5
votes
2answers
748 views

Is there a known formula for fractional derivative of cot x?

I wonder if there any established formula for fractional derivative of a function $\pi \cot (\pi x)$. I derived the following expression: $(\pi \cot (\pi ...
5
votes
1answer
408 views

Is Zeta function discrete-analytic?

Let's define discrete-analytic functions as functions that are equal to their Newton series expansion: $$f(x) = \sum_{k=0}^\infty \binom{x-a}k \Delta^k f(a)$$ My question is whether $\zeta(s,q)$ ...
1
vote
0answers
707 views

Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta

The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation. In the discrete calculus there is ...
7
votes
4answers
1k views

Non-vanishing of zeta(s), Re(s)=1, without complex analysis?

Say you are allowed to use Fourier analysis, complex variables, Euler-Maclaurin, etc., but no complex analysis - no holomorphic continuations, no definition of analytic function, and, in particular, ...
20
votes
3answers
880 views

Universality of zeta- and L-functions

VoroninĀ“s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...