Farey Series and the Riemann Hypothesis III.

In this paper we shall make a further study on the “the equivalence problem,” i.e., equivalent conditions to the Riemann hypothesis in terms of Farey series by developing a rather analytical (-arithmetical) method to establish unexpected short interval results, namely $$\frac{1} {5} - ,\frac{1} {6} - $$ results, therewith simultaneously clarifying the underlying reasons for $$\frac{1} {2} - ,\frac{1} {3} - \frac{1} {4} - $$ results obtained in Part I. [ABSTRACT FROM AUTHOR]