Type: Article

Irrationality of Lambert series associated with a periodic sequence

Luca, Florian; Tachiya, Yohei;

Abstract:

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