Journal of the Ramanujan Mathematical Society

Volume 28, Issue 1, March 2013 pp. 113--139.

Transcendence of series of rational functions and a problem of Bundschuh

Authors: Chester Weatherby
Author institution: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6

Summary: We investigate the transcendental nature of the sums A where A(x),B(x) is an algebraic valued periodic function, and the sum is over integers n which are not zeros of B(x). We offer a new method of evaluating these sums using only elementary techniques. In some cases we relate these sums to a celebrated theorem of Nesterenko and a conjecture of Schneider and obtain concrete as well as conditional transcendence results. These results include progress on a problem of Bundschuh regarding the sum for integer values of s.

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