On the Pathway to the Riemann Hypothesis
ABSTRACT :
The celebrated Riemann Hypothesis (RH) is the prediction that all the non-trivial zeros of the zeta function ζ ( s ) lie on a vertical line in the complex s-plane at Re( s ) =1/2 (the critical line). This has been an unsolved conundrum for the last 160 years. In this lecture a new approach from an entirely different direction is taken to resolve the hypothesis. It is shown that the location of the zeros of the zeta function is best determined by studying another function (which is called F(s)), which has poles at exactly the places where the zeta function has zeros. A study of the analyticity of F(s), leads to the study of an infinite sequence (the “factorization -sequenceâ€) of numbers each of which is either +1 or -1. It is shown that the analyticity of F(s), depends on the behavior of this factorization - sequence as the number of terms N in the sequence tend to infinity. It is then demonstrated that the Riemann Hypothesis would be true if the sequence behaves like a random sequence of N coin tosses (or like a 1-dimensional random walk of N steps), as N tends to infinity. The proofs, surprisingly could all be done form first principles. In this lecture, special efforts will be made so that it can be understood by non-specialists and by people in engineering or science.