)

I've been trying to find a question similar to this, but have failed to find any. The full statement is let $\zeta = e^{\frac{2\pi i}{3}}$, and use the fact that $(2 \zeta + 1)^2 = -3$ and quadratic ...
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22 views

Why does the following summation equivalence hold true?

$$ \sum_{i=1}^{n} \sum_{j=i}^{n}(j-i)=\sum_{i=1}^{n} \sum_{j^{\prime}=0}^{n-i} j^{\prime}=\sum_{i=1}^{n} \sum_{j^{\prime}=1}^{n-i} j^{\prime} $$ Can someone explain to me how we got from the first to ...
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25 views

Evaluating a certain generalized quadratic gauss sum

Let $p \geq 2$ and $0 \leq s \leq 2p-1$ be integers. Set $e(y) = \exp( 2 \pi i y )$. What's the trick in evaluating \begin{align} \sum_{r = 0}^{p-1} e\left( \frac{-r(r + s)}{p} \right) %\...
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2answers
182 views

Prove that $\sum _{x=0}^{p-1}e^{\frac {2\pi ix^{2}}{p}}={\sqrt {p}} $ , $ p \equiv 1{\pmod {4}}$

Prove that : $$\sum _{x=0}^{p-1}e^{\frac {2\pi ix^{2}}{p}}={\begin{cases}{\sqrt {p}}&{\text{if}}\ p\equiv 1{\pmod {4}}\\i{\sqrt {p}}&{\text{if}}\ p\equiv 3{\pmod {4}}\end{cases}}$$ where $p$ ...
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1answer
26 views

Gauss sums and Dirichlet characters

Currently, I'm attending an Analytic Number Theory course, and in the lecture notes I've come across the following statement: Does anyone know how to prove this, or at least can give me a reference? ...
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1answer
34 views

Distribution of Gauss sums on a circle in the Argand plane

It is a fairly well-known fact that the Gauss sum of a non-trivial character $\chi$ modulo a prime $p$ is always a complex number with an absolute value of $\sqrt{p}$. In other words, when the Gauss ...
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36 views

Need help in proving an inequality in lecture notes [duplicate]

While studying class notes in number theory I am unable to deduce this inequality. ( It is used in chapter gauss sums). Assume ( q, m) be gcd of q and m and all s, q, m be positive integers . It ...
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1answer
71 views

A special type of Gauss sum

In the work of my thesis I came up with a problem that is elementary, but I can't figure out its proof. Let $p$ be an odd prime, let $(\mathbb{Z}/p^n\mathbb{Z})^\times$ denote the multiplicative ...
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29 views

Multiplying two Gauss sums

In Goldfeld's book on automorphic representations, he computes the local root number of every ramified place of a character $\omega$ to be the Gauss sum $$\sum_{\substack{j=1\\(j,p)=1}}^{p^r}\omega(j)...
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2answers
149 views

Subfield of degree $p$ of $\mathbf{Q}_p(\zeta_{p^2})$

In this answer, Keith Conrad claims that a generator of the subfield of degree $p$ over $\mathbf{Q}_p$ of $\mathbf{Q}_p(\zeta_{p^2})$ is given by $$\sum_{a^{p-1}\equiv 1\bmod{p^2}} \zeta_{p^2}^a.$$ I ...
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16 views

Show that $\sum_{j}\chi_{\pi}^3(j)\zeta^{qj} \equiv \chi_{\pi}(q)g(\overline{\chi_{\pi}}) \pmod q.$

Hi I'm currently working on the proof of the Law of Biquadratic reciprocity from Ireland & Rosen's book on 'A Classical Introduction to Modern Number theory' and was stuck on the proof of ...
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1answer
58 views

Reference Request - Gauss Sum of a Quadratic Character

Let $N\geq 2$ be a natural number, then let $\chi: (\mathbb{Z}/N\mathbb{Z})^\times \to \{\pm 1 \}$ be a quadratic character. Consider the Gauss sum $$\tau(1,\chi) = \sum_{x\in (\mathbb{Z}/N\mathbb{Z})^...
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27 views

Another Gaussian sum

I'm trying to evaluate $$ \sum_{m'}\exp(-i\pi(m'+\alpha)/M) $$ where $\alpha=0,1,\dots,2M$. There are two questions. The first is the claim is that if $M$ is even and $$ \sum_{m'=1}^M\exp(-i\pi(m'...
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28 views

a general formula of certain Gauss sum

Let $k=\mathbb{F}_q$ be the finite field with $q=p^r$ elements. Assume that $q$ is odd. Let $\psi$ be a nontrivial additive character of $k$. For $a,b\in k^\times$, consider the Gauss sum $$G(a,b)=\...
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4answers
93 views

Multiple sum variable notation

I have a question regarding multiple sum variables. Is it correct to rewrite this sum like this ? $$ \sum_{1\leq i<j\leq n}^{}\frac{i}{j} = \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{i}{j} $$ and how ...
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1answer
34 views

How can one compute power sums with negative exponent?

