Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.5

Partial Franel Sums

R. Tomás
CH 1211 Geneva 23


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).

Full version:  pdf,    dvi,    ps,    latex    

Received September 2 2021; revised version received September 5 2021; September 12 2021; January 2 2022; January 11 2022. Published in Journal of Integer Sequences, January 12 2022.

Return to Journal of Integer Sequences home page