5
votes
1answer
236 views
Closed form for derivatives $\zeta^{(n)}(1/2)$
According to mathworld
41,42. "Derivatives $\zeta^{(n)}(1/2)$ can also be given in closed form"
with example for the first derivative.
What is the closed form? References?
The …
15
votes
2answers
553 views
zeta(3) in terms of derivatives of zeta at 1/2 and pi
Got numerical support that for odd $n$, $\zeta(n)$ might be
expressed in terms of the derivatives of $\zeta(\frac12)$.
Based on More Zeta Functions for the Riemann Zeros, Andre Vo …
3
votes
1answer
225 views
zeta(2k+1) is a rational multiple of pi^{2k} zeta'(-2 k) ?
Probably this is well know and elementary and will delete it, but couldn't find it on the web.
Got a sketch of proof and numerical evidence that
$\zeta(2k+1)$ is a rational multi …
3
votes
0answers
133 views
What is the (fractional) half-derivative of zeta at $s=0$ (and how to compute it)?
(I asked this in MSE before but there was only a general reference which did not help for my specific question)
I think I understood the concept of fractional derivatives …
10
votes
3answers
567 views
Good uses of Siegel zeros?
The short version of my question goes: What is known to follow from the existence of Siegel zeros?
A longer version to give an idea of what I have in mind: The "expectional zeros" …
15
votes
3answers
1k views
$\prod_{n=1}^{\infty} n{}^{\mu(n)}=\frac{1}{4 \pi ^2}$
When I tested this in Mathematica, I had expected it to say it did not converge. However, I got this:
$$\prod_{n=1}^{\infty} n{}^{\mu(n)}=\frac{1}{4 \pi ^2}$$
This indicates t …
1
vote
2answers
264 views
Does there exist a closed form for the factors of this infinite product ?
Assume $s,a \in \mathbb{C}, a \pm in \ne 0$.
The following infinite product nicely converges and can be expressed in a closed form:
$$\displaystyle \prod_{n=1}^\infty \left(1- \f …
3
votes
0answers
144 views
Alternating sums of the non-trivial zeros of $\zeta(s)$.
It is known that the infinite sum of the non-trivial zeros $\rho_n =\beta + \gamma_ni$ of $\zeta(z)$, when taken in pairs that are either conjugated or reflexive (they give the sam …
4
votes
1answer
361 views
Is there an explicit expression for the imaginary part of some non-trivial zero of zeta,
in terms of well-known constants, such as say $\gamma$ or $\pi$ say ?
0
votes
1answer
156 views
Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?
[This question is copied from math.stackexchange, it didn't get answers so far]
For some exercises with (divergent) summation of the Stieltjes constants,see also MSE I'm tryi …
15
votes
2answers
1k views
On the Universality of the Riemann zeta-function
Hi,
I have a question regarding the universality property of the Riemann zeta-function. I am no expert on this, so I'd be glad for any relevant reference.
First, recall Voronin's …
4
votes
1answer
246 views
Is there a connection between the closed forms of these two infinite products?
Take the following two infinite products that have closed forms.
Assume: $\gamma_n > 0 \in \mathbb{R};s,a,x \in \mathbb{C}; x \ne 0,a \pm ix\gamma_n \ne 0$
The first product:
$ …
3
votes
2answers
452 views
Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$
Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\pri …
4
votes
3answers
410 views
A closed form of infinite products of complex zeros involving $\Im(\rho_n)$. Does a proof of this closed form imply RH?
Building on this question scaling the imaginary part of $\rho$s in infinite products, I like to conjecture that:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} \right …
25
votes
3answers
2k views
Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?
What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples: …
