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Thanks a great deal for your comments. You have been reading almost the very first version of the paper. The correct version of the paper is: https://www.rese

• K_Eswaran 7y ago
  The Link to Item (4) was missed it is given below: (4) https://www.researchgate.net/project/The-Dirichlet-Series-for-the-Liouville-Function-and-the-Riemann-Hypothesis https://www.researchgate.net/project/The-Dirichlet-Series-fo...
• K_Eswaran 7y ago
  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
• gjm11 7y ago
  The version I linked to was (so far as I could tell) the latest on the arXiv, posted a couple of years after the first version. I've now looked at the latest version and it seems to me it still has much the same problems as before, just hidden a little better. As you say, your argument depends on having the Liouville series behave like a series of independent coin tosses. But you never really show that it does. In your "Appendix IV" you argue, e.g., that if m,n are coprime then lambda(m),lambda(n) are "independent". Strictly speaking, that's nonsense: independence is a relationship between _random variables_ and lambda(m),lambda(n) are not random variables at all. There might be some theorem that says something like "if you consider a long enough series of these, then their statistics are very close to those you'd get from independent random variables", but your argument doesn't show anything at all like that. (You do argue -- the argument seems clearly unsound to me[1], but the conclusion, or something like it, is probably correct -- that for fixed L and large enough N you can't deduce lambda(N) from lambda(N-1),...,lambda(N-L). But this has nothing at all to do with the relevant notion of statistical independence.) [1] Why is it unsound? Well, several things look wrong with it. Here's one of them: you say, in effect, that you can only work out lambda(N) from earlier values if you have lambda(some divisor or k). But clearly that isn't true; e.g., if N is a multiple of 3 then you have lambda(N) = lamda(2N/3). The "arithmetical proof" in Appendix V is exactly the same as before, and remains unsound for the same reason: it doesn't distinguish between "if L(n) ~ Cn^a then a=1/2" and "L(n) ~ Cn^1/2". The second of these is in fact clearly false, as I said before, because for some n L(n) is very close to zero. The errors being made here are not, it seems to me, the sort of superficial ones that just indicate glitches in the exposition of a basically-sound underlying argument, and that need patching over. They indicate fundamental misunderstandings of how this stuff works. A couple of other wrong things -- but, to be clear, the point isn't that a few specific statements are wrong, but that you don't have a firm grasp of just what it is you need to prove: In section 3.1 of your "pathway" document you say that it will be enough if the function lambda satisfies three conditions: (1) +1 and -1 equally common, (2) sequence not periodic, (3) can't predict new values of lambda from previous values. This is simply not true. For instance, consider a sequence defined as follows. Start with the sequence +-++--++++---- etc. (i.e., runs of length 1,1,2,2,4,4,8,8, etc.) This has properties 1 and 2, and it "almost" has property 3. Now perturb it by flipping each sign with probability 1/8 (either actually randomly, or using a deterministic-but-random-like sequence -- e.g., if you consider that the actual lambda(n) form such a sequence then you can flip the sign of the n'th term iff lambda(3n), lambda(3n+1), lambda(3n+1) are all +1). This still satisfies properties 1 and 2, and now clearly satisfies property 3 as well. But it doesn't at all satisfy the condition you need, of growth no faster than about sqrt(n): if you pick n=3.2^k-2 (i.e., at the end of a run of +) then the sum is approximately n/4. In section 5.2 of your main paper, you give two slightly more refined versions of the "twin" argument from older versions of the paper: it's not just that we can pair up numbers so that lambda(n),lambda(n') have opposite signs, but that we can divide the positive integers into subsets (the "towers") each of which has alternating signs. Or: ... into subsets which we can pair up with one another so that corresponding numbers in paired subsets have opposite signs. But these conditions, even taken together, don't have the consequence you want. Suppose we have any sequence of +,- at all, provided it has infinitely many of both signs; it can be decomposed into subsets satisfying those conditions, as follows. We start with all "towers" (of course they aren't "towers" in precisely your sense) T1, T2, ..., empty, and a "next sign needed" for each tower of +,-,+,-,+,-, etc. Now go through our sequence in order: n=1,n=2,n=3,..., assigning each number in turn to the "highest-priority" tower it's eligible for. Highest-priority means smallest value of (tower number) times (1 + number of numbers already assigned to tower). I claim we always assign infinitely many numbers to each tower. If not, consider the towers that only ever get finitely many numbers, and suppose tower k is the one for which (tower number) times (1 + max numbers ever assigned) is minimal. Then once n is large enough tower k will have highest priority, and we have both infinitely many + and infinitely many -, so tower k will get another number assigned to it, contradiction. I profoundly disagree with what you say about trust. "Nullius in verba" (https://en.wikipedia.org/wiki/Nullius_in_verba https://en.wikipedia.org/wiki/Nullius_in_verba)! Being a serious researcher doesn't mean that your work is exempt from checking or immune to serious error. I'm happy to trust that you mean what you say and are trying to get the mathematics right, but no one is entitled to be trusted to have actually got it right. (Incidentally, "philanderer" doesn't mean what I think you think it means. A philanderer is a man who is in the habit of flirting with (or more-than-flirting with) a lot of women. Perhaps you're looking for a word like "dilettante"?)