Partial Franel Sums

We derive analytical expressions for the position of irreducible fractions in the Farey sequence FN of order N for a particular choice of N, obtaining an asymptotic behavior with a lower error bound than in previous results when these fractions are in the vicinity of 0/1, 1/2, or 1/1. Franel’s famous formulation of Riemann’s hypothesis uses the summation of distances between irreducible fractions and evenly spaced points in [0, 1]. We define “partial Franel sum” as a summation of these distances over a subset of fractions in FN and we demonstrate that the partial Franel sum in the range [0, i/N], with N = lcm(1, 2, . . . , i), grows strictly slower than O(log N).