(t = 2,3,4,...) are irrational; here \phi and \sigma are Euler's function, and the sum of divisors. The proof depends on a general Lemma 1 on irrational series, and on these properties: (1) There are only O(x) integers n satisfying \phi(n) \leq x (or \sigma(n) \leq x). (2) There are only o(x) integers n \leq x for which \phi(k) = n (or \sigma(k) = n) has a solution k.
Lemma 1 is obtained as a special case of the more general Lemma 4: Let {a_k} and {b_k} be two infinite sequences of integers \geq 0 such that a_k \leq k^s and b_k \leq k^s (s a constant). Let f(n) and g(n) denote the number of positive a_k and b_k with 1 \leq k \leq n; assume that f(n) ––> oo as n ––> oo. Let there exist an infinite sequence of integers {m_i} such that

Finally assume there exists a constant c > 0 as follows: When i₁ and i₂ are consecutive suffixes with b_i1b_i2 > 0, and when i₁+cx < i₂, then there is a k with i₁+x < k < i₂+cx such that a_k > 0. Under these conditions all series

are irrational.
From Lemma 4, the author deduces Theorem 2: Let {n_k} be a strictly increasing sequence of positive integers such that limsup_{n ––> oo} {n_k \over k^l} = oo (l > 0 a constant). If t \geq 2 is an integer, then the series sum_{k = 1}^oo t^-n_k cannot have as its sum an algebraic number of degree \leq 1.
Reviewer:  K.Mahler
Classif.:  * 11J72 Irrationality
Index Words:  Number Theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag

Books	Problems	Set Theory	Combinatorics	Extremal Probl/Ramsey Th.
Graph Theory	Add.Number Theory	Mult.Number Theory	Analysis	Geometry
Probabability	Personalia	About Paul Erdös	Publication Year	Home Page