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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How to determine $\Gamma_{15}$?
For context,yesterday I asked How to determine $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4,\Gamma_5$?.
Now I learned of new theorem that made me curious:
Theorem:
Let $m,n \in \mathbb{Z}$ such that $gcd(m,n)=...
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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 ...
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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 ...
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M/V Multiplicative NT : Theorem 11.3 and the Siegel zero
Two questions regarding Theorem 11.3 in the book of Montgomery & Vaughan Multiplicative Number Theory on the section "Case 4. Quadratic $χ$, real zeros.":
First the book supposes there ...
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Extension of Dirichlet Divisors in Gauss Circle Integral Point Problem
Dirichlet Divisors
A well-known problem in number theory is to study the sum of divisor functions $d(n)$:
$D_2(x)=\sum_{n\leq x} d(n)$,
so $D_2(x)$ can be expressed as the number of the integer points ...
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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 $L(s, χ)$ ...
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Multiplication of Real Characters
At the beginning of Sec 9.3. of Montgomery/Vaughan's Multiplicative NT the following comes without explanation:
Suppose that χ is a character modulo q, that $q = q_1q_2 $, $ (q_1,
> q_2) = 1$, $...
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Exercise 1 Section 9.2. Montgomery/Vaughan's Multiplicative NT
I am trying to show that for any integer $a$, $$e(a/q) =
\sum_{d|q, d|a} \dfrac{1}{ϕ(q/d)} \sum_{χ \ (mod \ q/d)} χ(a/d) τ(χ).$$ First I considered the case $(a,q)=1$ and the mentioned equality holds. ...
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Lemma 9.11 Montgomery/Vaughan's Multiplicative NT
In the last step of the proof of Lemma 9.11 in the book Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan I couldn't understand the two equalities of the ...
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How to show that gcd of two quasiperiods is a quasiperiod?
Definition 3.1.1 in page 25 of this book is the definition of quasiperod and Proposition 3.1.3. shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about ...
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Application of Dirichlet theorem and Dirchlet density
I'm reading Serre's A course in Arithmetic and I have the following question about Proposition 14 in Chapter VI. It uses this version of Dirichlet's theorem: Let $m\ge 1$, $(a,m) = 1$. Let $P_a$ be ...
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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 ...
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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 ...
