Nov 25, 2018 - where the zeta function ζ(s) has non-trivial zeros), to determine the configuration ... as Nâ â), determines the truth of the Riemann Hypothesis.
A Quick Road-Map (Overview) of the Proof This brief note gives a Road-Map (Overview) of the paper, in order that the pathway to the proof becomes quickly and clearly visible to the reader. Firstly, in this paper, we investigate the analytical properties of a complex function F(s) = (2s)/(s) (where s is a complex variable, and F(s) has poles exactly at those positions where the zeta function (s) has non-trivial zeros), to determine the configuration of the non-trivial zeros of (s) and show that they all lie on the critical line Re(s)=1/2, as Riemann had hypothesized. Our path soon leads us to investigate the sum of a certain sequence of integer functions (n). By definition, the function (n) takes the value +1 if n has an even number of prime factors, else it is equal to -1 if n is a prime or has an odd number of prime factors. It so turns out that the sum L(N) of the first N values of (n) (viz. n=1,2,3,..N), called Summatory Liouville function, plays a very critical role in the proof of the Riemann Hypothesis. It is then shown, that the asymptotic (large N) behaviour of L(N) determines the strip within which the zeros lie. To be more precise, we prove a theorem, by adapting Littlewood’s method, to show that if for large N, |L(N)| behaves as C Na , (where the exponent a is greater than or equal to 1/2 but less than 1), then F(s) is non-analytic in the strip centred on the line Re(s)=1/2 with half width |a – ½| . Our task is then reduced to proving a=1/2. Alternately stated, this means the asymptotic behaviour of the L(N), if shown to satisfy |L(N) | ≤ N , ε 0 as N ∞, (or more precisely for every ε >0, L(N).N 0 as N ∞), determines the truth of the Riemann Hypothesis. It is then shown that the Liouville sequence {(n), n=1,2,3,….} (obtained by factorizing every integer n ) behaves like a particular instance of an 1-D random walk, for large N: so indeed |L(N) | ≤ N , follows from the theory of random-walks, and the law of the iterated logarithm is invoked to show ε 0 as N ∞, thus proving the Riemann Hypothesis. To do this, we examine the sequence {(N)} = {(1),(2), (3),..……,(n), … n=1,2,3,….N}. Since each (n)= +1 or -1, it is then observed that the sequence { (N)} superficially appears to look like the result of an experiment when a coin was tossed N times (with H=1 and T=-1). But it is well known that if one tosses a coin N times and then if one counts the excess, k(N), of Heads over Tails in N tosses, then the expectation of |k(N)|=C . N 1/2, for large N. Similarly, the distance s(N) traversed in a random walk has the same expression: expectation of |s(N)|=C. N1/2, Then since L(N) counts the excess of +1’s over -1’s in the sequence {(N)}, then comparing: |L(N) | = C Na with |k(N)|=C . N1/2 or with |s(N)|=C. N1/2, one is lead to conclude that one can expect the result a=1/2, i.e. the expectation of |L(N)|=C . N1/2 if the (n) behaves like a coin toss for large arbitrary values of n. This will only happen if the (n) (which is an arithmetical function) satisfies the following three criteria (i) Equal probabilities: for a randomly chosen n, (n) has an equal probability of being+1 or-1 (just like a coin toss) (ii) The sequence {(N)} will not have a cycle and (iii) Unpredictability : that is there exists no finite integer k, such that (n) is predictable from its previous k values, for arbitrary n. We
then prove by using the properties of prime factorization of integers that the (n) obey the 3 criteria and hence the (n) behaves statistically like coin tosses and L(N) is like a random walk and hence it is correcti to deduce that the exponent a=1/2. However, as the natural number sequence is merely one realization of a random walk, for the particular case we can only say that |L(N) | ≤ N , where ε represents the deviation from expectation. We then invoke the Law of the Iterated logarithm Khinchin and Kolmogorov to show that ε 0 as N ∞, thus proving the Riemann Hypothesis. The paper contains alternative proofs of some of the crucial theorems for resolving RH, as well as reports on some of the extensive numerical computations in support of some of the theorems. In particular, the criteria (ii) and (iii) for the coin-toss can be replaced by a single one of the mutual independences of the (n)’s in any finite sub-sequence of the natural numbers, which is proved, for large n, in Appendix IV of the paper using elementary arithmetic principles.
K. Eswaran, November 25, 2018
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Since the aforesaid criteria are all that is required to derive the random walk expression s(N) for large N, (see S. Chandrasekhar (1943)), the same criteria are then sufficient to deduce the expression L(N) for large N.
