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Let $\zeta(s)$ denote the non-continued, complex-valued zeta function defined on the complex plane where $\Re[s]>1$. Let $F(s)$ denote the continued zeta function defined on the entire complex plane (except possibly at $s=1$.) Is there not a way to write $F(s)$ explicitly in terms of $\zeta(s)$? My first inclination is to write something like...

$F(s) = \left\{\begin{array}{ll}\zeta(s) & \Re[s]>1 \\ 1 - \zeta(1 - s) & \Re[s] < 0 \\ ? & 0 \leq \Re[s]\leq 1 \end{array}\right.$

...but this may not be an accurate observation of the symmetry, if any, of the continued zeta function across the critical line. Furthermore, I do not know what to put for the question mark in the equation above, assuming anything could go there at all.

Is there anything reflective of the truth in what I've tried to write above? Is there an explicit formulation of the continued zeta function? How is the implicit formulation used in practice; say, to graph the continued zeta function?

Spencer Parkin
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  • Uh... well, you have some reflection formulas... – Simply Beautiful Art Apr 13 '17 at 22:47
  • https://en.wikipedia.org/wiki/Riemann_zeta_function#The_functional_equation – Simply Beautiful Art Apr 13 '17 at 22:49
  • There are formulas for $\zeta(s)$ which converge in the critical strip $0<\Re(s)<1$. One of the simplest ones is $\zeta(s)=\frac{1}{1-2^{1-s}}\sum _{k=1}^\infty\frac{(-1)^{k-1}}{k^s}$. There are also globally convergent series such as the ones defined at https://en.wikipedia.org/wiki/Riemann_zeta_function#Globally_convergent_series. – Steven Clark Jun 22 '19 at 18:28

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The short answer is "yes".

For $\mathrm{Re} s < 0$, one uses the functional equation of the zeta function. This says that if we define $$ \Lambda(s) = s(1-s) \pi^{-s/2} \Gamma(s/2) \zeta(s),$$ then $$\Lambda(s) = \Lambda(1-s).$$ You can relate $\zeta(s)$ to $\zeta(1-s)$ (times a power of $\pi$ and a quotient of Gamma functions) in this way, and so the values of $\zeta(s)$ for $\mathrm{Re} s < 0$ are easily attainable from the values for $\mathrm{Re} s > 1$.

Within the critical strip, there are a variety of ways to try to understand the zeta function. You can check the answers to this related question or my answer to a related question for thoughts involving the eta function $\eta(s)$ (which is like an alternating zeta function) or the "approximate functional equation".

davidlowryduda
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  • Thank you for the response. So I see that even for $\Re[s]<0$, my equation was wrong? It's hard for me to see how we arrive at your $\Lambda(s)$ function from the functional equation for $F$, but I'll take your word for it for now. I think the reflection formula you mentioned can be found here: https://en.wikipedia.org/wiki/Reflection_formula Since it implicitly defines $\zeta$ for $0\leq \Re[s]\leq 1$, it's not immediately clear how to use it to compute complex values in the critical strip. – Spencer Parkin Apr 13 '17 at 23:49