All Questions
Tagged with dirichlet-series calculus
12
questions
6
votes
1
answer
154
views
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\...
- 5,370
20
votes
5
answers
770
views
Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$
This is a math competition problem for college students in Sichuan province, China. As the title, calculate the limit
$$\lim_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}.$$
It is clear that the ...
- 915
2
votes
3
answers
508
views
Dirichlet series for square root of Riemann Zeta function
Can we obtain Dirichlet series for the function $\sqrt{\zeta(s)}$? Is it possible via Euler product for $\zeta(s)$?
1
vote
1
answer
152
views
How to show the abscissa of convergence for a Dirichlet series given by a mod 3 function
The first thing I need to do is to write the function
$$f(n) = \begin{cases} 1, & \mbox{if } n \ne 0 \mbox{ mod 3} \\ -2, & \mbox{if } n\equiv 0 \mbox{ mod 3} \end{cases}$$ in the form of a ...
- 87
0
votes
1
answer
99
views
Prove that this type of alternating series admits this supremum.
Let $a_k\geq 0$ be a decreasing to zero numerical sequence. How how can we prove this inequality ?
$$ \left|\sum_{k=n+1}^{\infty} (-1)^k a_k\right| \leq |a_n|$$
It may have something to do with ...
- 7
1
vote
1
answer
602
views
Dirichlet function and continuity
i can't solve it, hoping you can help.
let $f(x)=2x-1+(x-a)^2D(x)$ -- D(x) is dirichlet function.
a)prove that it exists for every x.
b)let $x_0\neq 1$, and let $ f(x) = \begin{cases}
...
- 1,114
2
votes
1
answer
362
views
How to prove this Dirichlet series identity?
On this website, within a proof of the prime number theorem, they boldly make the following claim:$$-\frac{\zeta'(x)}{\zeta(x)}=\sum_n \frac{\Lambda(n)}{n^x}$$
Where $\Lambda(x)$ is the von Mangoldt ...
- 4,954
17
votes
1
answer
2k
views
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
4
votes
1
answer
516
views
Does the "alternating" harmonic series where only prime terms are negative converge?
We know that the harmonic series $\sum \frac{1}{n}$ diverges, yet the alternating harmonic series $\sum \frac{(-1)^n}{n}$ converges.
Euler famously gave a proof of the infinitude (and of the "density"...
- 4,204
2
votes
1
answer
224
views
Sequence Involving Dirichlet Function
The question I have to prove is the following:
Let $D(x)$ be Dirichlet Function:
$$D(x) = \begin{cases}1 & x\in \Bbb Q \\ 0 & x \notin \Bbb Q \end{cases}$$
Let $(a_n)_{n=1}^{n\to\infty}$ be ...
- 2,791
2
votes
2
answers
236
views
Does this Dirichlet series converge to zero?
Consider the periodic Dirichlet series that has this iterative definition:
$$\text{a1}=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-\frac{2}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}+\frac{2}{\sqrt{6}}...
- 7,448
2
votes
2
answers
258
views
Summing Lerch Transcendents
The Lerch transcendent
is given by
$$
\Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}.
$$
While computing $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}
\sum_{p=1}^{\infty}\frac{(-1)...
- 18.6k