This question is partly inspired by David Stork's recent question about the enigmatic complexity of number theory. Are there algebraic systems which are similar enough to the integers that one can formulate a "Riemann hypothesis" but for which the Riemann hypothesis is false? One motivation for constructing such things would be to illustrate the barriers to proving the Riemann hypothesis, and another one would be to illustrate how "delicate" the Riemann hypothesis is (i.e., that it's not something that automatically follows from very general considerations).
I've run across various zeta functions over the years, but I seem to recall that either the Riemann hypothesis is probably/provably true, or the zeta function is too unlike the classical zeta function to yield much insight.
More generally, what happens if we replace "Riemann hypothesis" with some other famous theorem or conjecture of number theory that seems to be "delicate"? Can we construct interesting systems where the result fails to hold?
Best Answer
One way of making "fake integers" explicit is a Beurling generalized number system, which is the multiplicative semigroup $Z$ generated by a (multi)set $P$ of real numbers exceeding $1$; lots of research has been done on the relationship between the counting function of $P$ (the Beurling primes) and the counting function of $Z$ itself. In this context, it is certainly known that the Riemann hypothesis can fail; see for example this paper of Diamond, Montgomery, and Vorhauer.
If this is not "similar enough to the integers", then I think you should more clearly define what you mean by that phrase.