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We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units. [ABSTRACT FROM AUTHOR]

Let A be an abelian variety. The abelian Ax–Lindemann theorem shows that the Zariski closure of an algebraic flow in A is a translate of an abelian subvariety of A. The paper discusses some conjectures on the usual topological closure of an algebraic flow in A. The main result is a proof of these conjectures when the algebraic flow is given by an algebraic curve. [ABSTRACT FROM AUTHOR]

In 1994, Brüdern and Fouvry [1] initiated the investigation of Lagrange's four squares theorem with almost prime variables. In this paper, we prove that every suffi- ciently large integer, congruent to 4 modulo 24, can be represented as a sum of four squares of integers, each of which has at most four prime factors. Instead of the four-dimensional vector sieve developed by Brüdern and Fouvry [1], we establish this result by combining the threedimensional sieve and the switching principle. [ABSTRACT FROM AUTHOR]

It is well known that the orbit of a lattice in hyperbolic n-space is uniformly distributed when projected radially onto the unit sphere. In the present work, we consider the fine-scale statistics of the projected lattice points, and express the limit distributions in terms of random hyperbolic lattices. This provides in particular a new perspective on recent results by Boca, Popa, and Zaharescu on 2-point correlations for the modular group, and by Kelmer and Kontorovich for general lattices in dimension n = 2. [ABSTRACT FROM AUTHOR]

We investigate pairs of diagonal cubic equations with integral coefficients. For a class of such Diophantine systems with 11 or more variables, we are able to establish that the number of integral solutions in a large box is at least as large as the expected order of magnitude. [ABSTRACT FROM AUTHOR]

We study sums of a random multiplicative function; this is an example, of number-theoretic interest, of sums of products of independent random variables (chaoses). Using martingale methods, we establish a normal approximation for the sum over those n ≦ x with k distinct prime factors, provided that k = o(log log x) as x → ∞. We estimate the fourth moments of these sums, and use a conditioning argument to show that if k is of the order of magnitude of log log x then the analogous normal limit theorem does not hold. The methods extend to treat the sum over those n ≦ x with at most k distinct prime factors, and in particular the sum over all n ≦ x. We also treat a substantially generalised notion of random multiplicative function. [ABSTRACT FROM AUTHOR]

Let be open. We show that each homeomorphism satisfies . If we moreover assume that ƒ has finite distortion, then and ƒ–1 has finite distortion. The main ingredient is a new result on change of variables in integral (area and coarea formula) for such mappings. [ABSTRACT FROM AUTHOR]

Let ℱ( R) be a set of holomorphic functions on a Reinhardt domain R in a Banach sequence space (as e.g. all holomorphic functions or all m-homogeneous polynomials on the open unit ball of ). We give a systematic study of the sets dom ℱ( R) of all z ∈ R for which the monomial expansion of every ƒ ∈ ℱ( R) converges. Our results are based on and improve the former work of Bohr, Dineen, Lempert, Matos and Ryan. In particular, we show that up to any ε > 0 is the unique Banach sequence space X for which the monomial expansion of each holomorphic function ƒ converges at each point of a given Reinhardt domain in X. Our study shows clearly why Hilbert's point of view to develop a theory of infinite dimensional complex analysis based on the concept of monomial expansion, had to be abandoned early in the development of the theory. [ABSTRACT FROM AUTHOR]