Fast Computing Formulas for some Dirichlet L-Series
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915796809154560 |
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| author | Zuniga, Jorge |
| author_facet | Zuniga, Jorge |
| contents | For $χ_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,χ_k)$, expressed as linear combinations of Hurwitz $ζ$ functions, are found for $s=2,3$ and some selected values of $k$. By using a merged approach between the Wilf$-$Zeilberger method and a Dougall$'$s $_5H_5$ technique, new proven accelerated series of hypergeometric$-$type are derived for specific Hurwitz $ζ$ function values. These fast series that are computed by means of the binary splitting algorithm, enter into the reduced identities found producing very efficient formulas to compute selected $L-$function values. The new algorithms include $L_k(2)$ for $k = -4$ Catalan's constant, $-7, -8, -15, -20, -24$ together with $L_k(3)$ for $k = 1$ Apery's constant, $5, 8$ and $12$. Formulas were tested and verified up to 100 million decimal places for each $L-$value. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_12495 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fast Computing Formulas for some Dirichlet L-Series Zuniga, Jorge Number Theory 11M06, 11M35, 40-08, 33F10, 33C90 For $χ_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,χ_k)$, expressed as linear combinations of Hurwitz $ζ$ functions, are found for $s=2,3$ and some selected values of $k$. By using a merged approach between the Wilf$-$Zeilberger method and a Dougall$'$s $_5H_5$ technique, new proven accelerated series of hypergeometric$-$type are derived for specific Hurwitz $ζ$ function values. These fast series that are computed by means of the binary splitting algorithm, enter into the reduced identities found producing very efficient formulas to compute selected $L-$function values. The new algorithms include $L_k(2)$ for $k = -4$ Catalan's constant, $-7, -8, -15, -20, -24$ together with $L_k(3)$ for $k = 1$ Apery's constant, $5, 8$ and $12$. Formulas were tested and verified up to 100 million decimal places for each $L-$value. |
| title | Fast Computing Formulas for some Dirichlet L-Series |
| topic | Number Theory 11M06, 11M35, 40-08, 33F10, 33C90 |
| url | https://arxiv.org/abs/2601.12495 |