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| Format: | Preprint |
| Published: |
2018
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1802.07792 |
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| _version_ | 1866910406457425920 |
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| author | Tomas, Rogelio |
| author_facet | Tomas, Rogelio |
| contents | 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 $O(\log(N)δ_B(\log N))$, where $δ_B(x)$ is a decreasing function. Other partial Franel sums are also explored. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_07792 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Partial Franel sums Tomas, Rogelio Number Theory 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 $O(\log(N)δ_B(\log N))$, where $δ_B(x)$ is a decreasing function. Other partial Franel sums are also explored. |
| title | Partial Franel sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/1802.07792 |