Functions and Analysis


Zeros of the Riemann Zeta Function Within the Critical Strip and Off the Critical Line

Authors: Jonathan W. Tooker

In a recent paper, the author demonstrated the existence of real numbers in the neighborhood of infinity. It was shown that the Riemann zeta function has non-trivial zeros in the neighborhood of infinity but none of those zeros lie within the critical strip. While the Riemann hypothesis only asks about non-trivial zeros off the critical line, it is also an open question of interest whether or not there are any zeros off the critical line yet still within the critical strip. In this paper, we show that the Riemann zeta function does have non-trivial zeros of this variety. The method used to prove the main theorem is only the ordinary analysis of holomorphic functions. After giving a brief review of numbers in the neighborhood of infinity, we use Robinson's non-standard analysis and Eulerian infinitesimal analysis to examine the behavior of zeta on an infinitesimal neighborhood of the north pole of the Riemann sphere. After developing the most relevant features via infinitesimal analysis, we will proceed to prove the main result via standard analysis on the Cartesian complex plane without reference to infinitesimals.

Comments: 22 Pages. 1 color figure

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Submission history

[v1] 2019-12-02 11:50:41

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