All Questions
Tagged with dirichlet-series integration
13
questions
12
votes
1
answer
1k
views
Prove $\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }$
The paper mentioned a proposition:
$$
\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38}
K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }.
$$
Its equivalent is
$$
\int_{0}^{\infty}\vartheta_2(...
- 5,370
6
votes
1
answer
154
views
Integrals of Jacobi $\vartheta$ functions on the interval $[1,+\infty)$
I start from the following obvious observation, which is declared to be($q=e^{-\pi x}$):
\begin{aligned}
\int_{1}^{\infty}x\vartheta_2(q)^4\vartheta_4(q)^4
\text{d}x&=\int_{0}^{1}x\vartheta_2(q)^4\...
- 5,370
2
votes
1
answer
62
views
Step in Apostol's IANT on Dirichlet Series
I can't explain the following step in Apostol's IANT regarding Dirichlet Series.
Specifically, how does the magnitude of the following
$$\left | \int_a^b t^{s_0 -s -1} \right |$$
become the following?...
- 3,325
1
vote
3
answers
73
views
$\int_{0}^{\pi} D_{n}(y)dy=\frac{1}{2}$ Dirichlet
I need to calculate that $\int_{0}^{\pi} D_{n}(y)dy=\frac{1}{2}$ with $D_{n}(y)= \frac{1}{2\pi}\frac{\sin((n+\frac{1}{2})y)}{\sin(\frac{y}{2})}$ from Dirichlet.
Now I tried to do this with the known ...
- 875
4
votes
1
answer
239
views
Show that $\alpha \int_0^{\pi}X^2dx = \int_0^{\pi}(X')^2dx$ holds for $\alpha > 0$
Consider the equation in the form $X + \alpha X = 0$, with Dirichlet
boundary conditions at $x = 0$ and $x = π$.
a) Multiply the equation by X and integrate from 0 to π, then
integrate the first term, ...
- 169
2
votes
1
answer
1k
views
How can I show whether or not the Dirichlet function is integrable using the definition of upper and lower integrals?
If $f$ is a version of the Dirichlet function:
$f(x) = 1$ when $x$ is an irrational in $[0,1]$ and $f(x) = 0$ otherwise
How can I determine whether or not $f$ is integrable using the definition of ...
- 21
0
votes
1
answer
86
views
Taking the derivative inside the integral sign of Jensen or Lindelöf's integral representations related to the alternating Zeta function
Wikipedia's article for the Dirichlet eta function tell us from the section Integral representations what is the representation due to Lindelöf, and what is the representation for $(s-1)\zeta(s)$ due ...
user243301
1
vote
1
answer
110
views
Calculate $\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T\frac{\zeta(\frac{3}{2}+it)}{\zeta(\frac{3}{2}-it)}dt$ as $\sum_{n=1}^\infty\frac{\mu(n)}{n^3}$
Using a well known theorem for Dirichlet series can be justified that (for $T>0$) $$\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T\frac{\zeta(\frac{3}{2}+it)}{\zeta(\frac{3}{2}-it)}dt=\sum_{n=1}^\infty\...
user243301
5
votes
1
answer
137
views
L-series through integrals of rational functions
Recently I stumbled upon this short proof here:
$$L(1,\chi_2)=\sum_{j=0}^{+\infty}\left(\frac{1}{3j+1}-\frac{1}{3j+2}\right)=\int_{0}^{1}\frac{1-x}{1-x^3}\,dx=\int_{0}^{1}\frac{dx}{1+x+x^2}$$
so:
$$\...
- 1,380
0
votes
2
answers
190
views
Are these two naive upper bounds ok?
EDIT:
Is my work ok? Here, I am trying to show a uniform bound for the sum of $cos(n)$
$$|\sum_{n=1}^{N} cos(n)|$$
$$=\big |\sum \frac{e^{in} + e^{-in}}{2}\big|$$
$$\le \sum |\frac{e^{in} + e^{-in}...
- 1
0
votes
1
answer
3k
views
Proof that $\int f(x)\sin(Nx)\ dx \to 0$ as $N \to \infty$
I'm studying Fourier series out of Rudin's "Principals of Mathematical Analysis". In the proof that the Fourier series $s_N(f;x)$ converges pointwise to $f$, it assumes that at a point $x$, there is ...
- 2,110
2
votes
1
answer
272
views
Integral of a Dirichlet Series
I'm stuck at a problem of an exercise list... I'd like some help to solve it :)
The problem: Suppose that the Dirichlet Series
$$A(s)=\lim_{N \to \infty}\sum_{n=1}^Na(n)n^{-s}$$
has abscissa of ...
- 1,122
0
votes
2
answers
52
views
Define this Function on [0,1]
I need a function, $f$ that is bounded but not integrable on [0,1], but $f^2$. I am thinking I should modify the Dirichlet function and use it to create a function whose square is constant. I am stuck ...
- 1,995