Questions tagged [dirichlet-series]
For questions on Dirichlet series.
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Laplacian Dirichlet eigenvalues on a given domain
Let $\Sigma=[-1,1]\times[0,1]\cup[0,1]\times[-1,1]$ be an L-shape domain, over which I'm solving the Laplacian equation with Dirichlet boundary condition $$-\Delta f=\lambda f$$
I try applying the way ...
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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 ($(a,b)$ ...
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Dirichlet's Series - Convergence
Calculate the expression of the following Dirichlet's series:
$$ \dfrac{\zeta(s-1)}{\zeta(s)} = \sum_{n=1}^{\infty} \dfrac{\varphi(n)}{n^s} $$
$$ \dfrac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty} \dfrac{...
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How to find the sum of this infinite series
I am not sure how to evaluate the infinite sum:
$$\sum_{n=0}^\infty \frac{1}{(2n+1)^6}$$
Apparently, I can shift it to
$$\sum_{n=1}^\infty \frac{1}{(2n-1)^6}$$
which is supposed to be a well known sum ...
- 21
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Question on convergence of product and Dirichlet series representations of a function
Consider the following two representations of $f(s)$
$$f(s)=
\underset{K\to\infty}{\text{lim}}\left(\prod\limits_{k=1}^K \left(1-\frac{2}{\left.p_k\right.^s}\right)\right)\tag{1}$$
$$f(s)=\underset{N\...
- 7,631
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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 $\{...
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$\sum_{n=1}^{+\infty}\frac{\Lambda\left(n\right)\varphi\left(n\right)}{n^{s}}$ and Riemann Zeta function
Is it possible to write $\sum_{n=1}^{+\infty}\frac{\Lambda\left(n\right)\varphi\left(n\right)}{n^{s}}$, where $\Lambda(n)$ is the Von Mangoldt function and $\varphi(n)$ is the Euler totient function, ...
user168330
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Abscissa of convergence of a Dirichlet series with bounded coefficients and analytic continuation [closed]
If a Dirichlet series has coefficients +1 and -1 and an analytic continuation without poles (or zeros) to the right of Re(s) = 1/2, what can we say about it's abscissa of convergence?
Is it always at ...
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Perron's formula application - zeros of $L(s,\chi )$
After applying Perron's formula I have a complex integral involving something like
\[ \sum _{n=1}^\infty \frac {\mu (n)}{n^s}e\left (\frac {an}{q}\right ).\]
As usual ("usual" meaning e.g. ...
- 1,662
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Approximation of Dirichlet Series over Bernoulli Polynomials
In this post, the questioner asked about the behavior of a Dirichlet series over Bernoulli polynomials:
$$
\mathcal{B}(k,s) = \sum_{m\geq 1} \frac{B_k(m)}{m^s}.
$$
They show that this sum is equal to
$...
- 93
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Intuition behind theta function L-series correspondence
I know that we can analytically continue $L$-series by taking the Mellin transform of a sutiable theta function. Technically we need a transformation law for the theta function at $0$ and $\infty$ as ...
user427380
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Need help with finding Dirichlet generating function
I am unable to find the following generating function:
$$\tag{1}{\sum _{n=1}^{\infty } \frac{2^{-k_n} \mu \big(\frac{n}{2^{k_n}}\big)}{n^s}}$$
$\mu$ is Möbius function, $k_n$ is the highest integer ...
- 1,206
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Term-wise product of arithmetic functions and its Dirichlet generating function
If we know Dirichlet generating function F(s) of $f(n)$ and G(s) of g(n) we can express generating function of Dirichlet convolution of $f(n)$ and $g(n)$ as product of the two generating functions $F(...
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Question on analytic continuation of functions related to $\log\zeta(s)$ and $\frac{\zeta'}{\zeta}(s)$ that converge for $|\Re(s)|>1$
Consider the following two Dirichlet series
$$C_3(s)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N 1_{n=p^k}\, \log(n)\, n^{-s}\right),\quad\Re(s)>1\tag{1}$$
$$K_\Omega(s)=\underset{N\...
- 7,631
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Analytic continuation of Dirichlet series with sine in numerator
The following Dirichlet series converges for $\Re(s)\in (1,\infty)$ (constant $k\in\mathbb{R}$), which is evident since sine function in numerator is in interval $[-1,1]$.
$$f_1(s)=\sum _{n=1}^{\infty ...
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