ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11M06, 11M26, 11J99 Alcantara-Bode, Julio PUCP, IMCA Reorderings of some divergent series and applications Using some results of Skölem and Bang on Diophantine approximations it is shown that if f : IN → Cl is a function such that ∑_{n=1}^{∞} |f(n) − \frac{1}{n}| < ∞

and α ∈ [0, 1] is irrational then ∑_{n=1}^{∞} (f ([\frac{n}{α}]) + f ([\frac{n}{1−α}]) − f(n)) = −α ln α − (1 − α) ln(1 − α), where [x] denotes the integer part of x. This result is used to study some properties of the function J(β) = ∫_{0}^{1} ρ (\frac{1}{x}) ρ (\frac{β}{x} ) dx, β ∈ [0, 1], ρ(x) = x − [x] , that arises in Beurlings reformulation of the Riemann Hypothesis as a problem of Functional Analysis.