Using Maple to compare $\pi^2$ and the partial sums of $6\sum_{n=0}^{\infty}\frac{1}{n^2}$ I have noticed something that appears strange. For instance, let $S_{k}=6\sum_{n=0}^{k}\frac{1}{n^2}$ be the kth partial sum, for the first 50 digits we, have
9.8636074000893588188343481429332935379126206789621 ($S_{1000}$) 9.8696044010893586188344909998761511353136994072408 ($\pi^2$ )
Notice that the first three digits coincide (this is of course expected), but we have a strange coincidence of digits that are quite surprising. A table with the difference between the digits is given below (read from right to left!)
[-7, 2, 2, 7, 1, 7, 2, -9, -7, 0, -1, 0, 6, 2, 0, 4, 2, 4, 1, -3, -4, 1, -7, -5, -8, 8, -5, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 3, 0, 0, -6, 0, 0, 0], The zeros indicate that the digits coincide.
Below the tables for
$k=10^4$: [-7, 1, 4, 7, -1, 1, -3, -4, -1, 3, 4, 1, -1, -4, -1, 3, 4, 1, 7, -5, -1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 3, 0, 0, 0, -6, 0, 0, 0, 0]
$k=10^5$: [-4, 9, 2, 6, 1, 7, 5, -2, -7, -5, 2, 7, 5, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -9, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 4, -1, 0, 0, 0, 0]
$k=10^6$: [-1, 3, 4, 1, -3, 6, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -7, 1, 0, 0, 0, 0, 4, 9, -1, 0, 0, 0, 0]
Notice that in the table above the initial (from right to left) expected sequence of zeros has length 4, but we can see a sequences with 11 and 12 consecutive zeros! Can someone explain this surprising abundance of coinciding digits?
I have checked that a similar phenomenon occurs for other values of Riemann's $\zeta$ function. For instance: If we compare $\zeta(3)$ and the partial sums of $\sum_{n=1}^{\infty} \frac{1}{n^3}$ we have, for 100 digits and $k=10^{11}$ the following table
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, -2, 7, -4, -3, -3, -3, -3, 7, 6, -4, -3, -3, 7, -4, 7, -4, -3, -3, -3, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
so we have the first 21 digits that coincide (ok, expected) and "much later" 21 that don't coincide followed by 19 that coincide.
Is this a well known phenomenon?
