All Questions
Tagged with dirichlet-series zeta-functions
35
questions
9
votes
3
answers
2k
views
How does Wolfram Alpha know this closed form?
I was messing around in Wolfram Alpha when I stumbled on this closed form expression for the Hurwitz Zeta function:
$$
\zeta(3, 11/4) = 1/2 (56 \zeta(3) - 47360/9261 - 2 \pi^3).
$$
How does WA know ...
- 5,295
0
votes
0
answers
62
views
"Mollifier" of the Dirichlet L-function
I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
0
votes
0
answers
20
views
$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 0$?
Let $p_n$ be the $n$ th prime number.
Let $f(s)$ be a Dirichlet series defined on the complex plane as :
$$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 1 + \frac{2^{-s}}{2}+ \frac{3^{-s}}{3} + \...
- 16.4k
-1
votes
1
answer
279
views
Explore the relationship between $\sum\limits_{n = 1}^{2x} \frac{1}{{n^s}^x}$ and $\sum\limits_{n = 1}^{2x-1} \frac{(-1)^{n-1}}{{n^s}^x}$ [closed]
I am trying to find an algorithm with time complexity $O(1)$ for a boring problem code-named P-2000 problem. The answer to this boring question is a boring large number of $601$ digits. The DP ...
user1236730
0
votes
1
answer
99
views
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$?
if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann ...
- 43
2
votes
1
answer
73
views
How to prove the following Dirichlet-series/geometric-series idenity, step by step process?
$$\frac{\zeta(s)}{\zeta(hs)} =\prod_p\left(\frac{1-\frac{1}{p^{hs}}}{1-\frac{1}{p^{s}}}\right) =\prod_p\left(1+\frac{1}{p^s}+\cdots +\frac{1}{p^{(h-1)s}}\right)=\sum_{n\in S_h}\frac{1}{n^s}$$
What is ...
- 103
1
vote
2
answers
343
views
Does every Dirichlet series admit an analytic continuation? If so, to what extent?
The identity theorem for analytic continuation shows that uniqueness of analytic continuations of functions is very easy to characterize. This helps us a lot when we are trying to extend functions ...
- 2,461
0
votes
1
answer
80
views
proving that dirichlet series has non negative coefficients and does not converge for all $s\in\mathbb{C}$
given $Z(s)=\zeta^2(s)\zeta(s+it)\zeta(s-it)$ I need to prove that Z(s) is represented by a dirichlet series with non negative coefficients whiche does not converge for all $s\in\mathbb{C}$.
I have ...
- 13
0
votes
0
answers
144
views
What is the abscissa of convergence of the series $\sum\limits_{n=1}^{\infty} (-1)^n \frac {1} {n^s}\ $?
What is the abscissa of convergence of the series $\sum\limits_{n=1}^{\infty} (-1)^n \frac {1} {n^s}\ $?
In the lecture note our instructor claimed that the abscissa of convergence of the above ...
- 211
0
votes
0
answers
74
views
Is this argument sufficient to show alternating zeta series converges for $Re(s)>0$?
The alternating zeta series
$$ \eta(s) = \sum\frac{(-1)^{n+1}}{n^s} $$
is known to converge for $\sigma=\Re(s)>0$.
Question: There are many proofs offered but I wondered if the following simple (...
- 3,325
1
vote
0
answers
91
views
Is there a name for this generalization of the Riemann Zeta Function?
The usual Riemann Zeta function is
$$
\zeta(s) = \sum_n \frac{1}{n^s}
$$
Suppose we modify the denominator instead to
$$
\zeta_?(s) = \sum_n \frac{1}{(n\sqrt{1+Bn^2})^s}
$$
We can do some elementary ...
- 8,000
0
votes
0
answers
31
views
Regularity of Epstein zeta Z|h,0|(A; s) in h
My question considers the regularity of the Epstein zeta function in one of it modules. Let $h\in \mathbb R^d$ and $A\in \mathbb R^{d\times d}$ positive definite (we can also assume $A=I$ for ...
- 1
0
votes
0
answers
113
views
The formula some simple zeros of the Dirichlet eta function
Let $s$ be a complex variable with $\Re(s)>0$. The Dirichlet eta function $\eta(s)$ is defined by $$\eta(s)=(1-2^{1-s})\zeta(s)$$ where $\zeta(s)$, of course, is the Riemann zeta function. We know ...
- 372
6
votes
1
answer
150
views
Abelian group zeta function
Let $s \in \mathbb{C}$. What's known about
$$\zeta_{\mathrm{ab}}(s) := \sum_G \frac{1}{o(G)^s} \tag{1}$$
where the sum is over all finite abelian groups $G$ up to isomorphism?
By the primary ...
- 7,631
2
votes
1
answer
1k
views
Proving the relation between the Dirichlet eta function and the Riemann zeta function [closed]
The problem I am trying to solve is: I need to prove the relation between the Dirichlet eta function and the Riemann zeta function $\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$.
But I have no clue ...
- 31