Questions tagged [gauss-sums]
For questions on Gauss sums, a particular kind of finite sum of roots of unity.
130
questions
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Question on a result on Gauss sums and Dirichlet characters
In Modular Forms by Toshitsune Miyake, at page 80,
I am stuck at lemma $3.1.1$
Assume $W $ is a Gauss sum.
Lemma 3.1.1. Let $\chi$ be a primitive Dirichlet character mod $m$.
(1) $\sum_{a=0}^{m-1} \...
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1
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Gauss sum over imaginary quadratic field
Define the Gauss sum
$$ G\left( \frac{a}{m}\right) = \sum_{n(\text{mod}\ m)} e\left( \frac{a n^2}{m}\right) ,$$
then I know the following result:
For any $a,m$ with $(2a,m) = 1$,
$$ G\left( \frac{a}{...
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$\chi(n)=\frac{\tau_k(\chi)}{\sqrt k}\sum_{m=1}^k \bar\chi(m)e^{-\frac{2\pi imn}{k}}$
[Analytic Number Theory - Florian Luca and Jean Marie De Koninck, chapter 14, question 14.5]
A character is called primitive modulo $k$ if it is not induced from any divisor $d<k$, or in other ...
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Question on a Gaussian sum and Jacobi sum identity
I've tried, unsuccessfully, to solve exercise 8.5 from Ireland and Rosen's A Classical Introduction to Modern Number Theory. The exercise asks to prove
\begin{align}g(\chi)^2=\frac{J(\chi,\rho)g(\chi^...
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0
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Help with summing characters
Let $p \equiv 1(n)$. Let $\chi$ denote a character from $\mathbb{F}_p \rightarrow \mathbb{C}-\{0\}$. Then, in Ireland and Rosen's A Classical Introduction to Number Theory, while trying to find the ...
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Question about Gauss Sums
I'm reading Davenport's Multiplicative Number Theory, specifically chapter 2 in which we calculate the sum
$$G=\sum_{m=1}^{q-1}\left(\frac{m}{q}\right)e_q(m)$$
where $e_q(m)=\exp(2\pi i m/q)$.
He ...
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A Relation between Dirichlet L functions for Quadratic Character and Gauss sums
This is problem 8 of Chapter 16 in the book A Classical Approach to Modern Number Theory by Ireland and Rosen.
Let $g(\chi)$ be the classical Gauss sum : $\sum_{x = 1}^{p-1} \chi(x) \zeta^{x} $, $\chi$...
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1
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How does Ireland-Rosen reduce the coefficient of $z^{\frac{p-1}{2}}$ into this neat form in Kronecker proof of Gaussian sums?
Was reading Ireland-Rosen intro to number theory and ran into an expression on page 75 I couldn't make sense of.
They assert that (it is easy to see that) the coefficient of $z^\frac{p-1}{2}$ in the ...
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2
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A problem with showing that $g_a^2=(-1)^{\tfrac{p-1}{2}}p$ for quadratic Gauss sums $g_a$
Let $p>2$ be prime, and let $a\in\Bbb Z$. Also, let $\left(\frac{n}{p}\right)$ denote the Legendre symbol.
Then let the quadratic Gauss sum be
$$g_a=\sum_{x=1}^{p-1}\left(\frac{x}{p}\right)\zeta_p^{...
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38
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Gauss sum involved in the Hecke action on classical Hilbert modular forms
Let $F=\mathbb{Q}(\sqrt{D})$ be a real quadratic field and consider the classical Hilbert modular forms over $F$. Let $\varepsilon_0>1$ be the fundamental unit of $F$ and write $d=\varepsilon_0\...
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Proof of twisted Poisson summation formula using 'smoothed' Dirichlet characters?
Over at mathoverflow there is a question regarding the twisted Poisson summation formula:
$$
\sum_{n \in \mathbb{Z}} \chi(n) f \left( \frac{nx}{q} \right) = \frac{K}{x} \sum_{n \in \mathbb{Z}} \...
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3
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On the tribonacci constant with $\cos(2\pi\,k/11)$, plastic constant with $\cos(2\pi\,k/23)$, and others
(This post is indebted to Oscar Lanzi.)
Part I. $\color{blue}{p = 11}$
The equation,
$$\big(4\sin[3t] - \tan[t]\big)^2 = 11$$
seems to have five solutions, given by $t = \frac{2\pi\,k}{11}$ for $k = 1,...
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Quadratic Gauss sums and Fourier Analysis
Lately I have been reading this very interesting book Mathematics++: Selected Topics Beyond the Basic Courses by Kantor, Matoušek and Šámal, especially Chapter 3 on Fourier analysis.
In Section 5: ...
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1
answer
79
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Tate gamma factor as a principal value integral
Let $F$ be a local field, $\chi$ a multiplicative character of $F^{\ast}$, and $\psi$ an additive character of $F$. The gamma factor $\gamma(s,\chi,\psi)$ is defined by means of the local functional ...
