Higher Euler-Kronecker Constants

The coefficients that appear in the Laurent series of Dedekind zeta functions and their logarithmic derivatives are mysterious and seem to contain a lot of arithmetic information. Although the residue and the constant term have been widely studied, not much is known about the higher coefficients. In this thesis, we study these coefficients $\gamma_{K,n}$ that appear in the Laurent series expansion of $\frac{\zeta_K'(s)}{\zeta_K (s)}$ about $s=1$, where $K$ is a global field. For example, when $K$ is a number field, we unconditionally prove certain arithmetic formulas satisfied by these coefficients and we give bounds for them under GRH. Analogous bounds for function fields of curves defined over a finite field are also shown. \vspace{2mm}\\ We also study the distribution of values of higher derivatives of $\mathcal{L}(s,\chi)= L'(s, \chi)/L(s, \chi)$ at $s=1$ where $\chi$ ranges over all non-trivial Dirichlet characters with a given large prime conductor $m$. In particular, we compute moments, i.e. the average of $P^{(a,b)}(\mathcal{L}^{(n)}(1, \chi))$, where $P^{(a,b)}(z) = z^a \overline{z}^b$ and study their asymptotic behaviour as $m \rightarrow \infty$. We then construct a density function $M_{\sigma}(z)$, for $\sigma= $ Re$(s)$ and show that for Re$(s) > 1$ $$\text{Avg}_{\chi} \Phi(\mathcal{L}'(s, \chi)) = \int_{\mathbb{C}} M_{\sigma}(z) \Phi(z) |dz| $$ holds for any continuous function $\Phi$ on $\mathbb{C}$.

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