{"product_id":"congruences-for-l-functions-von-j-urbanowicz-kenneth-s-williams","title":"Congruences for L-Functions","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 \u0026lt; k \u0026lt; Idl\/8, gcd(k, d) = 1, gives ~ (-It(e) ~ (~) =:O(mod2n). eld o\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9780792363798\"\u003e\u003ch3\u003e\u003c\/h3\u003e\u003c\/div\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9789048154906\"\u003e\u003ch3\u003e\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Hardcover - 9780792363798","offer_id":50726462150,"sku":"9780792363798","price":53.49,"currency_code":"EUR","in_stock":true},{"title":"Softcover - 9789048154906","offer_id":39415545004125,"sku":"9789048154906","price":53.49,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/d023c7f4-61ae-4e6f-893c-38baa0a92c5b.jpg?v=1772088565","url":"https:\/\/shop.autorenwelt.de\/products\/congruences-for-l-functions-von-j-urbanowicz-kenneth-s-williams","provider":"Autorenwelt Shop","version":"1.0","type":"link"}