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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.

Dirichlet characters are multiplicative homomorphisms from $(\mathbb Z/N\mathbb Z)^\times$ to $\mathbb C^\times$, extended by $0$ to all of $\mathbb Z/N\mathbb Z$, and then composed with the quotient map $\mathbb Z\to \mathbb Z/N\mathbb Z$.

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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 $a, b$ coprime to $m$. I'm interested in these…
pregunton
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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 $a + km, k \in \mathbb{N}$. I would like to know what is the…
Nicolás A.
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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}\frac{f(n)}{\sqrt…
Alann Rosas
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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…
Zhan
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Evaluating double integral connected to Dirichlet L function

Recently I have been working on the following question: Let \begin{equation*} I_n:= \int\int_{[0,1]^2}\frac{(3x)^n(1-x^3)^n(3y)^n(1-y^3)^n}{(1+xy+x^2 y^2)^{2n+1}}dxdy. \end{equation*} Define \begin{equation*} L(\chi_{-3}, 2) :=…
Tipeg
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Bound for $L(\sigma+it,\chi)$ when $\sigma\geqslant 1/2$

Let $|t|\geqslant 3$. I don't know the bound for $L$-functions when $\chi$ is a primitive character modulo $q$. It seems to me that $$\zeta(\tfrac{1}{2}+it)\ll_{\varepsilon} |t|^{1/6+\varepsilon}$$ and similarly…
W. Wongcharoenbhorn
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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 $a_n=r_2(n)$, $\varphi(t)=\frac{1}{t}$, rememebering that…
Desco
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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 $A(n)=\sum_{x\leq t\leq n}\chi(t)$, then I know $|A(n)|\ll 1$. I think that will…
Emma Glaslow
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Twists of Holomorphic Forms by Principal Characters

So I know that if $f \in \mathscr{M}_{k}(N,\chi)$ has Fourier expansion $$f(z) = \sum_{n \ge 0}a(n)e^{2\pi inz},$$ and $\psi$ is a primitive character modulo $M$, then the twist $$(f \otimes \psi)(z) = \sum_{n \ge 0}a(n)\psi(n)e^{2\pi inz}$$ lives…
Laan Morse
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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) $\sum_{a=0}^{m-1} \chi(a) e^{2π\iota ab/m} =\overline{ \chi(b)} W(x)$…
Ricci Ten
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How to determine $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4,\Gamma_5$?

I started learning about Dirichlet Characters. Here is what I learned so far: Definition: Let $m \in \mathbb{N}$. We call a function $\chi:\mathbb{Z} \rightarrow \mathbb{C}$ a Dirichlet Character mod $m$ if the following…
NTc5
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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 defined as a completely multiplicative function…
Thomas Anton
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Proposition 16.5.4 in Ireland-Rosen

We aim to show that if $\chi$ is a complex Dirichlet character mod $m$, then $L(1, \chi) \neq 0$. Assuming otherwise, we easily prove that if $F(s) = \prod_{\chi}L(s, \chi)$, where the product is over all Dirichlet characters mod $m$, then $F(1) =…
Johnny Apple
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Why $L(s, χ)$ is nonzero for $s$ real and $χ$ complex?

A quick question... In Section 11.1 of the book of Montgomery & Vaughan's Multiplicative Number Theory when studying the case $χ$ complex it doesn't suppose there can be a real zero for $L(s, χ)$ but when studying the case $χ$ quadratic it says "If…
Ali
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Number of primitive Dirichlet characters of certain order and of bounded conductor

Writing $q(\chi)$ for the conductor of a Dirichlet character $\chi$, one can show using Mobius inversion that $$\#\{\text{$\chi$ primitive Dirichlet characters}\,:\,q(\chi)\leq Q\}\sim cQ^2.$$ My question is how to find an asymptotic expression for…
user1296122
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