Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distibution of values of higher derivatives of $L'(s,χ)/L(s,χ)$

Samprit Ghosh

TL;DR

This work studies the value distribution of derivatives of the logarithmic derivative of Dirichlet L-functions, focusing on the average behavior of $\mathcal{L}(s,\chi)=L'(s,\chi)/L(s,\chi)$ and extending to higher derivatives. It constructs density functions $M_{\sigma, P}$ by leveraging the uniform distribution of local characters on finite tori $T_P$ and shows that, for $\sigma>1$, there exists a limit density $M_{\sigma}$ such that $\mathrm{Avg}_\chi Φ(\mathcal{L}'(s,\chi)) = \int_{\mathbb C} M_{\sigma}(w) Φ(w) |dw|$ for continuous $Φ$, with a Fourier counterpart $\tilde{M}_{\sigma}(z) = \mathrm{Avg}_\chi ψ_z(\mathcal{L}'(s,\chi))$. Extending to higher derivatives is pursued via Faà di Bruno expansions, defining $g_{\sigma,r,P}$ and $M_{\sigma,r,P}$, but the density construction faces obstructions: non-injectivity of the mapping for $m\ge2$ causes the method to require substantially larger $\sigma$ (e.g., $σ>2.93$ for $m=2$), and the approach is unlikely to yield densities for $$ near 1. The paper also discusses extending results to $\tfrac{1}{2}<σ\le1$ under GRH with admissible test functions, noting that this route does not straightforwardly generalize to higher derivatives and remains work in progress.

Abstract

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 $σ= $ Re$(s) > 1$. We then discuss how things evolve for higher derivatives.

Distibution of values of higher derivatives of $L'(s,χ)/L(s,χ)$

TL;DR

This work studies the value distribution of derivatives of the logarithmic derivative of Dirichlet L-functions, focusing on the average behavior of and extending to higher derivatives. It constructs density functions by leveraging the uniform distribution of local characters on finite tori and shows that, for , there exists a limit density such that for continuous , with a Fourier counterpart . Extending to higher derivatives is pursued via Faà di Bruno expansions, defining and , but the density construction faces obstructions: non-injectivity of the mapping for causes the method to require substantially larger (e.g., for ), and the approach is unlikely to yield densities for $\tfrac{1}{2}<σ\le1$ under GRH with admissible test functions, noting that this route does not straightforwardly generalize to higher derivatives and remains work in progress.

Abstract

In this article we study the value distribution theory for the first derivative of the logarithmic derivative of Dirichlet -functions, generalizing certain results of Ihara, Matsumoto et. al. related to ``-functions" for Re. We then discuss how things evolve for higher derivatives.
Paper Structure (6 sections, 11 theorems, 52 equations)

This paper contains 6 sections, 11 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Distribution Functions : Some Background
  3. Construction of $M_{\sigma, P}$ functions
  4. A note on higher derivatives
  5. Obstruction to generalization to higher derivatives.
  6. Remarks on extending the results to $\frac{1}{2} < \sigma \leq 1$.

Key Result

Theorem 1.1

(Ihara) For $K$ as above and for $\sigma =$ Re$(s) >1$, there exists a real valued function $M_{\sigma} : {\mathbb C} \rightarrow \mathbb{R}$ satisfying, $M_{\sigma} (w) \geq 0, \;$ is $C^{\infty}$ in $w$ and $\; \int_{{\mathbb C}} M_{\sigma}(w) \; |dw| = 1$, such that holds for any continuous function $\Phi$ of ${\mathbb C}$. Moreover, where $\tilde{M}_{\sigma}(z)$ comes from the Fourier transf

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • ...and 17 more