When Vera heard Lu Zhou's praise, she smiled brightly.
This was undoubtedly the best compliment she had ever been given.
As she stood next to Lu Zhou, she said in a low voice, "Your guess is correct, Collatz conjecture is a number theory problem. It's also a complex analysis problem…"
In as early as 1994, L.Berg and G.Meinardus proved that the 3n+1 conjecture was equivalent to the function equation h(z3) = h(z^6)+{h(z2)+λh(λz2)+λ2h(λ2z2)}/3z (where λ=e^(2πi/3)). This could be expressed through the unit disc {z:|z|<1} as h(z)=h0+h1z/(1−z) (where h0 and h2 are complex constants).
In 1998, D.Schliecher used this foundation to prove that any integral function in the form of h(z) results in g(z) = z/2 + (1-cos(πz)(z+1/2)/2+1/π(1/2-cos(πz)sin(πz)+h(z)sin^2(πz)).
