2007/04/26 by S. M. Gonek, Gonek, S. M. · 1 voice · 1 citation
Mathematics · #11M26 #Advanced Mathematical Identities #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.CV #math.NT
paper · pdf · doi:10.48550/arxiv.0704.3448
openalex publication_date 2007/04/26 · arxiv published 2007/04/26 · arxiv updated 2007/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the approximation by products is good in this region, the zeta-function has at most finitely many zeros in it. We then construct a parameterized family of non-analytic functions with this same property. With the possible exception of a finite number of zeros off the critical line, every function in the family satisfies a Riemann Hypothesis. Moreover, when the parameter is not too large, they have about the same number of zeros as the zeta-function, their zeros are all simple, and they "repel". The structure of these functions makes the reason for the simplicity and repulsion of their zeros apparent and suggests a mechanism that might be responsible for the corresponding properties of the zeta-function's zeros. Computer evidence suggests that the zeros of functions in the family are remarkably close to those of the zeta-function (even for small values of the parameter), and we show that they indeed converge to them as the parameter increases. Furthermore, between zeros of the zeta-function, the moduli of functions in the family tend to twice the modulus of the zeta-function. Both assertions assume the Riemann Hypothesis. We end by discussing analogues for other L-functions and show how they give insight into the study of the distribution of zeros of linear combinations of L-functions.