Questions tagged [dirichlet-character]

Dirichlet characters appear in Dirichlet $L$-functions, in Gauss sums, and in other arithmetical generating functions. They are not exactly group characters, but are extensions by $0$ of such.

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Exponential Generating Function for Dirichlet Character

I am working on a differential equation problem, and have ended up with the following form: $$g(x; N) = \sum_{n=0}^{\infty} \frac{\chi_0(n)}{n!} x^n,$$ with $\chi_0(n)$ the principal Dirichlet ...
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What does $\sum_{p|n} \chi_{-4}(p)$ count?

Let $\chi_{-4}$ be the Dirichlet character defined by $$ \chi_{-4}(m) = \begin{cases} 1, & m\equiv 1 \mod 4 \\ -1, &m \equiv 3 \mod 4\\ 0, &m \equiv 0 \mod 2.\end{cases}$$ I know that $$...
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"Mollifier" of the Dirichlet L-function

I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
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What are trivial and non-trivial Dirichlet characters?

How to define them properly? And where to find free official documentation about it on the web please? I've been looking around for clear definitions for more than one hour now, without success. I ...
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Evaluate $L(1, \chi) = \sum_{n=1}^\infty \frac{\chi_5(n)}{n},$ for $\chi$ mod $5$

My HW question is: Evaluate the series $$L(1, \chi_5) = \sum_{n=1}^\infty \frac{\chi_5(n)}{n},$$ where $\chi_5$ is the unique nontrivial Dirichlet character mod $5$. My work is: \begin{align*} ...
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Evaluate $L(1, \chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n}$ for $\chi$ mod $3$

Here is the homework question I am working on: Evaluate (as a real number) the series $$L(1, \chi_3) = \sum_{n=1}^\infty \frac{\chi_3(n)}{n},$$ where $\chi_3$ is the unique nontrivial Dirichlet ...
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Show that $\sum_{n\leq x}\frac{f(n)}{\sqrt n}=2L(1,\chi)\sqrt{x}+O(1)$, where $f(n)=\sum_{d\vert n}\chi(n)$.

Exercise 2.4.4 from M. Ram Murty's Problems in Analytic Number Theory asks us to show that for $\chi$ a nontrivial Dirichlet character $(\operatorname{mod} q)$, we have the estimate $$\sum_{n\leq x}\...
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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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Proving Dirichlet character is primitive

Question: Let $m \in \mathbb{Z}_{\neq 0, 1}$ be squarefree integer. Define $N \in \mathbb{N}$ as $$N := \begin{cases} |m| &\ \text{if}\ m \equiv 1\pmod 4\\ 4 |m| &\ \text{if}\ m \equiv 2,3 \...
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If two characters agree on almost all primes then are they induced by same primitive characters?

Suppose $\chi_1$ and $\chi_2$ are Dirichlet characters moduli $q_1$ and $q_2$ and suppose they are induced by same primitive character. If $p$ is a prime greater than both $q_1$ and $q_2$ then ...
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Two questions about Dirichlet's characters and roots of unity.

Let $G$ be the units of $\mathbb Z/ q \mathbb Z.$ Let $\chi$ denote Dirichlet character on $G.$ I have two questions. Let $a \in G$ with order $k, $Then it is easy to see that $\chi(a)$ is a $k$ th ...
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Converting polylogarithms to Dirichlet L functions

When trying to simplify polylogarithms evaluated at some root of unity, namely $\text{Li}_s(\omega)$ for $\omega=e^{2\pi i ~r/n}$, it is reasonable to convert it to Hurwitz zeta functions or Dirichlet ...
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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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The p-adic value of Dirichlet L-function at 1

I'm reading the book An introduction to p-adic L-functions recently. The book can be found in https://warwick.ac.uk/fac/sci/maths/people/staff/cwilliams/lecturenotes/lecture_notes_part_i.pdf. In 4.4, ...
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What's the idea of Dirichlet’s Theorem on Arithmetic Progressions proof?

