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The Riemann Zeta functional equation is defined as follows.


(1) $\quad\zeta (s)=f(s)\,\zeta(1-s)\,,\quad f(s)=2^s\pi^{s-1}\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma (1-s)$


Note that $|f(s)|=1$ along the critical line where $s=\frac{1}{2}+i\,t$, but I've been investigating other values of $s$ for which $|f(s)|=1$ which is equivalent to $|\zeta(s)|=|\zeta(1-s)|$.


The following figure illustrates a contour plot of $|\zeta(s)|=|\zeta(1-s)|$ where the horizontal axis represents $\Re(s)$ and the vertical axis represents $\Im(s)$. Note there's a closed path which is symmetric about the critical line and real axis where $|\zeta(s)|=|\zeta(1-s)|$. The two red dots at the intersections of the closed path and the critical line are at $s=\frac{1}{2}+i\,6.28984$ and $s=\frac{1}{2}-i\,6.28984$.


Contour Plot of |\zeta(s)|=|\zeta(1-s)|

Figure (1): Contour Plot of $|\zeta(s)|=|\zeta(1-s)|$


Question (1): Can the value $6.28984$ be expressed as a function of mathematical constants?


Since I originally posted question (1) above, I've noticed that $\pm 6.28984$ is associated with the location of the minima and maxima of the Riemann-Siegel theta function.


The following table illustrates points along the closed path illustrated in Figure (1) where $\Re(s)\in\mathbb{Z}$.


Table (1): $\quad\begin{array}{ccc} \Re(s) & \Im(s) & \text{} \\ -16 & -0.511623 & 0.511623 \\ -15 & -0.945774 & 0.945774 \\ -14 & -1.5719 & 1.5719 \\ -13 & -2.12883 & 2.12883 \\ -12 & -2.66579 & 2.66579 \\ -11 & -3.17309 & 3.17309 \\ -10 & -3.6505 & 3.6505 \\ -9 & -4.09581 & 4.09581 \\ -8 & -4.50702 & 4.50702 \\ -7 & -4.88189 & 4.88189 \\ -6 & -5.21802 & 5.21802 \\ -5 & -5.51293 & 5.51293 \\ -4 & -5.76403 & 5.76403 \\ -3 & -5.96879 & 5.96879 \\ -2 & -6.12484 & 6.12484 \\ -1 & -6.23013 & 6.23013 \\ 0 & -6.28319 & 6.28319 \\ 1 & -6.28319 & 6.28319 \\ 2 & -6.23013 & 6.23013 \\ 3 & -6.12484 & 6.12484 \\ 4 & -5.96879 & 5.96879 \\ 5 & -5.76403 & 5.76403 \\ 6 & -5.51293 & 5.51293 \\ 7 & -5.21802 & 5.21802 \\ 8 & -4.88189 & 4.88189 \\ 9 & -4.50702 & 4.50702 \\ 10 & -4.09581 & 4.09581 \\ 11 & -3.6505 & 3.6505 \\ 12 & -3.17309 & 3.17309 \\ 13 & -2.66579 & 2.66579 \\ 14 & -2.12883 & 2.12883 \\ 15 & -1.5719 & 1.5719 \\ 16 & -0.945774 & 0.945774 \\ 17 & -0.511623 & 0.511623 \\ \end{array}$


The following table illustrates points along the closed path illustrated in Figure (1) where $\Im(s)\in\mathbb{Z}$.


Table (2): $\quad\begin{array}{ccc} \Im(s) & \Re(s) & \text{} \\ -6 & -2.82313 & 3.82313 \\ -5 & -6.66187 & 7.66187 \\ -4 & -9.22159 & 10.2216 \\ -3 & -11.3476 & 12.3476 \\ -2 & -13.2329 & 14.2329 \\ -1 & -14.9193 & 15.9193 \\ 0 & -16.4061 & 17.4061 \\ 1 & -14.9193 & 15.9193 \\ 2 & -13.2329 & 14.2329 \\ 3 & -11.3476 & 12.3476 \\ 4 & -9.22159 & 10.2216 \\ 5 & -6.66187 & 7.66187 \\ 6 & -2.82313 & 3.82313 \\ \end{array}$


Question (2): Can any of the $\Im(s)$ values in Table (1) or any of the $\Re(s)$ values in Table (2) be expressed as functions of mathematical constants?


Question (3): What is the length and area of the closed path illustrated in Figure (1)?


Question (4): Assuming $s$ is on the closed path illustrated in Figure (1), can $\Im(s)$ be written as a function of $\Re(s)$ or vice-versa?


Question (5): Would a proof that there is no point $s$ in the critical strip where $|f(s)|=1$, $\Re(s)\ne\frac{1}{2}$, and $\Im(s)>c$ for some $c\in\mathbb{R}$ constitute a proof of the Riemann hypothesis, or has it been shown that for any value of $c\in\mathbb{R}$ there is a point $s$ in the critical strip where $|f(s)|=1$, $\Re(s)\ne\frac{1}{2}$, and $\Im(s)>c$ for which $\zeta(1-s)$ is not a zero of the Riemann zeta function?


I've noticed successive derivatives of the Riemann zeta function $\zeta(s)$ also seem to exhibit analogous results. The following figure illustrates contour plots of $|\zeta^{(n)}(s)|=|\zeta^{(n)}(1-s)|$ for $n=0$, $n=1$, $n=2$, and $n=3$ in blue, orange, green, and red respectively where the horizontal axis represents $\Re(s)$ and the vertical axis represents $\Im(s)$.


Contour Plots of |\zeta^{(n)}(s)|=|\zeta^{(n)}(1-s)|

Figure (2): Contour Plots of $|\zeta^{(n)}(s)|=|\zeta^{(n)}(1-s)|$


I believe there's a single closed contour where $|\zeta(s)|=|\zeta(1-s)|$ associated with the first trivial zeta zero, and a pair of closed contours where $|\zeta(s)|=|\zeta(1-s)|$ associated with each subsequent trivial zeta zero. I believe in many cases (if not all) there are analogous closed contours associated with $|\zeta^{(n)}(s)|=|\zeta^{(n)}(1-s)|$.

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  • $\begingroup$ $\log f(s)$ has a simple explicit formula and the curves where $\Re(f(s)) = 0$ are close to those obtained from the Stirling approximation. The functional equation just says that $\theta(s) =\Gamma((s+1/2)/2) \pi^{-(s+1/2)/2} \zeta(s+1/2)$ is even, you can't deduce anything about the RH from that (the basic properties of $\theta(s)$ tell us the density of zeros) $\endgroup$ – reuns Nov 22 '18 at 20:42
  • $\begingroup$ Why don't you plot $\Re(f(s))$ and its Stirling approximation $\endgroup$ – reuns Nov 25 '18 at 22:23
  • $\begingroup$ @reuns Originally my primary motivation for this question was a desire to clarify question (4) and it's answer in my mind which you addressed in your first comment above. The closed paths where $|\zeta^{(n)}(s)|=|\zeta^{(n)}(1-s)|$ were initially somewhat of a curiosity to me, but I think I now understand the source of these closed paths. I've haven't yet taken the time to look into the Stirling approximation which I've seen mentioned a number of times, but I suppose it's about time I did. $\endgroup$ – Steven Clark Nov 25 '18 at 23:08
  • $\begingroup$ In my previous comment I meant question 5 instead of question 4. $\endgroup$ – Steven Clark Nov 26 '18 at 15:08

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