|
|
#2
09-20-2005, 04:22 AM
|
|||
|
|||
Re: Millenium problems?
[ QUOTE ]
anyone want to try and explain Riemanns Hypothesis?? [/ QUOTE ] I'll assume you want to know why the Riemann Hypothesis is important. The Riemann zeta function is a function on the complex plane whose analytic proprties capture information about the integers. If you are interested in properties of the integers, you can sometimes translate questions about the integers into questions about the Riemann zeta function, <font color="white">the analytic continuation of</font> 1/1^z+ 1/2^z + 1/3^z + 1/4^z + ... = (1 + 1/2^z + 1/4^z + 1/8^z...)(1 + 1/3^z + 1/9^z + 1/27^z + ...)(1 + 1/5^z + 1/25^z + ...)... For example, zeta(1) = infinity since the harmonic series diverges. That tells you that there are infinitely many primes, since otherwise the product formula would give a number. Other properties of primes can be encoded in the properties of the Riemann zeta function. That there are no zeros along the line Re(z)=1 allows you to prove the Prime Number Theorem, which says that the number of primes less than n is roughly n/(log n). Functions on the complex plane share many properties with polynomials or rational functions. To understand a complex function, you often would like to know the location of the zeros and poles. Knowing that the nontrivial zeros of the Riemann zeta function lie on the line of symmetry (Re(z)=1/2) would tell us a lot more about the integers, from the distribution of primes to the distribution of quadratic residues in modular arithmetic. Mathematicians study complex functions describing the properties of other systems than the integers. The analogues for extensions of the integers such as the number of the form a+bsqrt(2) are called L-functions, and there is a generalized Riemann hypothesis for these. Andre Weil proved an analogue for finite fields. 
|
 |
|
|
Linear Mode
