Partial Franel sums

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.