Questions tagged [dirichlet-series]
For questions on Dirichlet series.
88
questions
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Is series $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent?
I've done the following exercise:
Is series $\displaystyle\sum^{\infty}_{n=1}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent?
My approach:
We're going to use the Dirichlet's criterion for ...
- 4,225
14
votes
2
answers
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The Definite Integral Problem (with a twist)?
The Definite Integral Problem (with a twist)
In the Riemann integral one essentially calculates the area by splitting the area into $N$ rectangular strips and then taking $N \to \infty$.
Here's ...
- 1,177
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votes
4
answers
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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
$$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - ...
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1
vote
2
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Sines and cosines of angles in arithmetic progression [closed]
Prove that if $\phi$ is not equal to $2k\pi$ for any integer $k$, then
$$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$
...
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votes
2
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How are values of the Dirichlet Beta function derivative derived?
Wolfram Mathworld gives the following values for the beta function derivative.
$$\beta'(-1) = \frac{2K}{\pi},\quad \beta'(0) = \ln \left[\frac{\Gamma^{2}(\frac{1}{4})}{2\pi\sqrt{2}} \right],\quad \...
- 1,526
24
votes
3
answers
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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 ...
9
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3
answers
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Showing $\sum\frac{\sin(nx)}{n}$ converges pointwise
I do not understand how one can say using "Dirichlet conditions" that $\sum_{n=1}^{\infty}\dfrac{\sin(nx)}{n}$ is pointwise convergent. I know the proof for $x=1$ but how can one show it is ...
- 13k
18
votes
1
answer
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Derivative of Riemann zeta, is this inequality true?
Is the following inequality true?
$$\gamma -\frac{\zeta ''(-2\;n)}{2 \zeta '(-2\;n)} > \log (n)-\gamma$$
This for $n$ a positive integer, $n=1,2,3,4,5,...$, and more precisely when $n$ approaches ...
- 7,448
8
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Average order of $\mathrm{rad}(n)$
Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing n. Or equivalently, $$\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ ...
user261687
8
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1
answer
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For all Dirichlet series, is $a_n$ unique to $f(s)$?
For any Dirichlet series, $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ is the sequence, $a_n$, always unique to $f(s)$? In other words, is it possible to show that $a_n$ is the only sequence that will ...
- 3,557
4
votes
2
answers
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Landau's Theorem, Dirichlet Series
An important theorem of Landau states that, given a Dirichlet Series f(s) with coefficients $a_n$ then
If $a_n\geq0$ for all values of n, the real point of the line of convergence is a singularity ...
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3
votes
3
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how to show that this complex series converge?
If $$\sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}}$$ Converges( s is real)
and $\operatorname{Re}(z)>s$.
Then $$\sum_{n=1}^{\infty} \frac{a_{n}}{n^{z}}$$
also converges. $a_n$ is complex sequence.
- 31
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0
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Question on convergence of a formula for the Dirichlet eta function $\eta(s)=(1-2^{1-s})\,\zeta(s)$
Question: Is it true that formula (1) below for $\eta(s)$ converges for $\Re(s)>-m$ as $N\to\infty$ and if so, is formula (1) somehow related to the derivation of formula (2) below for $\eta(s)$ ...
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0
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What is the limit of this Dirichlet series?
Background & Motivation
I'm trying to verify/disprove the conjectured formula of the weighted integral of $f(x)$: The Definite Integral Problem (with a twist)?
$$ \lim_{k \to \infty} \lim_{n \...
- 1,177
18
votes
4
answers
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$\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions
In a paper about Prime Number Races, I found the following (page 14 and 19):
This formula, while
widely believed to be correct, has not yet been proved.
$$
\frac{\int\limits_2^x{\frac{dt}{\ln t}...
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