Cargando…

Partial Franel Sums

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 famou...

Descripción completa

Detalles Bibliográficos
Autor principal: Tomás, R.
Lenguaje:eng
Publicado: 2022
Materias:
Acceso en línea:http://cds.cern.ch/record/2851291
_version_ 1780977110329851904
author Tomás, R.
author_facet Tomás, R.
author_sort Tomás, R.
collection CERN
description 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 O(log N).
id cern-2851291
institution Organización Europea para la Investigación Nuclear
language eng
publishDate 2022
record_format invenio
spelling cern-28512912023-08-09T10:54:02Zhttp://cds.cern.ch/record/2851291engTomás, R.Partial Franel SumsMathematical Physics and MathematicsWe 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 O(log N).arXiv:1802.07792oai:cds.cern.ch:28512912022
spellingShingle Mathematical Physics and Mathematics
Tomás, R.
Partial Franel Sums
title Partial Franel Sums
title_full Partial Franel Sums
title_fullStr Partial Franel Sums
title_full_unstemmed Partial Franel Sums
title_short Partial Franel Sums
title_sort partial franel sums
topic Mathematical Physics and Mathematics
url http://cds.cern.ch/record/2851291
work_keys_str_mv AT tomasr partialfranelsums