1
Physiology & Medicine / Re: How Do You Classify Corn?
« Last post by Bored chemist on Today at 15:54:40 »If it's in the form of cornflakes or popcorn, I class it as food. Otherwise I don't.
At this point, it should be obvious that direct attack on the problem is impossible.
But it doesn't stop us from trying anyway. We can start from the functional equation.Here are some interesting results which might be useful in solving the problem.When Riemann zeta function produces 0 result, ζ(s) = 0, at least one of these terms is 0Quotehttps://en.wikipedia.org/wiki/Riemann_zeta_function#Riemann's_functional_equation
This zeta function satisfies the functional equation
where Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product sin(πs/2)Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor.
1) 2s = 0 → s = -∞
2) πs-1 = 0 → s = -∞
3) sin(πs/2).Γ(1-s) = 0 → s ∈ {negative even numbers}
4) ζ(1-s) = 0 = ζ(s)
Point #3 gives trivial zeros, while point #4 gives non-trivial zeros.
when 1-s=s → 1=2s → s=1/2
But ζ(1/2) <> 0
The derivation above should be enough to show that Riemann's Hypothesis is true, ie. non-trivial zeros of Riemann's Zeta function has real part of 1/2.From above results, it can be inferred that for any point in the critical strip,
ζ(s*)=ζ(1-s) if and only if s is in critical line,
where s*=complex conjugate of s.4) ζ(1-s) = 0 = ζ(s)The fact thatBut we only care about the case where ζ(s) = 0.
ζ(s) = (ζ(s*))*
0* = 0
→ ζ(s) = ζ(s*) = ζ(1-s) = 0
when s* = 1-s → s+s* = 1
→ Re(s)+Im(s).i + Re(s)-Im(s).i = 1 → 2.Re(s) = 1
→ Re(s) = 1/2