An application of Kronecker's theorem to transcendence theory.
We shall establish full asymptotic expansions for the mean squares of Lerch zeta-functions, based on F. V. Atkinson's device. Mellin-Barnes' type integral expression for an infinite double sum will play a central role in the derivation of our main formulae.
For the Lerch zeta-function Φ(s,x,λ) defined below, the multiple mean square of the form (1.1), together with its discrete and Irbid analogues, (1.2) and (1.3) are investigated by means of Atkinson's [2] dissection method applied to the product Φ(u,x,λ)Φ(υ,x,-λ), where u and υ are independent complex variables (see (4.2)). A complete asymptotic expansion of (1.1) as Im s → ±∞ is deduced from Theorem 1, while those of (1.2) and (1.3) as q → ∞ and (at the same time) as Im s → ±∞ are deduced from Theorems...
For any real τ, a lim sup set WG,y(τ) of τ-(well)-approximable points is defined for discrete groups G acting on the Poincaré model of hyperbolic space. Here y is a 'distinguished point' on the sphere at infinity whose orbit under G corresponds to the rationals (which can be regarded as the orbit of the point at infinity under the modular group) in the classical theory of diophantine approximation.In this paper the Hausdorff dimension of the set WG,y(τ) is determined for geometrically finite groups...
Let be a non-empty subset of positive integers. A partition of a positive integer into is a finite nondecreasing sequence of positive integers in with repetitions allowed such that . Here we apply Pólya’s enumeration theorem to find the number of partitions of into , and the number of distinct partitions of into . We also present recursive formulas for computing and .
Let be irrational. Several authors studied the numberswhere is a positive integer and denotes the set of all real numbers of the form with restricted integer coefficients . The value of was determined for many particular Pisot numbers and for the golden number. In this paper the value of is determined for irrational numbers , satisfying with a positive integer .
Number fields can be viewed as analogues of curves over fields. Here we use metrized line bundles as analogues of divisors on curves. Van der Geer and Schoof gave a definition of a function on metrized line bundles that resembles properties of the dimension of , where is a divisor on a curve . In particular, they get a direct analogue of the Rieman-Roch theorem. For three theorems of curves, notably Clifford’s theorem, we will propose arithmetic analogues.