Kurt Mahler

An application of Jensen's formula to polynomials

In this note new proofs will be given for two inequalities on polynomials due to N. I. Feldman [1] and A. 0. Gelfond [2], respectively; these inequalities are of importance in the theory of transcendental numbers. While the original proofs by the two authors are quite unconnected, we shall deduce both results from the same source, viz. from Jensens integral formula in the theory of analytic functions.

Lectures on transcendental numbers

Existence and first properties of transcendental numbers.- Convergent laurent series and formal laurent series.- First results on the values of analytic functions at algebraic points.- Linear differental equations: The lemmas of Shidlovski.- Linear differential equations: A lower bound for the rank of the values of finitely many siegel E-functions at algebraic points.- Linear differential equations: Shidlovskis theorems on the transcendency and algebraic independence of values of siegel E-functions.- Applications of Shidlovskis main theorems to special functions.- Formal power series as solutions of algebraic differential equations.

An unsolved problem on the powers of 3/2 *

Let α be an arbitrary positive number. For every integer n ≦ 0 we can write where is the largest integer not greater than , i.e the integral part of , and r n is its fractional part and so satisfies the inequality

On lattice points in n-dimensional star bodies I. Existence theorems

Let F(X) = F(x1,..., xn) be a continuous non-negative function of X satisfying F(tX) = |t|F(X) for all real numbers t. The set K in n-dimensional Euclidean space Rn defined by F(X)⩽ 1 is called a star body. The author studies the lattices Λ in Rn which are of minimum determinant and have no point except (0, ..., 0) inside K. He investigates how many points of such lattices lie on, or near to, the boundary of K, and considers in detail the case when K admits an infinite group of linear transformations into itself.

On the fractional parts of the powers of a rational number (II)

About twenty years ago, in a note of the same title [2], I obtained the following result. THEOREM 1. Let u and v be relatively prime integers satisfying u> v ≥ 2 and let e be an arbitrarily small positive number. Suppose the inequality is satisfied by an infinite sequence of positive integers n 1 n 2 , … Then

Simultaneous Diophantine approximation

The simplest problems of Diophantine approximation relate to the approximation of a single irrational number 0 by rational numbers pjq, and the principal question is how small we can make the error 0 — pjq in relation to q for infinitely many approximations. It is well known that this question can be answered almost completely in terms of the continued fraction expansion of 0. It must be admitted that our knowledge of the relationship between the continued fraction expansion of 0 and other possible representations of 0 is very fragmentary, a striking enough example being the number tf/2. However, the continued fraction theory gives us many general results which are best possible. Thus every 0 admits an infinity of approximations satisfying

On the Approximation of Logarithms of Algebraic Numbers

A new identity is given by means of which infinitely many algebraic functions approximating the logarithmic function In x are obtained. On substituting numerical algebraic values for the variable, a lower bound for the distance of its logarithm from variable algebraic numbers is found. As a further application, it is proved that the fractional part of the number ea is greater than a-40a for every sufficiently large positive integer a.

On a class of non-linear functional equations connected with modular functions

Let p be a prime. This paper deals with solutions of functional equations in either formal Laurent series or in analytic functions. Examples connected to special modular functions are considered.

On algebraic differential equations satisfied by automorphic functions: To Bernhard Hermann Neumann on his 60th birthday

Anodes for current generating cells having good corrosion resistance, voltage stability and long life, especially in primary current generating cells, are described. The anodes comprise an anode composition having from about 20 to about 90 weight percent of the first constituent selected from the group consisting of iron and silicon and at least about 10 weight percent of a second constituent different from the first constituent and selected from the group consisting of molybdenum, tungsten, vanadium, niobium, titanium, phosphorus, and silicon. Anode compositions containing iron and a second constituent other than silicon may also contain up to about 10 weight percent silicon.

Remarks on a paper by W. Schwarz

Abstract The author reports on old work of his on the transcendency of functions satisfying functional equations like F(z 2 )= (1−z)F(z)−z 1−z He suggests a number of directions in which this work might possibly be extended.