Approach to Riemann Hypothesis by Combined Commensurable Step Function Approximation with Bonnet Method
- 1 Institut für Physik, Humboldt-Universität, Berlin, Germany
Abstract
To the Riemann hypothesis, we investigate first the approximation by step-wise Omega functions Ω( u ) with commensurable step lengths u 0 concerning their zeros in corresponding Xi functions Ξ( z ). They are periodically on the y -axis with period proportional to inverse step length u 0 . It is found that they possess additional zeros off the imaginary y -axis and additionally on this axis and vanish in the limiting case u 0 → 0 in complex infinity. There remain then only the “genuine” zeros for Xi functions to continuous Omega functions which we call “analytic zeros” and which lie on the imaginary axis. After a short repetition of the Second mean-value (or Bonnet) approach to the problem and the derivation of operational identities for Trigonometric functions we give in Section 8 a proof for the position of these genuine “analytic” zeros on the imaginary axis by construction of a contradiction for the case off the imaginary axis. In Section 10, we show by a few examples that monotonically decreasing of the Omega functions is only a sufficient condition for the mentioned property of the positions of zeros on the imaginary axis but not a necessary one.
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