Questions tagged [gauss-sums]
For questions on Gauss sums, a particular kind of finite sum of roots of unity.
107
questions
0
votes
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 ...
1
vote
0answers
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 ...
0
votes
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 ...
1
vote
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})\...
1
vote
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
$$S(a,q)=\sum_{...
2
votes
1answer
50 views
Prove a formula to evaluate Legendre Symbol $(\frac{-3}{p})$ using quadratic Gauss Sums (i.e. without using quadratic reciprocity)
I've been trying to find a question similar to this, but have failed to find any. The full statement is let $\zeta = e^{\frac{2\pi i}{3}}$, and use the fact that $(2 \zeta + 1)^2 = -3$ and quadratic ...
0
votes
0answers
22 views
Why does the following summation equivalence hold true?
$$
\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}
$$
Can someone explain to me how we got from the first to ...
0
votes
0answers
25 views
Evaluating a certain generalized quadratic gauss sum
Let $p \geq 2$ and $0 \leq s \leq 2p-1$ be integers.
Set $e(y) = \exp( 2 \pi i y )$.
What's the trick in evaluating
\begin{align}
\sum_{r = 0}^{p-1} e\left( \frac{-r(r + s)}{p} \right)
%\...
3
votes
2answers
182 views
Prove that $\sum _{x=0}^{p-1}e^{\frac {2\pi ix^{2}}{p}}={\sqrt {p}} $ , $ p \equiv 1{\pmod {4}}$
Prove that :
$$\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}}$$ where $p$ ...
1
vote
1answer
26 views
Gauss sums and Dirichlet characters
Currently, I'm attending an Analytic Number Theory course, and in the lecture notes I've come across the following statement:
Does anyone know how to prove this, or at least can give me a reference? ...
2
votes
1answer
34 views
Distribution of Gauss sums on a circle in the Argand plane
It is a fairly well-known fact that the Gauss sum of a non-trivial character $\chi$ modulo a prime $p$ is always a complex number with an absolute value of $\sqrt{p}$.
In other words, when the Gauss ...
0
votes
0answers
36 views
Need help in proving an inequality in lecture notes [duplicate]
While studying class notes in number theory I am unable to deduce this inequality. ( It is used in chapter gauss sums).
Assume ( q, m) be gcd of q and m and all s, q, m be positive integers .
It ...
1
vote
1answer
71 views
A special type of Gauss sum
In the work of my thesis I came up with a problem that is elementary, but I can't figure out its proof.
Let $p$ be an odd prime, let $(\mathbb{Z}/p^n\mathbb{Z})^\times$ denote the multiplicative ...
0
votes
0answers
29 views
Multiplying two Gauss sums
In Goldfeld's book on automorphic representations, he computes the local root number of every ramified place of a character $\omega$ to be the Gauss sum
$$\sum_{\substack{j=1\\(j,p)=1}}^{p^r}\omega(j)...
3
votes
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 ...
1
vote
0answers
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 ...
0
votes
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
$$\tau(1,\chi) = \sum_{x\in (\mathbb{Z}/N\mathbb{Z})^...
2
votes
0answers
27 views
Another Gaussian sum
I'm trying to evaluate
$$
\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'...
1
vote
0answers
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
$$G(a,b)=\...
0
votes
4answers
93 views
Multiple sum variable notation
I have a question regarding multiple sum variables.
Is it correct to rewrite this sum like this ?
$$ \sum_{1\leq i<j\leq n}^{}\frac{i}{j} = \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{i}{j} $$
and how ...
0
votes
1answer
34 views
How can one compute power sums with negative exponent?
Is there a general method to find the sum of powers with negative exponent? For example:
$\sum_{i = 1}^{N} i^a$
with
$ a \in \mathbb{Z} - \mathbb{N}$
1
vote
2answers
77 views
Finding a formula for $(1^3)\cdot(n)+(2^3)\cdot(n-1)+(3^3)\cdot(n-2)+ \cdots + (n^3)\cdot(1)$
I need a formula for this sum:
$$(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\...
2
votes
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 \...
1
vote
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 ...
1
vote
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 $\gcd(n,q)=1$, ...
12
votes
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:
$$-\tan\frac{\pi}{5}+4\sin\frac{2\pi}{5}=\tan\frac{\pi}{5}+4\sin\frac{\pi}{5}=\sqrt{5+2\sqrt{5}}$$
$$-\tan\frac{2\pi}{7}+4\sin\...
2
votes
0answers
69 views
Quadratic Gauss Sum with Jacobi Symbol of even nonsquare free Modulus
Let $d, \ell$ be integers, $\left(\frac{\cdot}{\cdot}\right)$ be the Jacobi symbol, I would like to compute the following Gauss sum.
$$G_{\left(\frac{\cdot}{d}\right)}(\ell):=\sum_{\alpha\bmod(d)}\...
1
vote
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 ...
1
vote
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?
2
votes
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} \...
3
votes
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 ...
1
vote
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:
$$
\sum_{i=0}^{p^n -1}i^...
6
votes
2answers
1k views
Expected maximum of sub-Gaussian
I'm trying to answer the following question from the book high-dimensional probability:
Let $X_1,X_2,\dots$ be a sequence of sub-gaussian random variables, which are not necessarily independent. Show ...
0
votes
1answer
36 views
What is the reason for taking $\omega$ to be a primitive $q$-th root unity rather than taking any $q$-th root of unity?
Let $p$ and $q$ be two distinct odd primes. Let $\omega$ be a primitive $q$-th root of unity. Consider the sum
$$S = \sum_{x \in \Bbb {F_q}^*} \left ( \frac x q \right ) {\omega}^x.$$ Prove ...
1
vote
1answer
34 views
Show that $S^p = \sum\limits_{x \in \Bbb {F_q}^*} \left ( \frac x q \right ) {\omega}^{xp}.$
Let $a \in \Bbb Z$ and $p,q$ be primes. Define $\left (\frac a p \right )$ as follows $:$
$$\left(\frac{a}{p}\right) = \begin{cases}\;\;\,0&\text{ if }p \text { divides } a\\+1&\text{ if } a \...
0
votes
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 ...
1
vote
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 ...
3
votes
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 ...
-1
votes
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.
...
3
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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
votes
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 — ...
1
vote
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 ...
1
vote
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,
...
0
votes
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 ...
1
vote
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:
$$p(X_i=x_i)=\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x_i-\mu)...
0
votes
2answers
78 views
$ S^2= \left(\sum_{j=0}^{p-1} \epsilon^{j^2}\right)^2$ where $p$ are prime number, $\epsilon $ is primitive p-th root of unity.
Let $p$ are prime number, $\epsilon $ is primitive p-th root of unity.
Calculate:
$$ S^2= \left(\sum_{j=0}^{p-1} \epsilon^{j^2}\right)^2$$
$p=3;p=5$ the result are real number.
$\epsilon^{j^2}=\...
9
votes
0answers
209 views
How to prove that below quantity is purely imaginary?
How do I prove that the following quantity is purely imaginary:
$$\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?
