All Questions
Tagged with dirichlet-series real-analysis
25
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
$$
\int_{0}^{\infty}\vartheta_2(...
- 5,370
1
vote
1
answer
81
views
Convergence of sums in $\ell^p \implies \ell^{p-\epsilon}$
Supose $\displaystyle(b_n)_{n \in \mathbb{N}}$
is a sequence of positive real numbers that
$$\displaystyle\sum_{n \in \mathbb{N}}(b_n)^{2} <\infty.$$
Does exists some $\epsilon>0$ such that $\...
- 33
1
vote
1
answer
184
views
Finding the sum of a series using a Fourier series
I am stuck on how to calculate the value of the following sum:
$\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$
I am aware that you need to find the corresponding function whose Fourier series is represented ...
- 351
2
votes
1
answer
105
views
How to find the sum of this infinite series
I am not sure how to evaluate the infinite sum:
$$\sum_{n=0}^\infty \frac{1}{(2n+1)^6}$$
Apparently, I can shift it to
$$\sum_{n=1}^\infty \frac{1}{(2n-1)^6}$$
which is supposed to be a well known sum ...
- 21
2
votes
1
answer
462
views
On the abscissa of convergence of a Dirichlet series.
I am trying to find the abscissa of convergence of the Dirichlet series for the arithmetic function $|\mu(n)|$.
I have managed to show that $$\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^s}=\frac{\zeta(s)}{\...
user1021241
1
vote
1
answer
118
views
Uniform convergence about Dirichlet integral $f(s):=\int_1^\infty\frac{a(x)}{x^s}\,dx =\lim\limits_{T\to\infty}\int_1^T\frac{a(x)}{x^s}\,dx$
On page 87 of Ingham's book: The Distribution Of Prime Numbers, the author asserts the following results, but does not give proof.
Let $a(x)$ be a bounded and integrable function over any finite ...
- 915
2
votes
4
answers
157
views
Errors are decreasing in series $\sum_{n=1}^\infty(-1)^n/n^4$?
Let $v=\sum_{n=1}^\infty(-1)^n/n^4$ ($v$ for "value"), let $S=(\sum_{n=1}^m(-1)^n/n^4)_{m\in\mathbb Z_{\ge1}}$ be the partial sums, and let $e=(|S_n-v|)_{n\in\mathbb Z_{\ge1}}$ be the errors....
- 867
2
votes
1
answer
139
views
Establish a lower bound for the generalized alternating harmonic series
I'm given the following series:$$\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^p}$$
I need to show that the sum is greater than $1/2$ for every $p > 0$. For $p \ge1$ this is obvious, as it follows by ...
- 1,684
3
votes
2
answers
491
views
uniform convergence of Dirichlet eta function
For any $x\in\mathbb{R}$, define a series $$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{x}},$$ which is called the Dirichlet eta function from the complex analysis.
It converges pointwise on $(0,\infty)$ ...
- 1,354
8
votes
2
answers
401
views
On the sets of sums $\sum\limits_{n=1}^\infty\frac{a_n}{n^s}$ with $(a_n)$ periodic and integer valued, for different values of $s$ natural number
For every positive integer $s$, let $A_s$ denote the set of the sums of the converging series $\sum\limits_{n=1}^\infty\frac{a_n}{n^s}$ for every periodic sequence of integers $(a_n)$.
Then each $A_s$...
- 3,813
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
2
votes
0
answers
58
views
Convergence of product of series to zeta function
Consider the product of the following two partial series:
$$\left[\sum_{n=1}^{N} \frac{1}{n^s}\right] \cdot \left[\sum_{d=1}^{N} \frac{1}{d^{s^*}}\right]$$
When $\mathbf{Re}(s) > 1$, then in the ...
- 8,000
0
votes
0
answers
171
views
Usage of the Dirichlet kernel in a trigonometric series proof
So in attempting to prove the pointwise convergent result $$\lim_{N\to\infty} S_N(f)(x) = f(x)$$ provided $$\exists \space \delta>0 \space \text{s.t.} \space |f(x+t) - f(x)| \leq C|t| \text{ for } |...
- 360
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
5
votes
0
answers
207
views
Is $\sum\limits_{n = 1}^\infty (-1)^n\frac{H_n}{n}=\frac{6\ln^2(2)-\pi^2}{12}$ a valid identity? [duplicate]
A while ago I asked a question about an identity that I found while playing with series involving harmonic numbers. However, since the methods I used were not the focus of my question, I never ...
- 113