Races over the zeros of the Riemann zeta function

Suraj Singh Khurana

Decimal expansion refers to the representation of a number in decimal form, using a decimal point to separate the integer part of the number from its fractional part. The decimal expansion of a number can be obtained by dividing the number by powers of 10. For example, let's consider the number 1/3. Then we have the following repeating pattern,

However, for the famous constant pi (denoted as π), we have a more interesting decimal expansion, and is given by

It was shown by Lambert in 1760 that π is irrational and therefore the above decimal expansion will have no repeating pattern. The numbers occurring in the decimal expansion have been studied by many scientists for a long time. Due to this, there are many studies and research papers dedicated to the decimal expansion of pi. One can look at [1] for more details. On this page, we are interested in the numbers which represent the critical zeros of the Riemann zeta function ζ(s) (that is, those complex numbers s for which ζ(s) =0 where s is not a negative even integer). For those unfamiliar, these numbers are intimately linked with the distribution of prime numbers. For the sake of the interest of the reader let us remind you that finding the location of all the critical zeros will allow you to earn one million dollars. The first critical zero of ζ(s) has a real part equal to 1/2 and imaginary part ${\gamma }_1$ whose decimal expansion is given by

This was first calculated by the German mathematician Bernhard Riemann himself, who introduced the zeta function and showed its connection to prime numbers in his famous 1859 paper "On the Number of Primes Less Than a Given Magnitude". Just like π the decimal expansion of ${\gamma }_1$ does not "appear" to show any repeating pattern, but unfortunately, unlike π we still do not know whether this number is rational or irrational. If we denote ${\gamma }_{\mathrm{n}}$ to be the imaginary part of the n^th critical zero then it is expected that all the ${\gamma }_{\mathrm{n}}$ are irrational. The decimal expansion of these numbers have been a source of interest to several mathematicians. For details check [2]. Here we will search/explore for patterns in decimal expansions of fractional parts of ${\gamma }_{\mathrm{n}}$ by allowing ourselves to construct races which we define in the next section. A source of motivation is that since it is believed that the decimal expansion of these numbers have no repeating pattern, we should expect all the digits from 0 to 9 should occur equally likely for any decimal place. Therefore it is interesting to look for any irregularities, bias or conspiracy coming out of a fair process of playing with these numbers. For this purpose we construct a fair looking race which is based on the idea of of how many times a given digit appears in the decimal expansion of these numbers.

The race

We are interested in the frequency with which a given digit occurs in various decimal places of the decimal expansion of ${\gamma }_{\mathrm{n}}$ as we keep increasing n. To describe the race, let us consider five players named Player A, Player B, Player C, Player D, and Player E who will run in first, second, third, fourth, and fifth decimal place respectively. We are interested to know which Player collects the most number of a given digit as the Player runs vertically across ${\gamma }_{\mathrm{n}}$ in increasing order. See the below figure displaying the first five zeros with the players in their starting position.

More precisely let Race N be the contest where the players get a point every time N occurs as they run in their decimal place across the decimal expansions of ${\gamma }_{\mathrm{i}}$ till i is equal to a given n. In the above figure, if we consider Race 2 with n=5, then after the race completes, Player A gets 0 points, Player B gets 2 points, Player C gets 1 point, Player D gets 0 points and Player E gets 1 point. As the reader might have noticed that there could be ten possible races that can be held for a given n. Click on the races below (right-arrow icons) to display the performance of the players in all 10 races for n=1000. The reader can hover* over the paths in the graphs to find out more details and check the winner of each race.

Race results with n=1000:

Final result of each race

For the overall winner one can observe the number of wins for each player from the following table.

It is surprising to see Player B lost all the races and there is a tie between Player A and Player C for the winner position. The runner-up position is again a tie between Player D and Player E. My original guess was that all the players will win 2 races each. However these races were observed with n=1000, and therefore this will be interesting to see if there appears to be uneven distribution in number of wins among players for large values of n.

References

1. Berggren, Lennart, et al. Pi: A Source Book. Springer New York, 1997. Link
2. Papers on Zeros of the Riemann Zeta Function and Related Topics, by A. M. Odlyzko.
3. Andrew Odlyzko: Tables of zeros of the Riemann zeta function , by A. M. Odlyzko.