Questions tagged [dirichlet-series]
For questions on Dirichlet series.
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How to find the $\zeta$ representation of a $L$-series
Consider the following problem:
Show that for $s>1$:
$$\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}.$$
($\mu$ denotes the Mobius function)
My approach:
One may first note that the ...
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How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?
I'm considering the transfer-function
$$ t(x) = \log(1 + \exp(x)) $$
and find the beginning of the power series (simply using Pari/GP) as
$$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - ...
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Zeta Lerch function. Proof of functional equation.
so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following.
In the article "Note sur la function" by Mr. Mathias Lerch, a ...
2
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1
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Dirichlet series and Laplace transform
Let $\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}$ be a Dirichlet series. It can be represented as a Riemann-Stieltjes integral as follows:
$$\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}=\int_1^\...
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How does Wolfram Alpha know this closed form?
I was messing around in Wolfram Alpha when I stumbled on this closed form expression for the Hurwitz Zeta function:
$$
\zeta(3, 11/4) = 1/2 (56 \zeta(3) - 47360/9261 - 2 \pi^3).
$$
How does WA know ...
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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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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:
$$ \sum_{n=1}^{\infty} \frac{\sigma_0(n^2)}{n^s} = \zeta^2(s) \sum_{n=1}^{\infty} \frac{ \mu^2(n)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)} $$
$$ \sum_{n=1}^{...
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Building the theoretical foundation for generating functions - formal power series
I have read several documents on generating functions. I would like to inquire about two issues:
Among the materials I have read, some mention generating functions constructed from formal power ...
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Abscissa of convergence for a Dirichlet series
Let $\alpha \in \mathbb{Z}$ and $f(n) = n^{i \alpha n}$. What is the abscissa of convergence, $\sigma_c$, for the associated Dirichlet series, $\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$? Since $|f(n)| = 1$...
CommunityBot
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Perron's formula in the region of conditional convergence
I am a bit confused about the proof of Perron's formula. It states that for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ and real numbers $c > 0$, $c > \sigma_c$, $x > 0$ we have
$$\...
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3
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Power series for $\sum_{n=0}^\infty(-1)^n/n!^s$ (around $s=0$)
I'm looking for ways to compute the coefficients of the power series
$$
\sum_{n=0}^\infty\frac{(-1)^n}{n!^s}=\sum_{k=0}^{\infty}c_k s^k
$$
(a prior version of the question asked whether such an ...
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Residue of a Dirichlet Series at $s=1$
I have encountered this problem of determining the leading term in the Laurent expansion of a Dirichlet series. Let $d(n)$ be integers and consider the Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{...
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Dirichlet series with infinitely many zeros
Can a Dirichlet series have infinitely many zeros and be nonzero?
To be precise, by a Dirichlet series I mean a function of the form $s\mapsto \sum_{n\geq 1}\frac{a_n}{n^s}$ where the domain is the ...
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Proving the relation between the Dirichlet eta function and the Riemann zeta function [closed]
The problem I am trying to solve is: I need to prove the relation between the Dirichlet eta function and the Riemann zeta function $\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$.
But I have no clue ...
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Integrals of Jacobi $\vartheta$ functions on the interval $[1,+\infty)$
I start from the following obvious observation, which is declared to be($q=e^{-\pi x}$):
\begin{aligned}
\int_{1}^{\infty}x\vartheta_2(q)^4\vartheta_4(q)^4
\text{d}x&=\int_{0}^{1}x\vartheta_2(q)^4\...
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