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.
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.