Let us assume there exists some infinite order differential equation whose solution is:
$$ y= \sum_{n=1}^\infty A_n \exp(p_n^sx) $$
Where $p_n$ is the $n$'th prime. Substituting $ y=\exp(\lambda x)$ as a trail solution and factorizing. The differential equation must be simplified and factorized to:
$$ \prod_{j=1}^\infty (\lambda-p_n^s) = 0$$ $$ \implies \prod_{j=1}^\infty (1-\frac{p_n^s}{\lambda}) = 0$$
Expanding the above equation:
$$\implies 1 - (\sum_{i} p_i^s)\frac{1}{\lambda} + \sum_{i < j} (p_i p_j)^s \frac{1}{\lambda^2} - \sum_{i<j<k} (p_i p_j p_k)^s \frac{1}{\lambda^3} + \dots + \lim_{n \to \infty} \sum_{i<j<\dots<n}(-1)^n (p_i p_j \dots p_n)^s \frac{1}{\lambda^n} = 0 $$
Writing more compactly:
$$ 1+ \sum_{n=1}^\infty \sum_{x_i<x_j<\dots<x_n} (p_{x_i}p_{x_j} \dots p_{x_n})^s = 0 $$
We conjecture that $\sum_{x_i<x_j<\dots<x_n} (p_{x_i}p_{x_j}\dots p_{x_n})^s $ can be analytically continued as some function of $ P(s)$ where $P(s)$ is the prime zeta function
Hence, for $f_1(P(s)) = P(s)$
And $ f_2(P(s)) = P(s)^2 - P(2s)$
And so on ...
Where $f_n(P(s))$ is given by comparing the below equation to the equation before the analytic continuation:
$$ 1 + \frac{f_1(P(s))}{\lambda} + \frac{f_2(P(s))}{\lambda^2} + \dots = 0 $$
Writing it compactly:
$$ 1 + \sum_{i}^\infty \frac{f_i(P(s))}{\lambda^s} =0 $$
Note this can be thought as of the solution of a integral:
$$ y + f_1(P(s))\int_{-\infty}^x y(x) dx + f_2(P(s))\int_{-\infty}^x (\int_{-\infty}^k y(z) dz) dk + \dots$$
(To verify this subsitute $y=Ae^{2x}$) Writing it compactly:
$$ y + \sum_{i=1}^\infty f_i(P(s)) \int_{- \infty}^x (\int_{- \infty}^x (\dots( \int_{- \infty}^x y dx )dx)\dots \text{$i$ times})dx) = 0 $$
Strategy from Here
My idea was to take some asymptotic limit as $s \nearrow 1$ and rewrite this as a integral equation for $y^2$.
Since,
$$ y=\sum_{i=1}^\infty A_i \exp{p_i x} $$
Setting $A_1=0$ and squaring:
$$ y^2 = \sum_{i,j}A_{ij}' \exp{(p_i + p_j)x} = \sum_i^\infty B'_i e^{2ix} $$
By Goldbach's conjecture the rightmost side must contain all the even powers.
Questions
Is this a viable strategy or useful reformulation of Goldbach's conjecture? (Is it easier to show $y^2 =\sum_i^\infty B'_i e^{2ix} $as a solution)? Is the analytic continuation valid?