IRRATIONALITY OF LAMBERT SERIES ASSOCIATED WITH A PERIODIC SEQUENCE

Abstract

Let q be an integer with |q| > 1 and {an}n≥1 be an eventually periodic sequence of rational numbers, not identically zero from some point on. Then the number [Formula: see text] is irrational. In particular, if the periodic sequences [Formula: see text] of rational numbers are linearly independent over ℚ, then so are the following m + 1 numbers: [Formula: see text] This generalizes a result of Erdős who treated the case of m = 1 and [Formula: see text]. The method of proof is based on the original approaches of Chowla and Erdős, together with some results about primes in arithmetic progressions with large moduli of Ahlford, Granville and Pomerance.