Originally posted by BillyJoe
in simple layman's terms (hoping against hope).....What is The Riemann Hypothesis?
Roughly: if a number has
n digits, then, in the vicinity of that number, approximately every
nth number is prime.
Actually, that's off by a factor of 2.3 or so---in reality, there are fewer primes---because the average distance between primes near a number
x is approximately equal to the
natural logarithm of
x, i.e., the logarithm to the base
e = 2.718..., whereas the number of digits
x has is equal to the
common logarithm of
x, i.e., the logarithm to the base 10. The conversion factor between the two types of logarithm is the natural logarithm of 10, which is 2.302585... .
So, for example, near one million = 10<sup>6</sup>, where numbers have 6 decimal digits, the average distance between primes should be around (2.3)(6) = 13.8. And,
in fact, between 10<sup>6</sup> - 1000 and 10<sup>6</sup> + 1000, there are 140 primes, for an average distance of 2000 / 140 or about 14.3. So, it's pretty close.
(If you don't like all the 'approximately's and 'pretty close's, you could always try reading the
non-rough version. Just don't ask me too many questions about it, 'cause I don't quite understand it either.

)