Dynamics of zeros
This is a sub-page of page on the De Bruijn-Newman constant, and assumes all the notation from that page.
The entire functions [math]\displaystyle{ H_t(z) }[/math] obey the backwards heat equation
- [math]\displaystyle{ \displaystyle \partial_t H_t(z) = - \partial_{zz} H_t(z). }[/math]
If [math]\displaystyle{ H_t }[/math] has a simple zero at [math]\displaystyle{ z_j(t) }[/math], then by the implicit function theorem [math]\displaystyle{ z_j(t) }[/math] varies in a continuously differentiable manner (in fact analytic) for nearby times [math]\displaystyle{ t }[/math]. By implicitly differentiating the equation [math]\displaystyle{ H_t(z_j(t)) = 0 }[/math], we see that
- [math]\displaystyle{ \displaystyle \partial_t z_j(t) = \frac{\partial_{zz} H_t(z_j(t))}{\partial_z H_t(z_j(t))}. }[/math]
Being a simple zero, we have the Taylor expansion
- [math]\displaystyle{ \displaystyle H_t(z) = a (z-z_j(t)) + b (z-z_j(t))^2 + O( |z-z_j(t)|^3 ) }[/math]
for some complex numbers [math]\displaystyle{ a,b }[/math] with [math]\displaystyle{ a \neq 0 }[/math], and for [math]\displaystyle{ z }[/math] close to [math]\displaystyle{ z_j(t) }[/math]. In particular
- [math]\displaystyle{ \displaystyle \partial_z H_t(z) = a + 2 b (z-z_j(t))^2 + O( |z-z_j(t)|^2 ) }[/math]
- [math]\displaystyle{ \displaystyle \partial_{zz} H_t(z) = 2 b + O( |z-z_j(t)| ) }[/math]
TO BE CONTINUED
(See for instance [CSV1994, Lemma 2.4]. This lemma assumes that [math]\displaystyle{ t \gt \Lambda }[/math], but it is likely that one can extend to other [math]\displaystyle{ t \geq 0 }[/math] as well.)
