Euler constants from primes in arithmetic progression
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918146677407744 |
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| author | Languasco, Alessandro Moree, Pieter |
| author_facet | Languasco, Alessandro Moree, Pieter |
| contents | Many Dirichlet series of number theoretic interest can be written as a product of generating series $ζ_{\,d,a}(s)=\prod\limits_{p\equiv a\pmod{d}}(1-p^{-s})^{-1}$, with $p$ ranging over all the primes in the primitive residue class modulo $a\pmod{d}$, and a function $H(s)$ well-behaved around $s=1$. In such a case the corresponding Euler constant can be expressed in terms of the Euler constants $γ(d,a)$ of the series $ζ_{\,d,a}(s)$ involved and the (numerically more harmless) term $H'(1)/H(1)$. Here we systematically study $γ(d,a)$, their numerical evaluation and discuss some examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16547 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Euler constants from primes in arithmetic progression Languasco, Alessandro Moree, Pieter Number Theory Many Dirichlet series of number theoretic interest can be written as a product of generating series $ζ_{\,d,a}(s)=\prod\limits_{p\equiv a\pmod{d}}(1-p^{-s})^{-1}$, with $p$ ranging over all the primes in the primitive residue class modulo $a\pmod{d}$, and a function $H(s)$ well-behaved around $s=1$. In such a case the corresponding Euler constant can be expressed in terms of the Euler constants $γ(d,a)$ of the series $ζ_{\,d,a}(s)$ involved and the (numerically more harmless) term $H'(1)/H(1)$. Here we systematically study $γ(d,a)$, their numerical evaluation and discuss some examples. |
| title | Euler constants from primes in arithmetic progression |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.16547 |