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A note on the zeroes of the Fredholm series

Umberto Zannier, Francesco Veneziano

Abstract

The issue had been raised whether the Fredholm series $z+z^2+...+z^{2^n}+...$ has infinitely many zeroes in the unit disk. We provide an affirmative answer, proving that in fact every complex number occurs as a value infinitely many times.

A note on the zeroes of the Fredholm series

Abstract

The issue had been raised whether the Fredholm series has infinitely many zeroes in the unit disk. We provide an affirmative answer, proving that in fact every complex number occurs as a value infinitely many times.

Paper Structure

This paper contains 11 sections, 4 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some formulae
  3. A Ramanujan sum and related ones
  4. Preparing an approximate formula
  5. Proofs of main assertions
  6. About the function S(w)
  7. About the constants cm
  8. Concluding arguments
  9. Appendix by Francesco Veneziano
  10. Finding a zero of S(w)
  11. Some zeroes of the Fredholm series

Key Result

Proposition 1.1

Let $V=S({\mathcal{H}})\subset {\mathbb C}$ be the image of $S$ on ${\mathcal{H}}$. Then, for every open disk $U$ centered at $1$ and for every $v\in V$, the Fredholm function $f(z)$ assumes on $D\cap U$ infinitely many times the value $v$.

Figures (1)

  • Figure 1:

Theorems & Definitions (10)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • proof : Proof of Proposition
  • proof : Proof of Proposition
  • proof : Proof of Theorem for real values
  • proof : Proof of Theorem and Theorem
  • Remark 3.1
  • Remark 3.2