Let p be an odd prime such that the Greenberg conjecture holds for the maximal real cyclotomic subfield K1 of Q[ζp]. Let An = (C(Kn))p be the p-part of the class group of Kn, the n-th field in the cyclotomic tower, and let En, Cn be the global and cyclotomic units of Kn, respectively. We prove that under this premise, there is some n0 such that for all m ≥ n0, the class number formula (Em/Cm)p = |Am| hides in fact an isomorphism of Λ[Gal(K1/Q)]-modules.
We prove that sums of the form with f(X), g(X) ∈ ℤ[X] can be explicitly computed whenever f and g are subject to some certain conditions which are defined in the paper.
Fix f (t) ∈ Z[t] having degree at least 2 and no multiple roots. We prove that as k ranges over those integers for which the congruence f (t) ≡ 0 (mod k) is solvable, the least nonnegative solution is almost always smaller than k/(log k) c f . Here c f is a positive constant depending on f . The proof uses a method of Hooley originally devised to show that the roots of f are equidistributed modulo k as k varies.
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