Dirichlet’s Theorem on Arithmetic Progressions says that if $a, m$ are natural numbers such that $gcd (a,m) = 1$, then there are infinitely many prime numbers in the arithmetic progression $a + km, k \...
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Dirichlet Green's function outside the sphere

I am trying to find the Dirichlet Green's function for the spherical region centred at 0 and of radius a. I have found the Dirichlet Green's function inside the sphere by method of images, but what is ...
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Dirichlet L-Funktion for non-principal charakters at s=1

It seems that in my introducction to analytic number theorie they provide an easy proof that if $\chi\ne \chi_1$, $L(s,\chi)$ is an entire function of s. Thus I do not understand this proof. It says: $...
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$q$-expansion of Eisenstein series twisted by a Dirichlet character

Consider the Eisenstein series given by $$ G_{k}(\chi,z) = \sum_{m,n\in \mathbb{Z}, (n,N) = 1}\frac{\chi(n)}{(mz + n)^{k}} $$ where $\chi:(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow \mathbb{C}^{\...
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The Dirichlet character modulo a prime number

If I consider all Dirichlet characters modulo a prime number, let's say 5. Then I can have a form that looks like this: I found that $$\begin{align*} \chi_{5,4}^2&= \chi_{5,1} \\ \chi_{5,3}^4&...
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Is there a way to prove that $\left|\sum_{n \leqq x} \chi(n) e(\alpha n)\right| \ll_\epsilon \frac{x}{\log q}$ without using Burgess inequality? .

I am working in my undergrad thesis and I have came across the inequality $\left|\sum_{n \leqq x} \chi(n) e(\alpha n)\right| \ll_\epsilon \frac{x}{\log q}$, where $\chi$ is a dirichlet character ...
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What is the value of $L'(1,\chi)$ where $\chi$ is the non-principal Dirichlet character modulo 4?

I was trying to compute the following sum: $$\sum_{n\le x}{\frac{r_2(n)}{n}}$$ where $r_2(n)=\vert\{(a,b)\in\mathbb{Z}^2:a^2+b^2=n\}\vert$. Using Abel's summation formula with $a_n=r_2(n)$, $\varphi(t)...
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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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Positivity of partial Dirichlet series for a quadratic character?

Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
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Sum of Dirichlet characters

The question is: Let $\chi_1,\chi_2,\chi_3,\chi_4$ be Dirichlet characters in $(\mathbb Z/N\mathbb Z)^*$ and they are distinct pairwise. Can we proof that $\chi_1+\chi_2\neq\chi_3+\chi_4$? Or can we ...
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Is the principal character mod 1 considered primitive?

I am under the impression that the principal Dirichlet character $\chi_0\bmod{q}$ is primitive if and only if $q=1$. However, I read in Davenport's multiplicative number theory that the principal ...
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Dirichlet Characters for More General Rings

I was wondering if there was any value to generalising the definition of Dirichlet characters to more general rings in both domain and codomain. For example, a Dirichlet character mod $m$ can be ...
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Generalization for Riemann-Siegel theta function for Dirichlet L-functions

The Riemann-Siegel theta function $\theta(t)$ is well-behaved and satisfies $\zeta(\frac{1}{2}+it)=Z(t)e^{-i\theta(t)}$ for real $Z(t)$. Because of how smoothly $\theta(t)$ changes, $Z(t)$ changes ...
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Sum Over the Group of Dirichlet Character

I have the following question: Suppose $x=q^{o(1)}$, I want to prove the following estimate: $$\frac{1}{\phi(q)}\sum_{p_1, p_2\leq x}\sum_{\chi mod q}Re(\chi(p_1))Re(\chi(p_2))=\frac{1}{2}\pi(x)+o(1) \...
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LMFDB labeling of Dirichlet characters

I am trying to understand the LMFDB labeling of Dirichlet characters (see here), but I am not entirely sure what they mean by the "orbit index". To get things started, fix some Dirichlet ...
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Sum over primitive Dirichlet characters

Take $q,n\in \mathbb N$. I want to evaluate exactly \[ S_q(n):=\sum _{\chi }\tau (\chi )\chi (n)\] where the sum extends over primitive characters modulo $q$ and where $\tau (\chi )$ denotes the ...
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Looking for table of special values of the Dirichlet $L$-function

For double checking calculations I made I'd like to find a table of values of $L(-1,\chi_D)$ for small positive fundamental discriminats $D$. It there a table somewhere in the internet? Where? With $\...
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Why is the name "orthogonality"?

