Looks like this is responded by Raghavan and this is what I had seen on research gate comments
Dr. Kumar's response to the above:
I certainly know that the lambda - sequence is fixed and unalterable because each lambda(n) is obtained by factorizing n into prime factors and then defining lambda(n)=+1 if there are even factors else, lambda(n) =-1 if n is a prime or has odd prime factors.
In the paper for very long sequences, the lambda-sequence is treated as one instance of a hypothetical random walk. If this analogy is true then the magnitude of L(N), which is the sum of the first N terms of lambda(n)’s, (where N is very large) can be likened to the expected distance travelled by a random walker in N steps which is given by C .N^(1/2) (see S. Chandrasekhar(1943)).
However, for this analogy to be really meaningful and accurate, one must prove the lambda(n), for large and arbitrary n, must satisfy the criteria: (i) equal probabilities of being +1 or -1 , (ii) the lambda-sequence has no cycle and (iii) unpredictability.
In the paper I provide mathematical proofs for all the above criteria, after which one can deduce the asymptotic expression for L(N) as C. N^(1/2+e). We then invoke (i) Littlewoods Theorem 1 (proved in the paper) and then (ii) use Khinchin (1924) and Kolmogorov’s (1929) law of the iterated logarithm, for evaluating the bound ‘e’ and to show that e tends to zero as N tends to infinity, thus finally proving R.H.
One last comment: Herrington quotes Borwein’s statement as an “Equivalence to RH”, in actuality the condition stated by Borwein (2008) is only a necessary condition for RH to be true. The additional criteria (above) needs to be satisfied and hence need to be proved as done in my paper.
