is Riemann Hypothesis implied by the Cramer conjecture? #692 Correcting Math
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From: Archimedes Plutonium on 17 Jul 2010 13:40 Archimedes Plutonium wrote: > --- quoting from Wikipedia --- > Riemann's hypothesis is concerned with the zeroes of the ζ-function > (i.e., s such that ζ(s) = 0). The connection to prime numbers is that > it essentially says that the primes are as regularly distributed as > possible. From a physical viewpoint, it roughly states that the > irregularity in the distribution of primes only comes from random > noise. From a mathematical viewpoint, it roughly states that the > asymptotic distribution of primes (about 1/ log x of numbers less than > x are primes, the prime number theorem) also holds for much shorter > intervals of length about the square root of x (for intervals near x). > This hypothesis is generally believed to be correct. In particular, > the simplest assumption is that primes should have no significant > irregularities without good reason. > --- end quoting from Wikipedia --- > > Maybe, just maybe I have a proof of the Riemann Hypothesis. Funny how > a correction of the > Indirect Method Euclid Infinitude of Primes proof-- that the Euclid > Numbers of P-1 and P+1 > are necessarily primes in that method yields the proof of the Riemann > Hypothesis. > > The proof of the Legendre Conjecture is easily begot from the idea > that both n^2 +1 and > (n+1)^2 -1 lie between n^2 and (n+1)^2 and are prime using the > Indirect Euclid IP method. > > So if the Riemann Hypothesis is a further demand or restriction of > having a prime existing in a much smaller interval about number x, > well, the Indirect Euclid IP can fulfill that demand. > --- quoting Wikipedia on prime conjectures --- If Legendre's conjecture is true, the gap between any two successive primes would be (O sqrt p. In fact the conjecture follows from Andrica's conjecture. Harald Cramér conjectured that the gap is always much smaller, O(log2p); if Cramér's conjecture is true, Legendre's conjecture would follow. Cramér also proved that the Riemann hypothesis implies a weaker bound of (O sqrt p log p) on the size of the largest prime gaps. Legendre's conjecture implies that at least one prime can be found in every revolution of the Ulam spiral. --- end quote --- So I am under quite a pretty turn of events. There is a good possibility that the correct proof of the Infinitude of Primes Indirect Method, not only proves the Twin Primes and Polignac conjectures, but also the Mersenne primes infinity and Legendre primes but also and hopefully the Cramer conjecture and finally the Riemann Hypothesis. Who would have thought that? Noone in their wildest dreams would have thought that a tiny mistake in Euclid's IP proof Indirect was the key to solving the Riemann Hypothesis. It would go to show that some tiny mistake in mathematics can delay the progress of the subject for milleniums because the proof of Infinitude of Twin Primes cannot be obtained, nor the proof of the Riemann Hypothesis cannot be obtained unless we have a full and correct Euclid Infinitude of Primes Indirect method proof. To know that both P-1 and P+1 Euclid Numbers are necessarily prime in Indirect is what proves the Twin Primes and now, I suspect the Cramer conjecture and the Riemann Hypothesis. P.S. I do not think I need the combination of Mathematical Induction inside the Euclid Indirect IP for Cramer conjecture and Riemann Hypothesis. And where both conjectures require a single prime to exist between the smallest of intervals, the Euclid IP Indirect delivers two new primes in that smallest interval. If the interval is sqrt(p) about p, then the Euclid IP Indirect delivers two new primes of sqrt(p)+1 and p-1, provided I am understanding what the Cramer conjecture is. Archimedes Plutonium http://www.iw.net/~a_plutonium/ whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies |