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 Yearling
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May
18
comment Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta?
Anixx, it actually is defined for all indices in $\mathbb{C}$ of the polygamma. See link here: maplesoft.com/support/help/Maple/view.aspx?path=Psi I tested it in a Maple and it works fine. Was surprised to learn a function exists that links $\zeta(s)$ to its derivative. Through the functional equation you can then also link $\zeta'(s)$ and $\zeta'(1-s)$. However $\Psi^{s}(1)$ is defined in terms of $\zeta(s)$, so we won't gain much.
May
14
revised Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta?
Added formula linking zeta'(s) and zeta(s) (but there is a snag).
May
11
answered Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta?
May
2
asked Does this 'alternating' series with $\Lambda(n)$ converge for all $\Re(s)>0$?
Apr
16
asked Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?
Apr
9
answered How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?
Apr
9
revised How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?
Turned update into an answer.
Apr
9
revised How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?
Added solution to my problem.
Apr
8
asked How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?
Jan
2
comment How are Zeta function values calculated from within the Critical Strip?
Or through this extension: $$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(\sum _{n=1}^{\infty } \left( {\frac {n}{(n+1)^{s}}} + \frac{2\,s-1}{n^s} - {\frac {n-1}{\left( n-1 \right) ^{s}}}\right) \right), \qquad 0<\Re(s)<1$$ we get an even simpler (I believe the simplest) series-expression for: $$\displaystyle \zeta\left(\frac12\right) = \sum _{n=1}^{\infty } \left(\sqrt{n-1} -{\frac {n}{\sqrt{n+1}}}\right)$$
Dec
29
revised Analytic continuation of the Dirichlet $\eta(s)$ series to $\Re(s) \gt -1$. Why does this work?
Simplified the question (restricted to eta only) and included an addition.
Dec
29
comment Analytic continuation of the Dirichlet $\eta(s)$ series to $\Re(s) \gt -1$. Why does this work?
@Timbuc. That is indeed my question. Why does taking the simple average between two series induce the analytic continuation? The only two ways I found to continue the domain of $\eta(s)$ are on the Wiki page en.wikipedia.org/wiki/Riemann_zeta_function under the sections "Rising Factorial" and "Globally convergent series", but these are quite different continuations.
Dec
29
asked Analytic continuation of the Dirichlet $\eta(s)$ series to $\Re(s) \gt -1$. Why does this work?
Dec
28
revised Are these known telescoping series for $\zeta\left(\frac12\right)$?
Added explanation how I derived the second series from the first.
Dec
27
revised An infinite series that gives $f(s)=s$. How could it be explained more easily?
Added a possible explanation.
Dec
26
asked An infinite series that gives $f(s)=s$. How could it be explained more easily?
Dec
23
revised Are these known telescoping series for $\zeta\left(\frac12\right)$?
Added a simplified (more elegant) formula in the last section.
Dec
23
awarded  Yearling
Dec
23
asked Are these known telescoping series for $\zeta\left(\frac12\right)$?
Sep
2
revised Question about the zeros of $\zeta_{H}(s,a) \pm \zeta_{H}(s,1-a)$.
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