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Sum of the Multiplicative Legendre's symbol
Let $p$ a be odd prime show that $\displaystyle\sum_{a=1}^{p-2}\left(\frac{a(a+1)}{p}\right)=-1$.
Nota Bene : The $(\frac{a}{p})$ is $\textbf{Legendre Symbol}$
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Gauss sum over Legendre symbols combining residue and non-residue parts
I’m reading Davenport’s Multiplicative Number Theory and I’m currently in the second chapter, the subject of which is calculating the following sum:
$$
G= \sum_{n=1}^{q-1} \Bigr(\frac{n}{q}\Bigr )e_q(...
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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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1
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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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80
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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
$$G(a,b,c)=\sum_{n=0}^{...
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0
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Formula for twisted generalized quadratic Gauss sums
Let $q,a,b$ be integers, $q \geq 1$, $(q,a)=1$, and let $\chi$ be a Dirichlet character modulo $q$. Is there a formula for the sum
$$
G_\chi(a,b,q) := \sum_{n(q)} e^{2\pi i \frac{(an^2 + bn)}{q}} \chi(...
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1
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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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0
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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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1
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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 ...
user921584
3
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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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1
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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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1
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156
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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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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
$$S(a,q)=\sum_{...
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1
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Prove a formula to evaluate Legendre Symbol $(\frac{-3}{p})$ using quadratic Gauss Sums (i.e. without using quadratic reciprocity)
I've been trying to find a question similar to this, but have failed to find any. The full statement is let $\zeta = e^{\frac{2\pi i}{3}}$, and use the fact that $(2 \zeta + 1)^2 = -3$ and quadratic ...
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Why does the following summation equivalence hold true?
$$
\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}
$$
Can someone explain to me how we got from the first to ...
user804333
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2
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Prove that $\sum _{x=0}^{p-1}e^{\frac {2\pi ix^{2}}{p}}={\sqrt {p}} $ , $ p \equiv 1{\pmod {4}}$
Prove that :
$$\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}}$$ where $p$ ...
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1
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Gauss sums and Dirichlet characters
Currently, I'm attending an Analytic Number Theory course, and in the lecture notes I've come across the following statement:
Does anyone know how to prove this, or at least can give me a reference? ...
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1
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Distribution of Gauss sums on a circle in the Argand plane
It is a fairly well-known fact that the Gauss sum of a non-trivial character $\chi$ modulo a prime $p$ is always a complex number with an absolute value of $\sqrt{p}$.
In other words, when the Gauss ...
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1
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A special type of Gauss sum
In the work of my thesis I came up with a problem that is elementary, but I can't figure out its proof.
Let $p$ be an odd prime, let $(\mathbb{Z}/p^n\mathbb{Z})^\times$ denote the multiplicative ...
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Multiplying two Gauss sums
In Goldfeld's book on automorphic representations, he computes the local root number of every ramified place of a character $\omega$ to be the Gauss sum
$$\sum_{\substack{j=1\\(j,p)=1}}^{p^r}\omega(j)...
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2
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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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0
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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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1
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1
answer
430
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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
$$\tau(1,\chi) = \sum_{x\in (\mathbb{Z}/N\mathbb{Z})^...
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0
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Another Gaussian sum
I'm trying to evaluate
$$
\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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1
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0
answers
92
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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
$$G(a,b)=\...
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votes
4
answers
498
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Multiple sum variable notation
I have a question regarding multiple sum variables.
Is it correct to rewrite this sum like this ?
$$ \sum_{1\leq i<j\leq n}^{}\frac{i}{j} = \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{i}{j} $$
and how ...
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votes
1
answer
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Is there a general method to find $\sum_{i = 1}^{N} i^a$ [closed]
Is there a general method to find the sum of powers with negative exponent?
For example: $\sum_{i = 1}^{N} i^a$ with $ a \in \mathbb{Z} - \mathbb{N}$
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2
answers
80
views
Finding a formula for $(1^3)\cdot(n)+(2^3)\cdot(n-1)+(3^3)\cdot(n-2)+ \cdots + (n^3)\cdot(1)$
I need a formula for this sum:
$$(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
0
answers
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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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1
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2
answers
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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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0
answers
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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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5
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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:
$$-\tan\frac{\pi}{5}+4\sin\frac{2\pi}{5}=\tan\frac{\pi}{5}+4\sin\frac{\pi}{5}=\sqrt{5+2\sqrt{5}}$$
$$-\tan\frac{2\pi}{7}+4\sin\...
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0
answers
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Quadratic Gauss Sum with Jacobi Symbol of even nonsquare free Modulus
Let $d, \ell$ be integers, $\left(\frac{\cdot}{\cdot}\right)$ be the Jacobi symbol, I would like to compute the following Gauss sum.
$$G_{\left(\frac{\cdot}{d}\right)}(\ell):=\sum_{\alpha\bmod(d)}\...
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