Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the analytic continuation of Dirichlet series with missing digits

Jean-François Burnol

Abstract

We study the Dirichlet series associated with the integers whose radix-$b$ representation misses certain (fixed) digits. The existence of a meromorphic continuation to the entire complex plane, which was already well-known as a general fact valid for $b$-automatic Dirichlet series, is proven anew from a representation as an everywhere defined series with good convergence properties. A generating function related to the residues on the real axis is shown to be the multiplicative inverse of the moment generating function for the associated Cantor set in the unit interval. This makes the (normalized) residues some sort of generalized Bernoulli numbers.

On the analytic continuation of Dirichlet series with missing digits

Abstract

We study the Dirichlet series associated with the integers whose radix- representation misses certain (fixed) digits. The existence of a meromorphic continuation to the entire complex plane, which was already well-known as a general fact valid for -automatic Dirichlet series, is proven anew from a representation as an everywhere defined series with good convergence properties. A generating function related to the residues on the real axis is shown to be the multiplicative inverse of the moment generating function for the associated Cantor set in the unit interval. This makes the (normalized) residues some sort of generalized Bernoulli numbers.
Paper Structure (8 sections, 12 theorems, 102 equations)

This paper contains 8 sections, 12 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Representation by geometrically convergent series
  3. The case with N=1
  4. Analytic continuation (second method)
  5. Residues (I)
  6. The case of the zeta function
  7. Residues (II)
  8. Values at non-positive integers

Key Result

Proposition 2.1

For any $s = s_0 + it$, $t \in \mathbb{R}$, the Dirichlet series $K(s)$ diverges.

Theorems & Definitions (29)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Remark 2.1
  • proof : Proof of Lemma \ref{['lem:key']}
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 4.1
  • ...and 19 more