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For any odd prime $\ell$, let $h_{\ell }(-d)$ denote the $\ell$-part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Nontrivial pointwise upper bounds are known only for $\ell =3$; nontrivial upper bounds for averages of $h_{\ell }(-d)$ have previously been known only for $\ell =3,5$. In this paper we prove nontrivial upper bounds for the average of $h_{\ell }(-d)$ for all primes $\ell \geqslant 7$, as well as nontrivial upper bounds for certain higher moments for all primes $\ell \geqslant 3$. [ABSTRACT FROM PUBLISHER]

The article discusses the fractional parts of a n2 regarding its pair correlation. It notes that the fractional parts is uniformly distributed for any fixed real irrational fraction. It mentions that every fixed real irrational is represented by Diophantine. It further concludes that a Poisson property can be also phrased in terms of a sequence of correlation functions.

We show that a system of r quadratic forms over a p-adic field in at least 4r + 1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax-Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605-630] requires the characteristic to be large in terms of the degree of the field over Qp. The proofs use a p-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the problem of counting rational points of bounded height on rational cubic surfaces. For most such surfaces this question appears much too hard for current methods. We shall therefore examine a particular example, namely the surfaceformula herefor which the problem is tractable, though by no means trivial. This surface contains the lines X0=Xi=0, for i=1, 2, 3, which we shall exclude from consideration. We therefore defineformula herewhere an integer vector is said to be primitive if its components have no common factor. The function 5(H) counts points with the most naive height available. This suffices, however, to bring out the key features of the problem. We may observe at once that the condition x0dH is redundant, and that it suffices to assume that (x1, x2, x3) is primitive.Our principal result is the following theorem. [ABSTRACT FROM AUTHOR]