שמור ב:
| מחבר ראשי: | |
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| פורמט: | Preprint |
| יצא לאור: |
2000
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| נושאים: | |
| גישה מקוונת: | https://arxiv.org/abs/math/0010286 |
| תגים: |
הוספת תג
אין תגיות, היה/י הראשונ/ה לתייג את הרשומה!
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| _version_ | 1866908599938187264 |
|---|---|
| author | Holden, Joshua |
| author_facet | Holden, Joshua |
| contents | In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields k_{0}, using the values of the zeta function ζ_{k_{0}} at negative integers as our ``higher Bernoulli numbers''. In the case where k_{0} is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (The author would like to thank Henri Cohen for suggesting an analysis of the second formula.) We briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of k_{0}-irregularity (more generally, ``quadratic irregularity'') of a prime number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0010286 |
| institution | arXiv |
| publishDate | 2000 |
| record_format | arxiv |
| spellingShingle | Comparison of algorithms to calculate quadratic irregularity of prime numbers Holden, Joshua Number Theory Numerical Analysis 11Y40, 11Y60, 11Y16, 11B68 (Primary) 11R42, 11R29, 94A60, 11R18 (Secondary) In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields k_{0}, using the values of the zeta function ζ_{k_{0}} at negative integers as our ``higher Bernoulli numbers''. In the case where k_{0} is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (The author would like to thank Henri Cohen for suggesting an analysis of the second formula.) We briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of k_{0}-irregularity (more generally, ``quadratic irregularity'') of a prime number. |
| title | Comparison of algorithms to calculate quadratic irregularity of prime numbers |
| topic | Number Theory Numerical Analysis 11Y40, 11Y60, 11Y16, 11B68 (Primary) 11R42, 11R29, 94A60, 11R18 (Secondary) |
| url | https://arxiv.org/abs/math/0010286 |