CARMA SEMINAR Speaker: Florian Luca, Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México Title: Linear independence of certain Lambert series Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:00 pm, Wed, 13th Mar 2013 Abstract: We prove that if $q\ne0,\pm1$ and $\ell\ge1$ are fixed integers, then the numbers $$1, \quad \sum_{n=1}^\infty\frac{1}{q^n-1}, \quad \sum_{n=1}^\infty\frac{1}{q^{n^2}-1}, \quad \dots, \quad \sum_{n=1}^\infty\frac{1}{q^{n^\ell}-1}$$ are linearly independent over $\mathbb{Q}$. This generalizes a result of Erdős who treated the case $\ell=1$. The method 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. This is joint work with Yohei Tachiya. [Permanent link]