Why does the first 100,000 zeroes of the Riemann Zeta function have double-digit sequence count discontinuities at 00,11,22,33,44,55,66,77,88,99?

Solution 1:

This has nothing to do with the Riemann zeta function, but is rather a property of random sequences of digits. (As a rule there's no reason to assume that there's anything significant about the decimal expansion of any number, unless there's some concrete reason to believe otherwise.)

In particular, assume that we have a random sequence of digits where all of 0, 1, 2, ..., 9 are equally likely. Then you expect any particular two-digit sequence to occur 1 time in 100 as a pair of consecutive digits. Most of the numbers at the web site you've linked to are five digits, then a decimal point, then nine more digits. So in each of these numbers there are 12 possible "slots" where, say, "42" can occur, four before the decimal point and eight after. Across all the numbers there are 1.2 million such "slots" (actually a few less because not all the numbers have five digits before the decimal point) and so you expect (1.2 million)/(100) = 12000 occurrences of "42". This is what you see in your plot, except for the deviation because the sequence of zeros cuts off at 74931.

However, how should your browser behave if the sequence "111" appears? You'll find that it counts that as a single occurrence of "11". But the analysis I just gave would count it twice. So the browser-based counts for a sequence of digits that can overlap itself, such as 11, will be lower. In the literature of combinatorics on words this phenomenon is called "autocorrelation".

Solution 2:

When counting all occurrences of digit pairs, I get this graph instead:

It does show the 75 cutoff and the no-leading-zero effect, but nothing about double digits appears special. If you count a digit pair at most once per line, Michael Lugo's explanation tells you why things get skewed.