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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?

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  • $\begingroup$ Possibly unrelated comment: If instead of strips you use discs of increasing radii then the property that you want is false but it has a reasonable fix: add proximity functions and use the first main theorem of Nevanlinna theory. If you allow different (fixed, finitely many, at least 3) choices of z then you don't need proximity functions (second main theorem). $\endgroup$
    – Pasten
    Nov 19, 2014 at 2:35

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