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$$\frac{1}{\frac{\zeta (2 k)}{\pi ^{2 k}}}$$

Consider the sequence of values generated for the above for increasing values of k. The sequence starts like $6,90,945,9450,93555,\frac{638512875}{691}...$

Now plot the values of this sequence. They appear exponential in nature. Log-ing the graph confirms this.

Now naturally one would want to do a regression on the data. Doing so with values up to k=2000 produces $0.99999e^{2.28946017}$. The larger k gets, the closer the coefficient gets to 1.

So several questions naturally arise from this experiment.

A) Why does this sequence have an exponential distribution.

B) What's significant about that 2.28946017.

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From the definition of $\zeta(s)$ we see that $\zeta(k)\to 1$ as $k\to\infty$. This implies $$ \frac{1}{\zeta(2k)/\pi^{2k}}\sim \pi^{2k}= 1\cdot e^{(2\log\pi)k}. $$ We have $2\log\pi\approx 2.28945977$.

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