# 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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### $\chi(n)=\frac{\tau_k(\chi)}{\sqrt k}\sum_{m=1}^k \bar\chi(m)e^{-\frac{2\pi imn}{k}}$

[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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### 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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### 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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### 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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### 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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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 ...