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"?)
May 29, 2019
I thank you for your detailed commentary and queries.
My replies are as follows.
to your point [1]:
The point I make is that knowing lambda(N) (N large) and possibly, lambda(N-1), lambda(N-2), ..., lambda(N-K) For some fixed K. It is not possible to predict ALL the other lambdas higher than N. This is proved in the paper and can be intuitively understood because one cannot predict when the next prime occurs. Thus lambda(n) behaves like c(N) where c(N) is the Nth toss of a coin tossing experiment.
It is only necessary to prove that lambda(N) behaves (approximately) like a coin toss for large N. As explained this requirement is from Littlewood's theorem (proved in the paper). It is not necessary that the lambdas are perfectly random, of course they are not. Example, lambda(2n)= -lambda(n)) but if you choose n very large say n=10^100, then 2n will be very far away from n. Thus the 'randomness' in a very large sequence of lambdas is still practically undisturbed; the unpredictable occurrence of new primes also contribute toward 'randomness' - this is especially so for very long sequence of length N; To prove RH we only need this behaviour for N near infinity to satisfy Littlewood's Theorem. (Appendix VI actually verifies this phenomena over very large sequences). In a paragraph below I show how if M is large it is impossible to predict lambda(M+1) knowing only lambda(M) with out actually calculating it by using all the info on primes below Sqrt(M).
Regarding your argument where you construct an example in your para [1], of a sequence of lambdas and flip the signs of lambdas according to your specific rule is not permissible. Because you must remember that the lambda(n) take on values +1 or -1 depending how many factors the particular integer n has. So you cannot 'flip' their signs arbitrarily or as per your prescription, without actually demonstrating that your prescription does not violate the rules of arithmetic or the rules of factorization. The lambda's are arithmetical functions and obey some rules. I suggest that you study the proof of unpredictability (independence). it will convince you. NOTE: If you have a method of predicting lambda(n) from (say) its k previous values (k fixed), then this (hypothetical) method should work for ALL n, clearly it then means there exists some function f(x1,x2,..xK) which predicts the value according to the hypothetical method. The paper proves that such a function cannot exist, (this is my second proof of independence; see page 13 para 2(a) of Main Paper or slides 54-58 in my IITM Lecture).
In the next paragraph (where you talk about towers), you again "suppose" the lambdas in the towers to have some properties - you cannot suppose these things without actually proving that your "suppositions" do not violate the rules of factorization and arithmetic that govern the values of lambda(n).
Each tower contains integers whose lambdas alternate + - + - etc. I know that there are infinite number of towers; this does not make any difference to the proof - they are all countably infinite. The important thing is that each integer occurs only once in some tower and does not occur in any other tower.
To prevent ourselves being inveigled by Cantor-like arguments (which imposes "pre-suppositions" on the properties of the lambdas), I suggest that we get out of this "mapping" mind set. I have given an independent proof (which should be studied) of equal probabilities which only uses induction and the assumption that each even integer is followed by an odd integer which precedes another even integer (i.e the number of even integers is equal to the number of odd integers). Because of the latter reason, I believe this proof is water-tight. This proof is given in:
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_...
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Regarding your last point, I agree I was just loosely stating matters when I put things like: "L(n) ~ Cn^1/2"; (sorry!).
The actual rigorous requirement for the proof of RH is that as N tends to infinity one must have Mod[ L(N)] < C. N^(0.5 + e), where e is small but positive. The inequality must hold in the limit for very large values of N not a particular value, so Littlewood's condition actually specifies a bound; you can read the statement of Littlewood's Theorem given in the paper.
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The better way to intuitively understand as to why the lambdas behave like coin tosses is to consider the following argument:
(a) Each integer, n, can be uniquely factorized into primes, (b) So the larger the value of n, the number and variety of primes become more, because there will be more number of primes. So if is a large number M, (say M=10^100) and if lambda(M)= +1, meaning M has an even number of prime factors, then it is impossible to say whether the next number (M+1) will have lambda(M+1) +1 or -1 without actually calculating it. You will actually see that the primes which factorize M are different from the primes that factorize (M+1), in fact they will be completely different. That is why the lambdas behave like "coin tosses".
Another way of "seeing it", is to compare a binomial distribution (of coin tosses) with a Gaussian distribution (this comparison is valid for large number of samples). It is well known that: if a random variable X (say height of a person) depends on very many unknowns variables like diet, height of parents, environment etc.. then the random variable X behaves like a Gaussian. The same for coin tosses (coin toss depends on the very many ways it can be tossed) and a similar thing happens here because the lambda(n) depends on the factorization of n involving many primes(variables).
In Appendix VI (I strongly recommend that you read Appendix VI in its entirety), I have taken very large sequences of consecutive lambdas and I have actually shown by doing a chi-squared fit and comparing each sequence with a binomial sequence of the same size, that they are indistinguishable (statistically speaking) from coin tosses. In fact if one considers, the sequence of the first 176 trillion consecutive lambdas, the sequence is indistinguishable from coin tosses. My paper gives an explanation (reason) for such a phenomena and this reason lead to the proof of RH.
As far as I can tell, I think I have answered your Queries; the answers are also contained in greater detail in Lecture 5 to 7 in the Lecture Series of Vinayak Eswaran, whose Link is given in the last paragraph of this note: "SEVEN LECTURES". I very strongly suggest you study them and then you can read the papers and Notes contained in "THE SUMMARY" and my "Pathway" (see my previous write-up for the links in Researchgate under my name).
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Yes "dilitannte" would have been the correct word. By using the word "trust" I only meant that you should trust me just for the duration of time that it would take you to have read my arguments and to have studied carefully my papers and the Notes (as suggested in the previous para) till the very end, with an open mind. Of course, after this is done, (but not before), I expect that my reasoning will be subject to the closest and strictest scrutiny.
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I thank You once again for all the interest you have taken.
SEVEN LECTURES:
Since this blog will be read by others. I just wish to say (for the curious non-expert): that my brother Vinayak Eswaran (formerly professor at IIT Kanpur and presently a professor at IIT Hyderabad), has taken it upon himself to upload a series of Seven Lectures entitled "Seven Lectures on the Proposed Proof by Kumar Eswaran of the Riemann's Hypothesis".See Link:
https://www.youtube.com/playlist?list=PLRsxymPrOKUAk3eXhK9FZdGAPmgGGrhfb&disable_polymer=true https://www.youtube.com/playlist?list=PLRsxymPrOKUAk3eXhK9FZ...
These lectures are set at a level which can be understood by a STEM undergraduate student.
Regards
K.E
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