Questions tagged [gauss-sums]

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

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Quadratic Gauss sums and Fourier Analysis

Lately I have been reading this very interesting book Mathematics++: Selected Topics Beyond the Basic Courses by Kantor, Matoušek and Šámal, especially Chapter 3 on Fourier analysis. In Section 5: ...
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Tate gamma factor as a principal value integral

Let $F$ be a local field, $\chi$ a multiplicative character of $F^{\ast}$, and $\psi$ an additive character of $F$. The gamma factor $\gamma(s,\chi,\psi)$ is defined by means of the local functional ...
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Sum of the Multiplicative Legendre's symbol

Let $p$ a be odd prime show that $\displaystyle\sum_{a=1}^{p-2}\left(\frac{a(a+1)}{p}\right)=-1$. Nota Bene : The $(\frac{a}{p})$ is $\textbf{Legendre Symbol}$
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Gauss sum over Legendre symbols combining residue and non-residue parts

I’m reading Davenport’s Multiplicative Number Theory and I’m currently in the second chapter, the subject of which is calculating the following sum: $$ G= \sum_{n=1}^{q-1} \Bigr(\frac{n}{q}\Bigr )e_q(...
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gauss sum alternative definition with Trace map

I am a bit confused with the Gauss sums. Let $\chi: \mathbb F_p^\times \to \mathbb C^\times$ a multiplicative character and extend it by defining $\chi(0)=0$ One way (probably the standard one?) to ...
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How to find the Gauss sum $\sum_{x=0}^{p-1} e^{2\pi i(ax^2+bx+c)/p}$?

How to find the sum: $\displaystyle \sum_{x=0}^{p-1} e^{2\pi i(ax^2+bx+c)/p}$ where p is prime ? I'm sure I need to use this: $\displaystyle \sum_{x=0}^{p-1} e^{2\pi i(x^2)/p}$ = $\sqrt{p}$ when $ p≡3(...
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$2^n$-th rational reciprocity laws

Let $p,q$ be odd coprime primes. We are familiar with the quadratic reciprocity law: $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{(p-1)(q-1)}{4}}$. This is generalized (see for ...
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Only Legendre symbol gives constant magnitude discrete Fourier transform

It is well-known that if we take the Fourier transform $\hat{g}$ of a (translated) Legendre symbol $g(x) = \pm(x+a|p)$, then $|\hat{g}(r)| = \sqrt p$ for all $r\neq 0$. I heard that the converse is ...
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Why does the quadratic Gauss sum $G(a,b,c)$ evaluate to 0 when $gcd(a,c)>1$?

On the Wikipedia page for the quadratic Gauss sum it states that it will evaluate to $0$ in the case that $\text{gcd}(a,c)>1$, except when $l|b$. The Gauss sum is defined as $$G(a,b,c)=\sum_{n=0}^{...
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Formula for twisted generalized quadratic Gauss sums

Let $q,a,b$ be integers, $q \geq 1$, $(q,a)=1$, and let $\chi$ be a Dirichlet character modulo $q$. Is there a formula for the sum $$ G_\chi(a,b,q) := \sum_{n(q)} e^{2\pi i \frac{(an^2 + bn)}{q}} \chi(...
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A little bit different quadratic Gauss sum

Could anyone give me a direction on how to demonstrate that $$ \sum\limits_{k = 0}^{N-1} e^{-i\frac{\pi}{N}(k+C)^2} = \sqrt{N} e^{-i\frac{\pi}{4}}, $$ if $N\in \mathbb{N}$ is even, for any $C \in \...
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Show that the quadratic Gauss sum can be simplified by completing the square

For natural numbers, $a,b,c$ the Gauss sum is defined as $$G(a,b,c)=\sum_n^{c-1}e^{2\pi i\frac{an^2+bn}{c}}$$ From the quadratic Gauss sum Wikipedia page it is given that for odd $c$ and $\text{gcd}(a,...
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Covariance of geometric Brownian motion via first and second moments and MGFs

My attempts: Substitute in the $Z(t)= X(t)Y(t)$ in $\mathrm{Cov}(X(t),Z(t))$. From that, I tried to split it into smaller covariances. Also trying to figure out how can the moment generating functions ...
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Number of solutions to $x^3 + y^3 + z^3 + t^3 = 0 \pmod p$

Let $p$ be a prime. Prove that the number of ordered quadruples $(x,y,z,t)$ with $x^3 + y^3 + z^3 + t^3 = 0 \pmod p$ is $p^3 + 6p^2 - 6p$ if $p\equiv 1 \pmod 3$ and $p^3$ otherwise. I can deal only ...
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On the property of Gauss sum

In the book of a classical introduction to modern number theory by Kenneth Ireland and Michael Rosen page 92 in the proof of proposition 8.2.2 of gauss sum, I don't understand how he gets this $ \sum_{...
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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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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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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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168 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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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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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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4 votes
2 answers
393 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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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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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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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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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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2 answers
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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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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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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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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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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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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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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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2 answers
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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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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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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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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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13 votes
5 answers
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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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2 votes
0 answers
123 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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1 vote
1 answer
104 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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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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2 votes
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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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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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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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8 votes
3 answers
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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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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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1 answer
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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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  • 4,735
0 votes
1 answer
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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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1 vote
0 answers
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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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3 votes
0 answers
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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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