Correction and Supplement to Approach for a Proof of Riemann Hypothesis by Second Mean-Value Theorem

Abstract

From the theorem 1 formulated in [1], a set of functions of measure zero within the set of all corresponding functions has to be excluded. These are the cases where the Omega functions Ω(u) are piece-wise constant on intervals of equal length and non-increasing due to application of second mean-value theorem or, correspondingly, where for the Xi functions Ξ(z) the functions Ξ(y)y are periodic functions on the imaginary axis y with z=x+i y . This does not touch the results for the Omega function to the Riemann hypothesis by application of the second mean-value theorem of calculus and the majority of other Omega functions in the suppositions, but makes their derivation correct. The corresponding calculations together with a short recapitulation of the main steps to the basic equations for the restrictions of the mean-value functions and the application to piece-wise constant Omega functions (staircase functions) are represented.