All Questions
Tagged with dirichlet-series special-functions
12
questions
5
votes
0
answers
101
views
Zeta Lerch function. Proof of functional equation.
so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following.
In the article "Note sur la function" by Mr. Mathias Lerch, a ...
0
votes
0
answers
86
views
Understanding the functional equation $\left( \frac{2}{\pi} \right)^s \sin \left( \frac{\pi s}{2} \right) \Gamma(s)L(s)= L(1-s).$
In the functional equation $L(s)$ is the Dirichlet Beta function which is defined as $L(s)= \sum_{n=0}^{\infty}\frac{\chi(n)}{n^s}$ where $\chi$ is a Dirichlet character of period 4.
Now I know that ...
- 728
1
vote
1
answer
120
views
Squared modulus of Dirichlet eta function.
I have a problem:
Find formula for $|\eta(x+iy)|^{2}$, where $\eta(x+iy)=\sum_{n\ge1}\frac{(-1)^{n-1}}{n^{x+iy}}$
I calculated it in two ways and i got a contradiction.
First method: $|\eta(x+iy)|...
- 1,382
8
votes
2
answers
1k
views
How are values of the Dirichlet Beta function derivative derived?
Wolfram Mathworld gives the following values for the beta function derivative.
$$\beta'(-1) = \frac{2K}{\pi},\quad \beta'(0) = \ln \left[\frac{\Gamma^{2}(\frac{1}{4})}{2\pi\sqrt{2}} \right],\quad \...
- 1,526
0
votes
1
answer
96
views
Is $\sum_{n=0}^\infty (-1)^n (2n+1)^{-s}$ expressible in terms of the zeta function?
I am interested if the alternating Dirichlet Lambda function $$\sum_{n=0}^\infty (-1)^n (2n+1)^{-s}$$
can be expressed in terms of the zeta function or another Dirichlet L-Function.
I know that for $...
- 4,027
2
votes
1
answer
362
views
How to prove this Dirichlet series identity?
On this website, within a proof of the prime number theorem, they boldly make the following claim:$$-\frac{\zeta'(x)}{\zeta(x)}=\sum_n \frac{\Lambda(n)}{n^x}$$
Where $\Lambda(x)$ is the von Mangoldt ...
- 4,954
6
votes
2
answers
419
views
Dirichlet $L$ functions at $s=2$
Let $\chi$ be a Dirichlet character and let $L(\chi,s)$ denotes its Dirichlet $L$-function.
What is the value of $L(2,\chi)$ ? Or simply, is $L(2,\chi)/\pi^2$ rational ?
Many thanks for your answer !...
- 1,538
2
votes
1
answer
242
views
Writing Dirichlet series in infinite product.
In Serre's $A \, Course\, In \,Arithmetic$, it says the following:
$\sum\limits_{n=1}^{\infty}c(n)/n^s= \prod\limits_{p \,\rm prime}\frac{1}{1-c(p)p^{-s}+p^{2k-1-2s}}$
$\Longleftrightarrow$
...
- 2,473
4
votes
1
answer
215
views
$\zeta_m(s)=\prod\limits_{p\nmid m} \frac{1}{\left(1-\frac{1}{p^{f(p)s}}\right)^{g(p)}}$ is a Dirichlet series with non-negative coefficients
Let $p$ be a prime number, $m$ be any integer, $f(p)$ be the order of $p$ in $(Z/mZ)^*$, $i.e.$ $p^{f(p)} \equiv 1 \pmod m$ with $f(p)$ smallest.
Let $g(p)=\frac{\phi(m)}{f(p)}$ is a integer where $\...
- 2,473
2
votes
1
answer
374
views
Convergence of the Fourier Transform of the Prime $\zeta$ Functions
I think I found a way to write the truncated Prime $\zeta$ function like this:
$$
P_x(s)=\sum_{p<x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n}
\sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}}
\...
- 18.6k
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
4
votes
3
answers
907
views
Approximate Riemann zeta function
Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$.
In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function.
My question is: Is there a Functional equation for ...
- 8,536