The range of the sum-of-proper-divisors function
Answering a question of Erdős, we show that a positive proportion of even numbers are in the form s(n), where s(n) = σ(n) - n, the sum of proper divisors of n.
The rank of hyperelliptic Jacobians in families of quadratic twists
The variation of the rank of elliptic curves over in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over admits infinitely many quadratic twists of rank . Most elliptic curves have even infinitely many twists of rank and examples of elliptic curves with infinitely many twists of rank are known. There are also certain density results. This paper studies the variation of the...
The Rational Invariants of the Tame Quivers.
The rational points close to a curve
The rational points close to a curve II
The rational points close to a curve III
The Rationality of Holomorphic Eisenstein Series.
The rationality of the Fourier coefficients of certain Eisenstein series on tube domains (I)
The rationality of the Poincaré series associated to the p-adic points on a variety.
The real 3x+1 problem
The real field with the rational points of an elliptic curve
We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.
The reciprocal of the beta function and Whittaker functions
In this paper we derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler’s beta function. This expression is similar in form to several well-known integrals for the beta function itself.We then apply our new formula to the study of Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral representations of “fundamental”...
The reciprocal super Catalan matrix
The reciprocal super Catalan matrix has entries [...] . Explicit formulæ for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, q-analogues are also presented.
The reciprocity obstruction for rational points on compactifications of torsors under tori over fields with global duality.
The reciprocity theorem for Dedekind-Rademacher sums
The reduced ideals of a special order in a pure cubic number field
Let be a pure cubic field, with , where is a cube-free integer. We will determine the reduced ideals of the order of which coincides with the maximal order of in the case where is square-free and .
The reduced Wittring
The Reduction of Positive Definite Quinary Quadratic Forms.
The Reduction of Positive Definite Quinary Quadratic Forms. (Short Communication).
The refined p-adic abelian Stark conjecture in function fields.