Some time ago I read part of a book in which the author made some conjectures outlining what kind of zero distribution is expected for functions representable by Dirichlet series with completely multiplicative coefficients in some half plane, based on statements about non-zero value distributions. Unfortunately I forgot the author's name, and have not found the book since.
The connection outlined was something along the lines of:
If the number of solutions to $f(s)=z$, $z\neq 0$, in a strip is of order $T$, then the number of zeros of $f(s)$ in the strip is $O(T)$.
I remember thinking that this is a Phragmen-Lindelof version of Rouche's theorem, but in fact it isn't because it states only an upper bound on the number zeros, and it doesn't assume anything about the minima on the boundary of the strip. Therefore I suppose there are some obstructions introduced by this lack of information.
I would like to know:
Is this actually a theorem for some class of functions? If so, how is it proved? If not, and if I've recalled the conjecture correctly, for and on what is it based?
