Special values of L-series, periodic coefficients and related themes

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Pathak, Siddhi
Dirichlet series , Transcendental number theory , Non-vanishing of L(1,f) , Elliptic functions , Special values of zeta-functions
This thesis is centered around the theme of special values of L-functions and other infinite series, which are often expected to be transcendental numbers. More specifically, we focus on the following two questions in various scenarios: a) expressing the values in terms of certain special functions, b) determining their arithmetic nature (i.e., whether they are rational or irrational, algebraic or transcendental). Motivated by the conjectures of S. Chowla and P. Erdos, we first study the L-series L(s,f) attached to a periodic arithmetical function f. Utilizing tools from transcendental number theory, we investigate the non-vanishing and the arithmetic nature of the values L(1,f) and L'(1,f). We introduce a probabilistic viewpoint towards the study of the values L(k,f) for any integer k greater than or equal to 1, especially in the case when f is an Erdos function. On a related note, we explore the irrationality of the values of Dedekind zeta-functions at positive integers using elementary means. We also initiate the study of the sum over lattice points, of a rational function. This opens new doors for future research.
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