On Sunday 01 October 2006 18:28, Soo Reams wrote: > Brian Beesley wrote: > > > > Doesn't this depend on the (AFAIK unproved) Riemann hypothesis? > > Not certain of this, but I think Pi(x) is defined as the number of > primes less than or equal to x. The Riemann hypothesis is concerned with > a particular approximation to Pi(x), a more accurate one than Pi(x) = x > / lg x.
Isn't it _perfectly_ accurate _unless_ RH is _false_? The problem being that the computation of pi(x) involves evaluating an infinite series of terms which converge rather slowly (especially for large x), so that any computed value is indeed an approximation. If we had a complete table of the zeros of the zeta function we could do this computation quickly and accurately; but then the existence of such a table would render the truth or otherwise of the Riemann hypothesis moot. > > > > This is where the possible interest is. If Allen's algorithm turns out to > > be genuinely different, proveably correct and returns the same results as > > Pi(x) then there may be a handle on proving Riemann. Or even the extended > > Riemann hypothesis! > > Hmm, unlikely. Allen's algorithm sounds like it's at least as > computationally intensive as finding the primes individually. So what? There are two points here: (1) Advance in mathematics often comes from linking seemingly unrelated fields; and, whilst an infinite amount of computation may still not be sufficient to generate a proof, many types of proof depend on a strictly finite (and sometimes surprisingly small) amount of computation. The trick is in finding the right combination of already proved results, manipulating these to derive something "new". A new algorithm for pi(x) might (but doesn't necessarily) result in a new basis for proving RH and/or a new method of manipulating known results to derive a proof. (2) A new method of computing pi(x) _may_ lead to a new method of computing the zeros of zeta(s) which _may_ result in the finding of at least one zero which isn't on the critical line. If this helps you to think about this, AFAIK no-one has yet managed to prove even that there at most a finite number of zeros off the critical line. Suppose there are two (or more) classes of zero, one (or more) on the critical line and one (or more) off it. So happens that the methods we have at the moment only find zeros on the critical line ... we can calculate the "first" trillion or so zeros and find them all on the critical line, as we have already done, generating evidence of the truth of RH without constituting a proof of any sort. I personally believe that RH is true - but this is only blind faith, which a demonstration by counterexample would shatter. Regards Brian Beesley _______________________________________________ Prime mailing list [email protected] http://hogranch.com/mailman/listinfo/prime
