# Interesting pattern in riemman zeta function

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

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.

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 $2\log\pi\approx 2.28945977$.