All Questions
Tagged with dirichlet-series sequences-and-series
98
questions
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32
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2 concise tables of “usual” series (mostly trigonometrics) and of "usual" L-series (Zeta, Eta, Beta...)
CONTEXT
Common series are usually described as infinite sums, written as consecutive terms ending with (…). Or they can be described using the $\sum_{}$ symbol and arguments usually including $(-1)^k$ ...
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How do we increase the region of convergence for the Riemann Zeta function (using Dirichlet Series form)?
The Riemann Zeta Function can be defined as: $\zeta(s)=\sum \frac 1 {n^s}$ for $s>1$. The series converges for $s>1$. wiki (https://en.wikipedia.org/wiki/Riemann_zeta_function) mentions that:
An ...
0
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1
answer
44
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For what values of $c$ is $\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}=0$ when $x=\exp \left(-\frac{\rho _1}{c}\right)$?
The alternating Dirichlet series, the Dirichlet eta function, can be written in the form: $\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}$
For what values of $c$ is $$\sum _{k=1}^{\infty } (-1)^{k+1}...
- 7,448
4
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Can we extend the Divisor Function $\sigma_s$ to $\mathbb{Q}$ by extending Ramanujan Sums $c_n$ to $\mathbb{Q}$?
It can be shown that the divisor function $\sigma_s(k)=\sum_{d\vert k} d^s$ defined for $k\in\mathbb{Z}^+$ can be expressed as a Dirichlet series with the Ramanujan sums $c_n(k):=\sum\limits_{m\in(\...
- 1,810
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2
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42
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Given a Dirichlet series that diverges, are there conditions to know when the modulus goes off to infinity?
I was working on a problem, and I had made the assumption that given a Dirichlet series
$$
L(s,f)=\sum_{n\geq 1}\frac{f(n)}{n^s}
$$
If I have some $\sigma\in\mathbb{C}$ such that $L(\sigma,f)$ ...
- 2,295
1
vote
1
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81
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Convergence of sums in $\ell^p \implies \ell^{p-\epsilon}$
Supose $\displaystyle(b_n)_{n \in \mathbb{N}}$
is a sequence of positive real numbers that
$$\displaystyle\sum_{n \in \mathbb{N}}(b_n)^{2} <\infty.$$
Does exists some $\epsilon>0$ such that $\...
- 33
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1
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67
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Can $\alpha$ be found for $\sum_{n=1}^{\infty}\frac{\sigma_0(n^2)}{\sigma_0(n)}\frac{1}{n^s}=\zeta(s)\sum_{n=1}^{\infty}\frac{\mu^2(n)\alpha }{n^s}$?
I was looking for a pattern among these below:
$$ \sum_{n=1}^{\infty} \frac{\sigma_0(n^2)}{n^s} = \zeta^2(s) \sum_{n=1}^{\infty} \frac{ \mu^2(n)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)} $$
$$ \sum_{n=1}^{...
- 908
6
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2
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269
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Write the sum $\sum\limits_{a \in \mathbb{N}}\sum\limits_{b \in \mathbb{N}} \frac{(a,b)}{a^sb^t}$ in terms of the Riemann zeta function
I have the following exercise, and I need some help:
Write the sum
$$\sum\limits_{a \in \mathbb{N}}\sum\limits_{b \in \mathbb{N}} \frac{(a,b)}{a^sb^t}$$ in terms of the Riemann zeta function ($(a,b)$ ...
- 91
2
votes
1
answer
105
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How to find the sum of this infinite series
I am not sure how to evaluate the infinite sum:
$$\sum_{n=0}^\infty \frac{1}{(2n+1)^6}$$
Apparently, I can shift it to
$$\sum_{n=1}^\infty \frac{1}{(2n-1)^6}$$
which is supposed to be a well known sum ...
- 21
0
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0
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49
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Uniformly convergent series manipulation
I get confused reading about L-series and there is a lemma on infinite series. The question should only concern about analysis and there should be no number theory involved. The lemma is below,
Let $\{...
- 558
2
votes
1
answer
122
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Is there any relationship between a Dirichlet series and the same series with the sequence "shifted" by one term?
Suppose
$$ F(s) := \sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}} $$
is a Dirichlet series for the sequence $a_{1}, a_{2}, \ldots\in\mathbb{C}$. Then let
$$ G(s) := \sum_{n=1}^{\infty} \frac{a_{n}}{(n+1)^{s}}...
- 2,461
0
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34
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How to write a convergent function as a dirichlet expansion?
Normally when using Dirichlet series, it is used as a generating function to prove certain results like Dirichlet's theorem. I'm wondering whether it's possible to write a function $f(s)$ as a ...
- 33
5
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2
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137
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Proof that the series $\sum_{n=2}^{\infty}\frac{[\Omega(n)]^\alpha}{n^2}$ converges
Let's consider the series
$$f(\alpha)=\sum_{n\gt1}\frac{[\,\Omega(n)\,]^\alpha}{n^2}$$
where $\Omega(n)$ denotes the number of prime factors of $n$ counted with their multiplicity and $\alpha\ge0$ is ...
- 1,828
4
votes
1
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223
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Proof that Dirichlet series $\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^2}=\frac{5}{2}$
So I want to prove the following:
$$\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^2}=\frac{5}{2},$$
where $\omega(n)$ is the number of distinct prime factors of $n.$
I computed it to $10^{10}$ and it does ...
- 655
6
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0
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198
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What do we know about the analytic continuations of Dirichlet series?
Let $s=\sigma+it$ be a complex number and define the function:
$$F(s)=\sum_{k=2}^{\infty}\frac{p_\pi(k)}{k^s}$$
Where $p_\pi(k)$ is the number of unordered factorizations of $k$, corresponding to OEIS ...