Is there a general method to find the sum of powers with negative exponent? For example: $\sum_{i = 1}^{N} i^a$ with $ a \in \mathbb{Z} - \mathbb{N}$
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2answers
77 views

Finding a formula for $(1^3)\cdot(n)+(2^3)\cdot(n-1)+(3^3)\cdot(n-2)+ \cdots + (n^3)\cdot(1)$

I need a formula for this sum: $$(1^3)\cdot(n)+(2^3)\cdot(n-1)+(3^3)\cdot(n-2)+ \cdots + (n^3)\cdot(1)$$ I have found this formula : $$1\cdot n +2\cdot(n-1)+3\cdot(n-2)+ \cdots +(n-1)\cdot 2 +n\...
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0answers
32 views

The sum $\sum_{t \not \mod - 1} \chi[t(t + 1)]$

It's from Multiplicative Number Theory, Davenport, pp. 24. In calculating $\tau^{2}$ where \[ \tau = \sum_{x = 1}^{q - 1} \chi(n)e_{q}(x), \] $\chi(n)$ Dirichlet character mod 3 and $e_{q}(x) = e^{2 \...
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2answers
5k views

Sum of even numbers N? [closed]

To sum first N numbers we can use this formula: $$1 + 2 + 3 + \ldots + n = \frac{(n (n + 1)}{2}$$ To sum even numbers we multiply this formula by 2: $$2 + 4 + 6 + \ldots + 2n = n (n + 1)$$ Lets ...
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0answers
67 views

Imprimitive Dirichlet characters as sum of additive characters

Let $\chi$ be a Dirichlet character modulo $q$. Let $e(x)=\text{exp}(2\pi i x)$ and $\tau(\chi)$ be the Gauss sum $\tau(\chi)=\sum_{m \,\text{mod}\, q}\chi(m)e(m/q)$. For any $n$ with $\gcd(n,q)=1$, ...
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5answers
555 views

Trig identities analogous to $\tan(\pi/5)+4\sin(\pi/5)=\sqrt{5+2\sqrt{5}}$

The following trig identities have shown up in various questions on MSE: $$-\tan\frac{\pi}{5}+4\sin\frac{2\pi}{5}=\tan\frac{\pi}{5}+4\sin\frac{\pi}{5}=\sqrt{5+2\sqrt{5}}$$ $$-\tan\frac{2\pi}{7}+4\sin\...
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0answers
69 views

Quadratic Gauss Sum with Jacobi Symbol of even nonsquare free Modulus

Let $d, \ell$ be integers, $\left(\frac{\cdot}{\cdot}\right)$ be the Jacobi symbol, I would like to compute the following Gauss sum. $$G_{\left(\frac{\cdot}{d}\right)}(\ell):=\sum_{\alpha\bmod(d)}\...
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1answer
79 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 ...
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0answers
50 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?
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0answers
89 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} \...
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1answer
122 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 ...
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0answers
39 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^...
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2answers
1k 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 ...
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1answer
36 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 ...
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1answer
34 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 \...
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1answer
76 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 ...
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0answers
16 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 ...
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0answers
46 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 ...
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1answer
52 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. ...
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0answers
61 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 ...
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1answer
51 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 ...
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1answer
140 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 ...
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1answer
39 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
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1answer
296 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 — ...
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1answer
83 views

How can I prove $Y=X+Z$ is a Gaussian Random Variable iff $Z$ is a Random Variable?

Let $X$ and $Y$ be Gaussian Variables. We know $Y=X+Z$. Let $X$ and $Z$ be independent. How can I prove $Y$ is a Gaussian Random Variable iff $Z$ is a Random Variable? Can I use $X$, $Z$ Orthogonal ...
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1answer
81 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, ...
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1answer
60 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 ...
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2answers
393 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)...
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2answers
78 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}=\...
9
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0answers
209 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?