I wonder what the "right" notion of "algebraic Dirichlet series" might be. Here I'm thinking of formal Dirichlet series $D(s)=\sum_{n\geq 1} a_n/n^s$, say with $a_n$ being rational numbers.

I'm trying to figure out an analogon for the rich class of algebraic formal power series $A(z)=\sum_{n\geq 0} a_n z^n$, that satisfy a polynomial relation $p(z, A(z))=0$, for some polynomial $p$ with integer coefficients.

The main question is, what should be the replacement of $z$? A good candidate seems to be $\zeta(s)$, i.e., maybe "algebraic" should mean "satisfying a polynomial equation $p(\zeta(s), D(s))$=0".

More generally, what would an interesting notion of "differentially algebraic Dirichlet series", i.e., is there a good candidate for a replacement of the derivative, that plays nicely together with the replacement for $z$. It would be particularly nice if this derivative would send $z^k$ to a linear combination of smaller powers of $z$.

Of course, most important are that such dependencies actually occur, so some nice examples beyond those on wikipedia:Dirichlet series would be very, very helpful.