arXiv subjects
Publications and source records attributed to Samprit Ghosh.
In this article, we study the value distribution theory for the first derivative of the logarithmic derivative of Dirichlet $L$-functions, generalizing certain results of Ihara, Matsumoto et al. related to ``$M$-functions'' for $\sigma = \operatorname{Re}(s) > 1$. We then discuss the main obstruction toward generalization to higher derivatives.
Let $\chi$ be a non-principal Dirichlet character, and let $L(s,\chi)$ be the associated Dirichlet $L$-function. We write $\mathcal{L}(s,\chi)$ for its logarithmic derivative $L'(s,\chi)/L(s,\chi)$. In this article, we prove arithmetic formulas for the higher derivatives $\mathcal{L}^{(r)}(1,\chi)$ and establish unconditional moment asymptotics for $P^{(a,b)}(\mathcal{L}^{(r)}(1,\chi))$ as $\chi$ runs over non-principal Dirichlet characters of large prime conductor. As an application, we show that these moment asymptotics imply positivity of the second Li coefficient of prime cyclotomic fields of sufficiently large conductor. Combined with a zero-free criterion in terms of this Li coefficient, this rules out the possible exceptional zero in a Stark-type zero-free region for these fields.
The higher Euler-Kronecker constants of a number field $K$ are the coefficients in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function about $s=1$. These coefficients are mysterious and seem to contain a lot of arithmetic information. In this article, we study these coefficients. We prove arithmetic formulas satisfied by them and prove bounds. We generalize certain results of Ihara.
Let $d>k$ be positive integers. Motivated by an earlier result of Bugeaud and Nguyen, we let $E_{k,d}$ be the set of $(c_1,\ldots,c_k)\in\mathbb{R}_{\geq 0}^k$ such that $\vert\alpha_0\vert\vert\alpha_1\vert^{c_1}\cdots\vert\alpha_k\vert^{c_k}\geq 1$ for any algebraic integer $\alpha$ of degree $d$, where we label its Galois conjugates as $\alpha_0,\ldots,\alpha_{d-1}$ with $\vert\alpha_0\vert\geq \vert\alpha_1\vert\geq\cdots \geq \vert\alpha_{d-1}\vert$. First, we give an explicit description of $E_{k,d}$ as a polytope with $2^k$ vertices. Then we prove that for $d>3k$, for every $(c_1,\ldots,c_k)\in E_{k,d}$ and for every $\alpha$ that is not a root of unity, the strict inequality $\vert\alpha_0\vert\vert\alpha_1\vert^{c_1}\cdots\vert\alpha_k\vert^{c_k}>1$ holds. We also provide a quantitative version of this inequality in terms of $d$ and the height of the minimal polynomial of $\alpha$.