～～黎曼假設～～

Allow me to use english to explain.

Zeta function is defined by Sum 1/n^s
Obviously this function is defined for s>1 as the convergence is well known.

However, in complex analysis, we can extend this function (as smooth as possible) to be defined for almost all complex number s. This is called analytic continuation. For example, y=sqrt(x) is defined for x>0 only, but we can extend it to be defined for x<0 also by y=-sqrt(-x). This extention is "smooth" in the critical value x=0. For Zeta function, the extension is "smooth" if the function is smooth at the critical line Res=1.

So how to define such analytic continuation? Mathematicians use functional equation that the function satisfies. It is some equations satisfied by the function, and it may make sense when the variable is outside the domain. just like g(n) = n!, this function satisfy the equation g(n)=n g(n-1). If we allow n to be any *real* number, then g(n) is extended to be the Gamma Function.

In the case Zeta Function, one of the functional equation is
http://upload.wikimedia.org/math/6/b/1/6b12b8be47efe5c40c58b5cbebaa09d2.png

Therefore we see that when s = -2, -4, -6... the sine term gives 0, and so Zeta(s) is 0. These negative even numbers are called trivial zero 明顯零點
You may ask, why s=2,4,6... are not zero from the functional equation? Because from the equation we have Gamma(1-s), and Gamma function is infinity when taken negative integer variable. You know, 0 * oo is undefined.

But these are not all the zeros. Mathematician also found other zeros of Zeta function, and it happens that they all lie in the "critical strip" 0

2007-01-12 18:13:53 補充：
Shit, Yahoo eat my symbol again......and it happens that they all lie in the " critical strip " 0 < Re s < 1. These other zeroes are called nontrivial zeros 非明顯零點. The Riemann hypothesis says these zeroes all lie on the line Re s = 1/2.

2007-01-12 21:49:01 補充：
As an example: after analytic continuation, Zeta(0) = 1 + 1 + 1 ... = -1/2, and Zeta(-1) = 1 + 2 + 3 ... = -1/12, these are important numbers in string theory.

I believe yahoo eat his symbol yet again.