Analytic number theory / PDE / Zeta function

Heat flow and the zero of polynomial-a approach to Riemann Hypesis

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this is a note after reading the blog:Heat flow and the zero of polynomial.

1.instead of consider the original version:

\partial_{zz}f(z,t)=\partial_tf(z,t).

consider the corresponding “equidistribution version” is also interesting:

\partial_{zz}f(z,t)=\theta(z,t)\partial_tf(z,t),especially \theta(z,t)=e^{2\pi i\alpha t},\alpha\in R-Q.

2.

where f(z)=z^n+a_{n-1}z^{n-1}+...+a_1z+a_0.

f(z,t)=\sum_{k=1}^n\sum_{0\leq m\leq k-2,2|k-m}\frac{k!}{m!(k-m)!}z^mt^{k-m}.

=\sum_{k=1}^m\sum_{0\leq m\leq k-2,2|k-m}C_k^mt^{k-m})z^mt^{k-m}

\sum_{m=0}^{n-2}(\sum_{k=m,2|k-m}^nC_k^mt^{k-m})z^m.

rescaling:

F_t:(z_1(t),...,z_n(t))\longrightarrow (\frac{z_1(t)}{t},...,\frac{z_n(t)}{t}).

F_t\cdot f(z,t)=\sum_{m=0}^{n-2}(\sum_{k=m,2|k-m}^nC_{k}^mt^{k-n})z^m.

\lim_{t\to \infty}F_t\cdot f(z,t)=\sum_{m=0,2|n-m}^{n-2}C_n^mz^m.(*)

even term \longrightarrow constant.(after renormelization)

odd term \longrightarrow 0(invariant).so at least the sum zeros of is invarient.

by the algebraic fundamental theorem,we have n zero \{z_1,...,z_n\}of (*).

until now,we already now if the n zeros is distinct,then because the energy is the energy is the same and the entropy is increase so \exists T>>0,\forall t_i,t_j>T,\{t>T|z_i(t)\} \cap \{t>T|z_j(t)\}=\emptyset.\lim_{t\to \infty}|z_i(t)|=\infty and \lim_{t\to \infty}arg(z_i(t))=z_i.

but how to know the information of the change of direction at “blow up” time?

1.change direction only at blow up.

2.energy invariant \sum_{1\leq i\neq j\leq n}\frac{1}{|x_i-x_j|^2}.

3.general philosophy

deformation some function under some evolution equation, such like heat equation,wave equation,shrodinger equation.and there is some conversion thing under the equation,and some quantity that could calculate directly such like the trace of spectral.

4.difficultis

this philosophy could generate to the analytic function case,but to make the limit case(I only know how ti deal with this now)coverage.we need very good control on the coefficient.

and to investigate the change of direction at blow up point maybe we need some knowledge about the burid group.

 

 

 

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