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+iy. 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.
Highlights
In [1], we considered Xi functions Ξ ( z ) with respect to their zeros which by means of Omega functions Ω (u ) are representable in the form (2.1)
By an improved treatment of the compatibility of the two conditions (2.24) for x ≠ 0, it became clear that the piece-wise constant Omega functions Ω (u ) with equal interval lengths of constancy corresponding to periodicity of the functions Ξ y on the imaginary axis have to be excluded from the theorem
After displacement of the critical line to the imaginary axis x = 0 in a Xi function Ξ ( z ) for easier work we come to a form (2.1) with the following special function Ω (u )
Summary
In [1], we considered Xi functions Ξ ( z ) with respect to their zeros which by means of Omega functions Ω (u ) are representable in the form (2.1). By an improved treatment of the compatibility of the two conditions (2.24) for x ≠ 0, it became clear that the piece-wise constant Omega functions Ω (u ) with equal interval lengths of constancy corresponding to periodicity of the functions Ξ (iy) y on the imaginary axis have to be excluded from the theorem. These Omega functions form a set of measure zero within the set of all possible Omega functions and this is not relevant for the Omega function to the Riemann hypothesis and does not spoil its proof.
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