In the digits of PI themselves there are no evidences that Benford's laws could be manifested.
That is why I decided to conduct the following experiment: I have broken the number PI and its first 1 million digits, in sequences of repeating odd numbers and repeating even numbers, and then added the sum of all the digits in that sequence.
Since single digits are evenly distributed in the digits of PI, I have only included sequences with $2$ digits and above.
What you get is the following (for the first $3.141592653589793238462$):
sequence : sum
31 : 4
159 : 15
26 : 8
535 : 13
8 : (single digit not included)
8 : (single digit not included)
9793 : 28
2 : (single digit not included)
846264 : 30
33 : 6
795 : 21
02884 : 22
1971 : 18
... All the way to the 1 million digit
I have then searched to see how many times each single digit number occurred in all the sums (assuming we reassemble all the sums into a large sequence and every digit can act as a first digit in a possible sequence)
1: 47981
2: 26817
3: 6102
4: 15534
5: 3258
6: 14976
7: 2943
8: 14962
9: 3126
(0 15035 which is not included in Benford's laws )
I am aware that we are no more directly discussing about the digits of PI, but rather about the sums built from these sequences but it raises my curiosity, and my question is broken in to the following:
A) I understand why there are more even numbers, but it seems as if the gap between an even number and its following odd number is approximately $* 5$ (slightly less between $2$ and $3$)
Why is that?
B) Even though $1$ is an odd number it still appears much more than all the other even numbers. Also we can clearly see a decrease in appearances, as we progress to the next number (in odd numbers and even numbers separately, and more evidently in the beginning).
It seems as if we could readjust the fact that even numbers occur more than odd numbers, than we would have a 1 appear the most, followed by 2, followed by 3 ...
When comparing to some of the graphs and stats on Wikipedia’s page for Benford’s laws, the results are pretty close looking (similar).
- Update:
From here on I have updated the question following a comment I have received:
"My feeling is there will always be more runs with length two than with length three, more with length three than with length four, and so on, and this will bias things in favor of small digits. Also, a bias for even digits over odd. Of course, this is all based on some randomness assumptions about the digits of π, none of which have been proved." – Gerry Myerson
Back to my updated question: Since there are only theories for Benford's laws, is it possible that Benford's laws in the physical world are manifested because there will always be most 1 pattern digits, following by 2 pattern digits that have more possibilities to begin in 1 (and then by 2...) when adding their sums, following by 3 pattern digits that have more possibilities to begin by 1 (and then by 2...) when adding their sums ....? (Assuming it can be proved that it is the case in the digits of pi)
C) Or are the similarities purely coincidental?