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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 $\zeta_{H}(s,a) \pm \zeta_{H}(s,1-a)$.
[Edit removed during grace period]
Sep
2
revised Question about the zeros of $\zeta_{H}(s,a) \pm \zeta_{H}(s,1-a)$.
Added meromorphic to be more precise about the type functions I am after.
Sep
2
asked Question about the zeros of $\zeta_{H}(s,a) \pm \zeta_{H}(s,1-a)$.
Aug
20
comment Do all complex zeros of $Li_s(z)\,- \, Li_{1-s}(z)$ get the shape $s=\dfrac12 + \dfrac{k \, \pi }{\,\ln(2)}\,i$ when $z \rightarrow 0^{-}$?
Thanks Antonio! That's it. Subtracting the power series at $z \rightarrow 0$ just reduces to $z^2(2^{āˆ’s}āˆ’2^{sāˆ’1})$. The result then immediately follows (and it also follows that these are the only roots and also none exist outside the strip). Much simpler than I thought...
Aug
19
asked Do all complex zeros of $Li_s(z)\,- \, Li_{1-s}(z)$ get the shape $s=\dfrac12 + \dfrac{k \, \pi }{\,\ln(2)}\,i$ when $z \rightarrow 0^{-}$?
Jul
2
awarded  Curious
Jun
9
comment Question about the convergence of an infinite series in all of $\mathbb{C}$.
Greg, $G(i)$ (and all $\Re(s)=0$) indeed also diverges when evaluating the series, however Maple does assign the value $-13.23372427475...$ (guess some form of analytic continuation is applied). Not sure how to calculate this number, but is seems to be the average value of an infinite amount of finite odd and finite even series.
Jun
9
comment Question about the convergence of an infinite series in all of $\mathbb{C}$.
Thanks Greg. I tested convergence with increasingly higher finite series and it also seems to work for the imaginary axis for $\Re(s) \ne 0$, however indeed difficult to prove.
Jun
9
accepted Question about the convergence of an infinite series in all of $\mathbb{C}$.
Jun
8
comment Question about the convergence of an infinite series in all of $\mathbb{C}$.
Ethan. Got it and corrected it in the question.