Partial Franel Sums
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
Full version: pdf,
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.
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