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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2506.10321 |
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| _version_ | 1866911671961780224 |
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| author | Zuniga, Jorge |
| author_facet | Zuniga, Jorge |
| contents | This work extends the results of the preprint Ramanujan type Series for Logarithms, Part I, arXiv:2506.08245, which introduced single hypergeometric type identities for the efficient computing of $\log(p)$, where $p\in\mathbb{Z}_{>1}$. We present novel formulas for arctangents and methods for a very fast multiseries evaluation of logarithms. Building upon a $\mathcal{O}((p-1)^{6})$ Ramanujan type series asymptotic approximation for $\log(p)$ as $p\rightarrow1$, formulas for computing $n$ simultaneous logarithms are developed. These formulas are derived by solving an integer programming problem to identify optimal variable values within a finite lattice $\mathbb{Z}^{n}$. This approach yields linear combinations of series that provide: (i) highly efficient formulas for single logarithms of natural numbers (some of them were tested to get more than $10^{11}$ decimal places) and (ii) the fastest known hypergeometric formulas for multivalued logarithms of $n$ selected integers in $\mathbb{Z}_{>1}$. An application of these results was to extend the number of decimal places known for log(10) up to 2.0$\cdot$10$^{12}$ digits (June 06 2025). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast Ramanujan--type Series for Logarithms. Part II Zuniga, Jorge Number Theory Numerical Analysis 65B10 33C20 33C90 90C11 65C05 91G60 This work extends the results of the preprint Ramanujan type Series for Logarithms, Part I, arXiv:2506.08245, which introduced single hypergeometric type identities for the efficient computing of $\log(p)$, where $p\in\mathbb{Z}_{>1}$. We present novel formulas for arctangents and methods for a very fast multiseries evaluation of logarithms. Building upon a $\mathcal{O}((p-1)^{6})$ Ramanujan type series asymptotic approximation for $\log(p)$ as $p\rightarrow1$, formulas for computing $n$ simultaneous logarithms are developed. These formulas are derived by solving an integer programming problem to identify optimal variable values within a finite lattice $\mathbb{Z}^{n}$. This approach yields linear combinations of series that provide: (i) highly efficient formulas for single logarithms of natural numbers (some of them were tested to get more than $10^{11}$ decimal places) and (ii) the fastest known hypergeometric formulas for multivalued logarithms of $n$ selected integers in $\mathbb{Z}_{>1}$. An application of these results was to extend the number of decimal places known for log(10) up to 2.0$\cdot$10$^{12}$ digits (June 06 2025). |
| title | Fast Ramanujan--type Series for Logarithms. Part II |
| topic | Number Theory Numerical Analysis 65B10 33C20 33C90 90C11 65C05 91G60 |
| url | https://arxiv.org/abs/2506.10321 |