Let $G$ be a finite abelian group and $\hat G$ be the character group of $G$.Let $G=\{a_1,a_2,...,a_n\}$ and $\hat G=\{f_1,f_2,...,f_n\}$.$a_1$ is the identity of $G$ and $f_1$ is the principal ...
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Unitary characters of $1$-torus

This may be really elementary but I'm lost. Let $S^1\subseteq\mathbb{C}$ be the unit circle, which is a group with complex multiplication. Let $\mathbb{T}:=\frac {\mathbb{R}}{\mathbb{Z}}$ be the $1$-...
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Polynomials whose fractional part behaves like a logarithm

For a given integer $m$, I'm looking for a classification of all polynomials $P$ with rational coefficients satisfying the logarithm-like condition $$P(ab)=P(a)+P(b) \pmod 1$$ for any integers $a, b$ ...
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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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How would one motivate/know to introduce the Dirichlet character in the formula for the number of lattice points on a circle of radius $\sqrt N$

Grant's masterful video https://www.youtube.com/watch?v=NaL_Cb42WyY&ab_channel=3Blue1Brown ("Pi hiding in prime regularities") describes a way of computing $\pi$ that ultimately leads us ...
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Primitive and Induced Dirichlet Characters

I'm reading Zagier's book on L-Series and Quadratic Forms and I'm stuck at the proof of the classification of real primitive Dirichlet characters. I understood the proof that $\widehat{(\mathbb{Z}/N\...
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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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Proof that the product of $L$-function over all Dirichlet characters is not less than 1 (from Fourier Analysis by Stein)

On page 266 of Fourier Analysis by Stein and Shakarchi, it is shown that If $s > 1$, then $$\prod_{\chi} L(s,\chi)\geq 1,$$ where the product is taken over all Dirichlet characters. In particular ...
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Confusion with notation for Dirichlet characters

While I have understood the properties of Dirichlet characters and how to construct tables for them, I am incredibly confused by the notation. Here are the notations in use in the book Introduction to ...
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Equivalent definition of primitive Dirichlet character

This is likely to be a stupid question, but I can't tell which part is going wrong. Let $\chi:(\mathbb{Z}/N)^\times\to S^1$ be a Dirichlet character mod $N$. I found the definition is that $\chi$ is ...
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Dirichlet L-series and Hecke L-series

I'm working on L-series (reading Rosen's book Number Theory in Function fields) and i read that Dirichlet $L$-series are supposed to be a special case of Hecke $L$-series, and i can't understand why ?
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$ \prod_{k=0}^\infty \prod_{j=1}^q ( 1- \chi(j) z/(kq+j) ) = (1-z) \prod_{k=2}^\infty ( 1 - \chi(k) z/k) $

Let $\chi$ be a primitive non-principal Dirichlet character with conductor $q>1$. The following equation $$ \prod_{k=0}^\infty\prod_{j=1}^q \biggl( 1- \chi(j) \frac{z}{kq+j} \biggr) = (1-z) \prod_{...
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Relationship between Dirichlet character and Legrendre symbol [closed]

I'm wondering whether you always can express a non trivial Dirichlet character by a Legendre symbol. And in case so, how would one explicitly do that? Or how does one connect the two things?
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Conductor of product of characters is the product of the conductors

First of all, I know that there are other questions similar to this, but these use other definitions or don't have a complete answer. I'm trying to prove that if $\chi_{1}$ and $\chi_{2}$ are two ...
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Calculate integral $ \int_{0}^{+\infty} \frac{x - \sin{\left(x \right)}}{x^{3}}\, dx $ using the Dirichlet integral

I try to calculate the integral: $$ \int_{0}^{+\infty} \frac{x - \sin{\left(x \right)}}{x^{3}}\, dx, $$ using the Dirichlet integral $$ \int\limits_0^{+\infty} \frac{\sin \alpha x}{x}\,dx = \frac{\pi}{...
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Understanding sum of Dirichlet characters

I'm trying to understand how Dirichlet Characters work, especially the sum of Dirichlet Characters. Concerning the definitions, we have: $\chi : G \longrightarrow \mathbb{C} ^{\times}$ is a ...
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Want to show $\sum_{n>N}n^{-1/2} \ll N^{-1/2}$

I feel like I can use the result $$\sum_{n=x}^N \frac{a_n}{n^s} = A(N)N^{-s} + s \int_x^N A(t)t^{-s-1}dt$$ where $s=1/2$ to verify the $\ll$ approximation. If I pick $A(n)=\sum_{x\leq t\leq n}\chi(t)$,...
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Dirichlet Characters, Characteristic Functions, and Orthogonality in Proof of Dirichlet's Theorem

I am confused by the justification of Equation 7 from https://sites.math.washington.edu/~morrow/336_14/papers/austin.pdf, which states that (substituting in the definitions and slightly modifying the ...
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Extending a Group Character to a Dirichlet Character

In https://math.mit.edu/classes/18.785/2015fa/LectureNotes17.pdf, just after Definition 17.4 on page 7, the author states that "Every $m$-periodic Dirichlet character $\chi$ restricts to a group ...
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