Tagged Questions
0
votes
0answers
87 views
$L(1+it,\chi)\neq 0 $ whenever $t \neq 0 \in \mathbb{R}$
I understand that the proof of the assertion in the title uses the same method which proves that zeta function satisfies $\zeta(1+it)\neq 0$, where the above $L$ is Dirichlet L-function.
I.e, you ...
5
votes
1answer
125 views
Exponentiation of a Dirichlet series
I'm trying to understand a proof in Chandrasekharan's Introduction to Analytic Number Theory. Specifically, the proof of the lemma on p.118 before Dirichlet's theorem on primes in arithmetic ...
2
votes
0answers
85 views
Dirichlet series 'shifted' by a polynomial
Let $F(x) \in \mathbb{Z}[x]$ and
$$
\xi(s) = \sum^\infty_{n=1}g(n)n^{-s}
$$
be the Dirichlet series associated an arithmetic function $g(n)$. Define a new Dirichlet series
$$
\xi_F(s) = ...
2
votes
1answer
149 views
Dirichlet character over Riemann zeta function
Let $\chi$ be a Dirichlet character mod q and let $$L(s,\chi)=\sum_{n\leq x} \frac{\chi(n)}{n^s}.$$ What is the value of $\displaystyle\lim_{s \rightarrow 1} \frac{L(s,\chi)}{\zeta(s)}$ for principal ...
4
votes
1answer
101 views
Relation between zeta function and Dirichlet L-function
Let $$H(s)=\frac{\zeta(s)}{\phi(q)} \sum_{\chi \mod{q}} L(s,\chi)=\sum_{n=1}^{\infty} \frac{h(n)}{n^s}$$ What is the smallest n (as a function of q) such that $h(n)\neq 1$?
3
votes
0answers
117 views
Dirichlet series represents an analytic function
Let $$T(x)=\sum_{n \leq x} t_n$$ and $T(X)=O(x^a)$ for $a \geq 0$. Now let $$F(s)=\sum_{n=1}^{\infty} \frac{t_n}{n^s}$$ What needs to be checked to prove that this Dirichlet series represents an ...
2
votes
2answers
86 views
Convergence of sum in proof that $\Phi(s) - \frac{1}{s-1}$ extends to $\Re(s) > \frac{1}{2}$
Definitions: $\Phi(s) = \displaystyle\sum_{p} \frac{\log p}{p^s}$ where $p$ denotes a prime number.
$\zeta(s) = \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^s}$ denotes the Riemann zeta function.
...
5
votes
0answers
169 views
How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
5
votes
2answers
378 views
An identity involving the Möbius function
$$\sum_{n=1}^{\infty}\frac{1}{n^s}\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=1$$
for $s>1$.
How do I prove this identity?
2
votes
2answers
103 views
A question about an identity involving Dirichlet characters
Let $\chi$ be a Dirichlet character $\bmod q$. We have
$$\sum_{n=0}^{\infty} (-1)^{n-1} \chi(n) n^{-s}=\prod_p ...
10
votes
2answers
309 views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...
16
votes
3answers
665 views
On Dirichlet series and critical strips
(I'll keep this one short)
Given a Dirichlet series
$$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$
where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero ...
