Questions tagged [dirichlet-series]
For questions on Dirichlet series.
563
questions
26
votes
2
answers
3k
views
Proving that $\frac{\pi}{4}$$=1-\frac{\eta(1)}{2}+\frac{\eta(2)}{4}-\frac{\eta(3)}{8}+\cdots$
After some calculations with WolframAlfa, it seems that
$$
\frac{\pi}{4}=1+\sum_{k=1}^{\infty}(-1)^{k}\frac{\eta(k)}{2^{k}}
$$
Where
$$
\eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}}
$$
is the ...
- 5,627
24
votes
3
answers
2k
views
On Dirichlet series and critical strips
(I'll keep this one short)
Given a Dirichlet series
$$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$
where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero ...
21
votes
4
answers
879
views
How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?
I'm considering the transfer-function
$$ 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 - ...
- 35k
20
votes
5
answers
771
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
18
votes
1
answer
2k
views
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
18
votes
4
answers
1k
views
$\# \{\text{primes}\ 4n+3 \le x\}$ 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.
$$
\frac{\int\limits_2^x{\frac{dt}{\ln t}...
- 18.6k
17
votes
1
answer
2k
views
Is series $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent?
I've done the following exercise:
Is series $\displaystyle\sum^{\infty}_{n=1}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent?
My approach:
We're going to use the Dirichlet's criterion for ...
- 4,225
16
votes
0
answers
625
views
Odd values for Dirichlet beta function
I would like to find a proof for the generating formula for odd values of Dirichlet beta function, namely: $$\beta(2k+1)=\frac{(-1)^kE_{2k}\pi^{2k+1}}{4^{k+1}(2k)!}$$
My try was to start with the ...
- 28.1k
15
votes
1
answer
845
views
An entire function interpolating $\mu(n)$
This is in order to repair the pdf and answers of this user.
$$f(x)=2\sum_{k\ge 0}\frac{x^{2k+1}}{\zeta(2k+2)}=2x\sum_{n\ge 1} \frac{\mu(n)/n^2}{1-x^2/n^2}, \qquad |x|<1$$ The RHS extends ...
- 78.4k
14
votes
2
answers
2k
views
The Definite Integral Problem (with a twist)?
The Definite Integral Problem (with a twist)
In the Riemann integral one essentially calculates the area by splitting the area into $N$ rectangular strips and then taking $N \to \infty$.
Here's ...
- 1,177
13
votes
3
answers
350
views
Looking for a closed form for $\sum_{k=1}^{\infty}\left( \zeta(2k)-\beta(2k)\right)$
For some time I've been playing with this kind of sums, for example I was able to find that
$$
\frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\left( \zeta(2k+1)-\beta(2k+1)\right)
$$
where
$$
\beta(x)=\sum_{k=...
- 5,627
13
votes
2
answers
444
views
Regularity of root spacing of $G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$
Define, on $\mathbb{C}$:
$$G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$$
A domain colored portrait of $G(z)$ (boxes are supposed to be negative signs):
suggests that the roots of $G(z)$ are ...
- 5,698
12
votes
1
answer
231
views
An asymptotic series for $\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right),\,n\to\infty$
Using empirical methods, I conjectured that$^{[1]}$$\!^{[2]}$
$$\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right)=1-\frac{2\pi^2}9\,4^{-n}+\frac{38 \,\pi ^4}{2025}\,4^{-2n}-\frac{...
- 47.3k
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
$$
\int_{0}^{\infty}\vartheta_2(...
- 5,370
12
votes
0
answers
473
views
How to interpret a strange formula about $\zeta'(s)/\zeta(s)$
I obtained a strange formula about $\zeta'(s)/\zeta(s)$
$$
\begin{split}
\frac{\zeta'(s)}{\zeta(s)}-(2\pi)^s&\sum_{\Im(\rho)>0} (-i\rho)^{-s}(2\pi)^{-\rho} e^{-i\pi \rho / 2} \Gamma(\rho)\;\;\...
- 78.4k