P.S.: The sequence that you have given cannot happen because of the following reasons: (i) It is cyclic and there cannot be any periodicity in the lambda seque

P.S.: The sequence that you have given cannot happen because of the following reasons: (i) It is cyclic and there cannot be any periodicity in the lambda sequence. (This is proved in the paper) (ii) The sequence that you choose goes as N/3, this means that F(s) cannot be extended to the left beyond Re(s)=1. In fact, if such as sequence is possible than we have a disproof of RH. (iii) The sequence does not have the statistics of a coin toss. See Appendix VI of the Main Paper whose link is in the summary. The chi-squared statistics will be different. (iv) I have given a second Proof of equal probabilities: which does not depend on mapping (but only uses induction): https://www.researchgate.net/publication/324828748_A_Simple_Proof_That_Even_and_Odd_Numbers_of_Prime_Factors_Occur_with_Equal_Probabilities_in_the_Factor-ization_of_Integers_and_its_Implications_for_the_Riemann_Hypothesis https://www.researchgate.net/publication/324828748_A_Simple_... This second proof should overcome your misgivings. To repeat: I suggest you read Item (3) in the "SUMMARY" link I provided this outlines the main argument of the proof and the basic steps. Just to put things in proper perspective and credibility: I wish to say that the fact that the lambda sequence behaves like coin tosses have been actually verified by computation (chi-square fitting) over very long sequences. And also over the sequence of the first 176 trillion integers! See Appendix VI of Main paper for details. Thank You for your interest. KE