The weak converse of Zeckendorf’s theorem

Weil Converse Theorem

Hecke generalized this equivalence, showing that an integral form has an associated Dirichlet series which can be analytically continued to C and satisfies a functional equation. Conversely, Weil showed that, if a Dirichlet series satisfies certain functional equations, then it must be associated to some integral form. Our goal in this paper is to describe this work. In the first three sections...

The Converse of the Individual Ergodic Theorem

converge almost everywhere to a finite limit f*(x). It then follows that the limit function/* is integrable and that/*(7x) =/*(x) almost everywhere. This result can be applied to certain cases in which the given measure m is not preserved by the transformation T. In order to discuss this application, we recall some terminology for measures and transformations. If (X, S) is a measurable space, a...

On the converse of Wolstenholme’s Theorem

The problem of distinguishing prime numbers from composite numbers (. . .) is known to be one of the most important and useful in arithmetic. (. . .) The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. Wilson’s Theorem states that if p is prime then (p− 1)! ≡ −1 (mod p). It is easy to see that the c...

The Converse Ostrowski Theorem: aspects of compactness

The Ostrowski theorem in question is that an additive function bounded (above, say) on a set T of positive measure is continuous. In the converse direction, recall that a topological space T is pseudocompact if every function continuous on T is bounded. Thus theorems of ‘converse Ostrowski’type relate to ‘additive (pseudo)compactness’. We give a di¤erent characterization of such sets, in terms ...

Ribet’s Converse to Herbrand’s Theorem