We begin by discussing approximations of the Riemann zeta-function by truncations of its Dirichlet series andĀ Euler product. We then construct a parameterized family of non-analytic approximations to the zeta-function. Every function in the family satisfies a Riemann hypothesis with the possible exception of a few zeros off the critical line. We show that when the parameter is not too large, the functions have roughly the same number of zeros as the zeta-function, their zeros are all simple, and they repel. In fact, if the Riemann Hypothesis is true, the zeros of these functions converge to those of the zeta-function as the parameter increases, and between zeros of the zeta-function the functions in the family tend to twice the zeta-function. For these and other reasons they may be regarded as models of the Riemann zeta-function. The structure of the functions explains the simplicity and repulsion of their zeros when the parameter is small. One might therefore hope to gain insight from them into the mechanism responsible for the corresponding properties of the zeros of theĀ zeta-function. |