# The Riemann Hypothesis Simplified 1.1.3

1. First distinguish between long-ranged (infinite primes) and short-ranged (finite primes) arithmetic progressions.

2. Next observe that both these kinds of progressions are associated to (spiral) wave-forms.

3. Key Observation: Long-ranged progressions are associated to wave-forms whose growth rates have square-root size, but that this is NOT true of short-ranged progressions.

4. Coin flipping: If a coin is fair, then there is a margin of error whose growth rate is approximately the square- root of the number of coin flips. One way to depict these imbalances is assigning 1 to heads, -1 to tails, summing the 1s and -1s, and registering the imbalances as departures in a positive or negative direction from the x-axis.

5. We call these pathways “random walks”, and if we average the furthest distance of a random walk from the beginning of the walk, we find that it converges to (2 Sqrt[x])/\[Pi].

6. We could imagine the average to be more or less than this square-root value: if it was more, then random walks would on average be more random, and if it was less, then random walks would on average be less random.

7. We can relate coin flipping and random walks to the Moebius function which assigns 1 and -1s to square-free integers.

8. In the same way that the 1s and -1s assigned to heads and tails are summed to produce a random walk, the Mertens function sums the 1s and -1s assigned to square-free integers to produce a random walk.

9. We could imagine this wave-form to have more or less amplitude. If the amplitude was more, then the random walk would be more random, and if it was less, then the random walk would be less random.

10. Riemann Hypothesis: From one point of view it says simply that the wave-form associated to the random walk of the Mertens function cannot be more nor less random than indicated by the square-root bounds.

11. By subjecting the Mertens function to the re-arrangements depicted, we see that this is the same as saying that error between the count of the primes and the primes is a random walk associated to a wave-form whose growth-rate has square-root size.

12. Key Observation: Given that progressions associated to spiral wave-forms whose growth-rates LACK square-root size are short-ranged (finite primes), it follows that the falsity of the hypothesis implies that there are a finite number of primes.

13. The Riemann Hypothesis may look like a statement about a fairly trivial abstract problem, of no great interest to anyone other than the puzzle enthusiast or the introspective pure mathematician, but its importance consists in the fact that it is a statement about the necessary and sufficient conditions of arithmetic continuity and therefore consciousness itself.

14. It follows from the Riemann Hypothesis that universes or dimensions in which the constants of nature are other than what they are in our universe (universes hostile to arithmetic continuity and consciousness) are not amenable to our mathematical-physics or even to our ordinary human understanding.

15. It also follows that these dimensions exist -in the form of short-ranged progressions- and that they determine what occurs in this dimension.