Multiplicative arithmetic of finite quadratic forms over Dedekind rings
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Letq(X) be a quadratic form in an even numberm of variables with coefficients in a Dedekind ringK. Let us assume that the setsR(q,a) = {N∈K^{m};q(N) = a} of representations of elementsa ofK by the formq are finite. Then certain multiplicative relations are obtained by elementary means between the setsR(q,a) andR(q,ab), whereb is a product of prime elementsρ ofK with finite coefficientsK/ρK. The relations imply similar multiplicative relations between the numbers of elements of the setsR(q,a), which formerly could be obtained only in some special cases like the case whenK = ℤ is the ring of rational integers and only by means of the theory of Hecke operators on the spaces of theta-series. As an application, an almost elementary proof of the Siegel theorem on the mean number of representations of integers by integral positive quadratic forms of determinant 1 is given.
Anatoli Andrianov^{1} ^{2}
Volume 131, 2021
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