Finite Euler products and the Riemann hypothesis

Abstract:

We investigate the approximation of the Riemann zeta-function by short truncations of its Euler product in the critical strip. We then construct a parameterized family of non-analytic functions that approximate the zeta-function to the right of the critical line. With the possible exception of finitely many zeros off the critical line, each function in the family satisfies a Riemann Hypothesis. Moreover, when the parameter is not too large, the functions in the family have about the same number of zeros as the zeta-function, their zeros are all simple, and the zeros “repel”. The structure of these functions makes the reason for the simplicity and repulsion of their zeros apparent. Computer calculations suggest that the zeros of functions in the family are remarkably close to those of the zeta-function, even for small values of the parameter. We show that if the Riemann Hypothesis holds for the Riemann zeta-function, then the zeros of these functions indeed converge to those of the zeta-function as the parameter increases and that, between consecutive zeros of the zeta-function, the functions tend to twice the zeta-function. Finally, we discuss analogues of the model for other L-functions and the insight they give into the distribution of zeros of linear combinations of L-functions.