Re: proving the Riemann hypothesis
- --- Shi Huang wrote:
>Not really. RH states that the *degree* of irregularity
> So proving the RH would prove that primes are both
> random and non random, which is the key property of
> being a prime or what make it interesting. The RH is
> meant to prove what we suspected of primes is true.
> If we can prove the yin yang duality of primes without
> using the RH, we basically would accomplish the same
> thing. Prove the duality of prime is the real goal
> and the RH is just one way of doing it. There may
> exist simpler ways. May this characterization of RH
> represent a reasonable assesment of the RH?
in the distribution of primes can be quantified, and that
its exponent is exactly 1/2.
It has already been proven that this "degree of irregularity"
in the distribution of primes is at least 1/2, and less than 1.
To the extent that I can understand what you're saying, I
think that the "yin yang duality" of primes is already proven.
Proving RH would simply constrain the irregularity to levels
which make the primes much easier to work with, but it doesn't
change the fundamental nature of the primes.
- For example 2 adjacent gaps cannot be equal if they aren't multiple of
6. For example the gap between 2 pairs of twins is at least 4. For
example each prime number has the form 2n+/-1, 3n+/-1, 4n+/-1, 6n+/-1.
Each pair of twins has the form 12n+-1, there are approximate formulas
for the nth prime and the number of primes < x and so on. You cannot
call all this random ore unpredictable. Of course the prime numbers are
distributed as regularly as possible, that's a tautology. In
mathematics everything is as regular as possible. Is pi random? Build
P=2,357111317192329 , and you have the same case as pi. Consider the
primes to be an irrational number, and there are no problems. If you
mean there is no formula f(n) which produces primes for each n, then
you are right. In this sense primes are random. (I am not quite sure
there is a formula p=[k^n^3] (H.W.Mills) which is said to produce only
prime numbers). If you define "formula" as an algorithm, as a
calculation instruction such as the sieve of Eratosthenes, then the
primes are not random but simply what they are. Perhaps the compound
numbers are random? Or are they only non-transparently complicated?