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By E.M. Chirka

One carrier arithmetic has rendered the 'Et moi, .. " si j'avait so remark en revenir, human race. It has placed good judgment again je n'y semis element aile.' Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded non­ The sequence is divergent; as a result we can be sense'. capable of do anything with it Eric T. Bell o. Heaviside arithmetic is a device for inspiration. A hugely beneficial instrument in an international the place either suggestions and non­ linearities abound. equally, all types of elements of arithmetic function instruments for different elements and for different sciences. employing an easy rewriting rule to the quote at the correct above one unearths such statements as: 'One carrier topology has rendered mathematical physics .. .'; 'One provider common sense has rendered com­ puter technological know-how .. .'; 'One provider class idea has rendered arithmetic .. .'. All arguably real. And all statements available this manner shape a part of the raison d'etre of this sequence.

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But we could also have taken the O-lattice L = EndA (P ), where P = Oα is any proper O-submodule of E, with α ∈ E ∗ . The resulting stabilizer M = AutA (P ) × O∗ /∆A∗ in G(k) is then the conjugate of M by the image of α in T (k) = E ∗ /k ∗ . This gives an action of the quotient group E ∗ /k ∗ · O∗ on the open compact subgroups M we have defined, and hence on their fixed lines (π ⊗ χ)M = v . We now turn to the construction of M in the case when the character χ is unramified. The Hermitian lattice L = OE + OE · w is determined by φ(w, w) in A − {0}, up to multiplication by NO∗E .

This was done by Oesterl´e in 1985 [O] who made Theorem 1 explicit. He proved that for CLASS NUMBER PROBLEM FOR IMAGINARY QUADRATIC FIELDS 27 (D, 5077) = 1, h(D) > 1 log |D| 55 1− p|D, p=|D| √ 2 p p+1 , which allowed one to solve the class number 3 problem. More recently, using the above methods, Arno [A] (1992), solved the class number four problem, and subsequently, work with Robinson and Wheeler [ARW] (1998), and work of Wagner [Wag] (1996) gave a solution to Gauss’ class number problem for class numbers 5, 6, 7.

By f (s) ∼ g(s) in a region s ∈ R ⊂ C we mean that there exists a small ε > 0 such that |f (s)−g(s)| < ε in the region R. Here we are appealing to the standard use of approximate functional equations which allow one to replace an L-function by a short (square root of conductor) sum of its early Dirichlet coefficients. This is the basis for the so called zero repelling effects (Deuring–Heilbronn phenomenon) associated to imaginary quadratic fields with small class number. For example, if h(D) = 1 and D → −∞, and D1 is a fixed discriminant of a quadratic field, then we expect that for Re(s) > 12 , L(s, χD1 )L(s, χD χD1 ) ∼ L(2s, χD1 ), which implies (see [Da]) that L(s, χD1 ) has no zeros γ + iρ with γ > 12 .

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