Wednesday, May 31, 2023

2:00 p.m. (sharp)

Zoom Web Conference/BA6183

PhD Candidate: Samprit Ghosh

Supervisor: V. Kumar Murty

Thesis title: Higher Euler-Kronecker constants

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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 prove, under GRH, $$\gamma_{K,n} \ll (\log (\log(|d_K|))^{n+1}$$

$d_K$ being the absolute discriminant of $K$.

Analogous bounds for the function field case are also shown. We prove (unconditionally) interesting arithmetic formulas satisfied by these constants.

We also study the distribution of values of higher derivatives of $\mathcal{L}(s,\chi)= L'(s, \chi)/L(s, \chi)$ at $s=1$ and $\chi$ 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$(s) > 1$

$$\text{Avg}_{\chi} \Phi(\mathcal{L}'(s, \chi)) = \int_{C} M_{\sigma}(z) \Phi(z) |dz| $$ holds for any continuous function $\Phi$ on $C$.

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The draft of the thesis can be found here: Thesis_Samprit_Ghosh

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