Congruences for L-Functions:
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Dordrecht
Springer Netherlands
2000
|
| Series: | Mathematics and Its Applications
511 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | In [Hardy and Williams, 1986] the authors exploited a very simple idea to obtain a linear congruence involving class numbers of imaginary quadratic fields modulo a certain power of 2. Their congruence provided a unified setting for many congruences proved previously by other authors using various means. The Hardy-Williams idea was as follows. Let d be the discriminant of a quadratic field. Suppose that d is odd and let d = PIP2· . . Pn be its unique decomposition into prime discriminants. Then, for any positive integer k coprime with d, the congruence holds trivially as each Legendre-Jacobi-Kronecker symbol (~) has the value + 1 or -1. Expanding this product gives ~ eld e:=l (mod4) where e runs through the positive and negative divisors of d and v (e) denotes the number of distinct prime factors of e. Summing this congruence for o < k < Idl/8, gcd(k, d) = 1, gives ~ (-It(e) ~ (~) =:O(mod2n). eld o |
| Physical Description: | 1 Online-Ressource (XII, 256 p) |
| ISBN: | 9789401595421 9789048154906 |
| DOI: | 10.1007/978-94-015-9542-1 |
Staff View
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| 500 | |a In [Hardy and Williams, 1986] the authors exploited a very simple idea to obtain a linear congruence involving class numbers of imaginary quadratic fields modulo a certain power of 2. Their congruence provided a unified setting for many congruences proved previously by other authors using various means. The Hardy-Williams idea was as follows. Let d be the discriminant of a quadratic field. Suppose that d is odd and let d = PIP2· . . Pn be its unique decomposition into prime discriminants. Then, for any positive integer k coprime with d, the congruence holds trivially as each Legendre-Jacobi-Kronecker symbol (~) has the value + 1 or -1. Expanding this product gives ~ eld e:=l (mod4) where e runs through the positive and negative divisors of d and v (e) denotes the number of distinct prime factors of e. Summing this congruence for o < k < Idl/8, gcd(k, d) = 1, gives ~ (-It(e) ~ (~) =:O(mod2n). eld o | ||
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Record in the Search Index
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| indexdate | 2024-07-10T01:21:15Z |
| institution | BVB |
| isbn | 9789401595421 9789048154906 |
| language | English |
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| spelling | Urbanowicz, Jerzy Verfasser aut Congruences for L-Functions by Jerzy Urbanowicz, Kenneth S. Williams Dordrecht Springer Netherlands 2000 1 Online-Ressource (XII, 256 p) txt rdacontent c rdamedia cr rdacarrier Mathematics and Its Applications 511 In [Hardy and Williams, 1986] the authors exploited a very simple idea to obtain a linear congruence involving class numbers of imaginary quadratic fields modulo a certain power of 2. Their congruence provided a unified setting for many congruences proved previously by other authors using various means. The Hardy-Williams idea was as follows. Let d be the discriminant of a quadratic field. Suppose that d is odd and let d = PIP2· . . Pn be its unique decomposition into prime discriminants. Then, for any positive integer k coprime with d, the congruence holds trivially as each Legendre-Jacobi-Kronecker symbol (~) has the value + 1 or -1. Expanding this product gives ~ eld e:=l (mod4) where e runs through the positive and negative divisors of d and v (e) denotes the number of distinct prime factors of e. Summing this congruence for o < k < Idl/8, gcd(k, d) = 1, gives ~ (-It(e) ~ (~) =:O(mod2n). eld o Mathematics Field theory (Physics) Functions of complex variables Functions, special Number theory Number Theory Field Theory and Polynomials Functions of a Complex Variable Special Functions Mathematik Kongruenz (DE-588)4164978-3 gnd rswk-swf L-Funktion (DE-588)4137026-0 gnd rswk-swf L-Funktion (DE-588)4137026-0 s Kongruenz (DE-588)4164978-3 s 1\p DE-604 Williams, Kenneth S. Sonstige oth https://doi.org/10.1007/978-94-015-9542-1 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Urbanowicz, Jerzy Congruences for L-Functions Mathematics Field theory (Physics) Functions of complex variables Functions, special Number theory Number Theory Field Theory and Polynomials Functions of a Complex Variable Special Functions Mathematik Kongruenz (DE-588)4164978-3 gnd L-Funktion (DE-588)4137026-0 gnd |
| subject_GND | (DE-588)4164978-3 (DE-588)4137026-0 |
| title | Congruences for L-Functions |
| title_auth | Congruences for L-Functions |
| title_exact_search | Congruences for L-Functions |
| title_full | Congruences for L-Functions by Jerzy Urbanowicz, Kenneth S. Williams |
| title_fullStr | Congruences for L-Functions by Jerzy Urbanowicz, Kenneth S. Williams |
| title_full_unstemmed | Congruences for L-Functions by Jerzy Urbanowicz, Kenneth S. Williams |
| title_short | Congruences for L-Functions |
| title_sort | congruences for l functions |
| topic | Mathematics Field theory (Physics) Functions of complex variables Functions, special Number theory Number Theory Field Theory and Polynomials Functions of a Complex Variable Special Functions Mathematik Kongruenz (DE-588)4164978-3 gnd L-Funktion (DE-588)4137026-0 gnd |
| topic_facet | Mathematics Field theory (Physics) Functions of complex variables Functions, special Number theory Number Theory Field Theory and Polynomials Functions of a Complex Variable Special Functions Mathematik Kongruenz L-Funktion |
| url | https://doi.org/10.1007/978-94-015-9542-1 |
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