Fast formulas for the Hurwitz values $ζ(2,a)$ and $ζ(3,a)$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914188429885440 |
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| author | Guillera, Jesús |
| author_facet | Guillera, Jesús |
| contents | We prove two fast formulas for the Hurwitz values $ζ(2,a)$ and $ζ(3,a)$ respectively with the help of the WZ method. In them $(a)_n$ denotes the rising factorial or Pochhammer's symbol defined by $(a)_0=1$ and $(a)_n=a(a+1)\cdots(a+n-1)$ for positive integers $n$. The Huwitz $ζ$ function is defined by $ζ(s,a)=ζ(0,s,a)=\sum_{k=0}^{\infty} (k+a)^{-s}$. In addition, we can use these fast evaluations to compute also in a rapid way Dirichlet values of the kinds $L_χ(2)$ and $L_χ(3)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01975 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast formulas for the Hurwitz values $ζ(2,a)$ and $ζ(3,a)$ Guillera, Jesús Number Theory We prove two fast formulas for the Hurwitz values $ζ(2,a)$ and $ζ(3,a)$ respectively with the help of the WZ method. In them $(a)_n$ denotes the rising factorial or Pochhammer's symbol defined by $(a)_0=1$ and $(a)_n=a(a+1)\cdots(a+n-1)$ for positive integers $n$. The Huwitz $ζ$ function is defined by $ζ(s,a)=ζ(0,s,a)=\sum_{k=0}^{\infty} (k+a)^{-s}$. In addition, we can use these fast evaluations to compute also in a rapid way Dirichlet values of the kinds $L_χ(2)$ and $L_χ(3)$. |
| title | Fast formulas for the Hurwitz values $ζ(2,a)$ and $ζ(3,a)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.01975 |