Partial Franel sums

TL;DR

This paper derives analytical expressions for the positions of fractions in Farey sequences, analyzes their asymptotic behavior, and explores partial Franel sums, providing new bounds and growth estimates related to Riemann hypothesis connections.

Contribution

It introduces new formulas for the positions of fractions in Farey sequences and analyzes the asymptotic growth of partial Franel sums with improved error bounds near key fractions.

Findings

Partial Franel sums grow as O(log(N)δ_B(log N)) in certain ranges.

New asymptotic bounds are established for fractions near 0/1, 1/2, and 1/1.

Analytical expressions for fraction positions in Farey sequences are derived.

Abstract

Analytical expressions are derived for the position of irreducible fractions in the Farey sequence $F_N$ of order $N$ for a particular choice of $N$. The asymptotic behaviour is derived obtaining 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 $[0,1]$. A partial Franel sum is defined here as a summation of these distances over a subset of fractions in $F_N$. The partial Franel sum in the range $[0, i/N]$, with $N={\rm lcm}(1,2,...,i)$ is shown here to grow as $O(\log(N)\delta_B(\log N))$, where $\delta_B(x)$ is a decreasing function. Other partial Franel sums are also explored.