Finite Euler products and the Riemann Hypothesis
| dc.creator | Gonek, S. M. | |
| dc.date | 2007-04-26 | |
| dc.date.accessioned | 2026-07-07T07:58:19Z | |
| dc.date.available | 2026-07-07T07:58:19Z | |
| dc.description | 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. | |
| dc.description | 4 figures | |
| dc.identifier | https://arxiv.org/abs/0704.3448 | |
| dc.identifier | http://arxiv.org/abs/0704.3448 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/127963 | |
| dc.subject | Number Theory | |
| dc.subject | Complex Variables | |
| dc.subject | 11M26 | |
| dc.title | Finite Euler products and the Riemann Hypothesis | |
| dc.type | text |