these pde's and the Dirichlet divsor problem

Question

I noticed that

$$t^2 \frac{\partial^3}{\partial t^3}\Delta_t(s)+s^2 \frac{\partial}{\partial s} \Delta_t(s)=0 $$

is satisfied by

$$\Delta_t(s)= - \sqrt{\frac{t}{s}}Y_1{(4\pi\sqrt{ts})}+ \sqrt{\frac{t}{s}}K_1(4\pi \sqrt{ts})$$

for $Y_1$ and $K_1$ Bessel functions.

I noticed this looked similar to the "error term" in the Dirichlet divisor problem

$$\Delta(t)= -\sum_{s \in \Bbb N} d(s) \sqrt{\frac{t}{s}}Y_1{(4\pi\sqrt{ts})}-\sum_{s \in \Bbb N} d(s)\sqrt{\frac{t}{s}}K_1(4\pi \sqrt{ts})$$

I know from number theory that the order of magnitude of $\Delta(t)$ as $t\to \infty$ is close to $O(t^{1/4+\epsilon}).$ Making this exact is the notorious Dirichlet divsor problem.

I wonder if there are techniques from the theory of pde's that I can use to show this asymptotic holds or at least get close.

I observed that

$$t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\partial s} \Psi_n(t,s)$$

has a solution

$$\Psi_n(t,s)=2 \sqrt{\frac{tn}{s}}K_1(2\sqrt{t ns})$$

where the coefficients are

$$b(n)=\big\lbrace 2,12,24,40,60,84, \cdot\cdot\cdot\big\rbrace.$$

which I noticed correspond to twice the pronic/oblong numbers (if you throw out the first term $2$).

I thought about writing out the coefficients of the sequence $r(n)$ (to be determined) for the PDE corresponding to $Y_1$ as we did for $b(n)$ corresponding to $K_1$.

Then through linearity of solutions we can obtain a PDE involving both $r(n)$ and $b(n)$:

$$t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \chi_n(t,s)+s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \chi_n(t,s)=\sum_{n=2}^\infty r(n)\frac{\partial}{\partial s} \chi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\partial s} \chi_n(t,s)$$

where $$\chi_n(t,s)=-2 \sqrt{\frac{tn}{s}}Y_1{(2\sqrt{tns})}+2 \sqrt{\frac{tn}{s}}K_1(2 \sqrt{tns})$$

Is $r(n)$ a known arithmetic sequence?

Answer 1

For

$$\Psi_n(t,s)=2 \sqrt{\frac{n t}{s}} K_1\left(2 \sqrt{t n s}\right)\tag{1}$$

one has

$$t^2 \frac{\partial^3\, \Psi_n(t,s)}{\partial t^3}=n\, s^2 \frac{\partial\, \Psi_n(t,s)}{\partial s}\tag{2}$$

s0

$$t^2 \frac{\partial^3}{\partial t^3}\left(\sum\limits_{n=1}^{\infty} \Psi_n(t,s)\right)=s^2 \sum\limits_{n=1}^{\infty} n\, \frac{\partial\, \Psi_n(t,s)}{\partial s}\\\qquad\qquad=s^2 \frac{\partial}{\partial s} \left(\sum_{n=1}^\infty \Psi_n(t,s)\right)+s^2 \sum_{n=1}^\infty (n-1)\frac{\partial\, \Psi_n(t,s)}{\partial s}\tag{3}$$

so $b(n)=n-1$.

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For $$\varphi_n(t,s)=2 \sqrt{\frac{n t}{s}} Y_1\left(2 \sqrt{t n s}\right)\tag{4}$$

one has

$$t^2 \frac{\partial^3 \varphi_n(t,s)}{\partial t^3}=-s^2 n \frac{\partial\, \varphi_n(t,s)}{\partial s}\tag{5}$$

so $$t^2 \frac{\partial^3}{\partial t^3}\left(\sum\limits_{n=1}^{\infty} \varphi_n(t,s)\right)=-s^2 \sum\limits_{n=1}^{\infty} n\, \frac{\partial\, \varphi_n(t,s)}{\partial s}\\\qquad=-s^2 \frac{\partial}{\partial s}\left(\sum\limits_{n=1}^{\infty} \varphi_n(t,s)\right)-s^2 \sum\limits_{n=1}^{\infty} (n-1)\, \frac{\partial\, \varphi_n(t,s)}{\partial s}\tag{6}$$

so $r(n)=b(n)=n-1$.