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

CERN

CH 1211 Geneva 23

Switzerland

**Abstract:**

We derive analytical expressions for the position of irreducible fractions
in the Farey sequence *F*_{N} 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 *F*_{N} 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*).

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