For questions on Dirichlet series.
1
vote
1answer
87 views
Euler product of Dirichlet Series
For $n$ a positive integer, let $f(n)$ be the squarefree part of $n$.
Find the Euler product for $\mathfrak D_{f}(s)$ where $\mathfrak D_{f}(s)$ is the Dirichlet Series of $f$.
5
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0answers
168 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 ...
3
votes
0answers
37 views
Dirichlet series' help
If $f(n)$ is an arithmetic function with $|f(n)|=1$, and $$\lim_{s\to+1} (s-1)\sum_{n=1}^\infty\frac{f(n)}{n^s}=0$$
Can I deduce that $$\lim_{s\to +1}(s-1)^2\sum_{n=1}^\infty\frac{f(n)\ln(n)}{n^s}=0$$
...
3
votes
0answers
46 views
Can that two double series representations of the $\eta$/$\zeta$ function be converted into each other?
By an analysis of the matrix of Eulerian numbers(see pg 8) I came across the representation for the alternating Dirichlet series $\eta$:
$$ \eta(s) = 2^{s-1} \sum_{c=0}^\infty \left( ...
3
votes
0answers
116 views
Dirichlet series represents an analytic function
Let $$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}$$ What needs to be checked to prove that this Dirichlet series represents an ...
2
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0answers
84 views
Dirichlet series 'shifted' by a polynomial
Let $F(x) \in \mathbb{Z}[x]$ and
$$
\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) = ...
1
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0answers
54 views
Can we express $x$ as a Dirichlet series?
Can we express $x$ as a Dirichlet series ?
Thus $x = a_0 + a_1 2^{-x} + a_2 3^{-x} + a_3 4^{-x} + ...$ where the $a_i$ are real numbers ? I do not know how to get a nondivergent solution.
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vote
0answers
52 views
How write Dirichlet character sums for the terms of the von Mangoldt function?
The way to separately write the terms of the von Mangoldt function $\Lambda$ as Dirichlet character sums seems to be:
$$\Lambda (1) = \sum\limits_{n=1}^{\infty } \frac{(e^{\Lambda (1)} \chi ...
0
votes
0answers
46 views
Limits as a representation of the Dirichlet function
I read that the Dirichlet function (1 if Rational, 0 else) can be written as:
What is the proof of that? Are those limits commutative? Is there any other closed formula for Dirichlet function? (With ...
0
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0answers
86 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 ...
