New Analytical Formulas for the Rank of Farey Fractions and Estimates of the Local Discrepancy

New analytical formulas are derived for the rank and the local discrepancy of Farey fractions. The new rank formula is applicable to all Farey fractions and involves sums of a lower order compared to the searched one. This serves to establish a new unconditional estimate for the local discrepancy of Farey fractions that decrease with the order of the Farey sequence. This estimate improves the currently known estimates. A new recursive expression for the local discrepancy of Farey fractions is also given. A second new unconditional estimate of the local discrepancy of any Farey fraction is derived from a sum of the Mertens function, again, improving the currently known estimates.