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Estimates for the number of zeros of shifted combinations of completed Dirichlet series

Pedro Ribeiro

Abstract

In a previous paper, Yakubovich and the author of this article proved that certain shifted combinations of completed Dirichlet series have infinitely many zeros on the critical line. Here we provide some lower bounds for the number of critical zeros of a subclass of shifted combinations.

Estimates for the number of zeros of shifted combinations of completed Dirichlet series

Abstract

In a previous paper, Yakubovich and the author of this article proved that certain shifted combinations of completed Dirichlet series have infinitely many zeros on the critical line. Here we provide some lower bounds for the number of critical zeros of a subclass of shifted combinations.
Paper Structure (11 sections, 11 theorems, 224 equations, 1 figure)

This paper contains 11 sections, 11 theorems, 224 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Lemmas for the proof of Theorem \ref{['theorem 1.1']}
  3. Proof of Theorem \ref{['theorem 1.1']}
  4. Outline of the Proof
  5. A suitable integral representation
  6. Studying the sign changes of $\left(\mathcal{G}_{p}(\theta_{M})\right)_{p\in\mathbb{N}}$
  7. The dominance of $\left(\mathcal{G}_{p}(\theta_{M})\right)_{p\in\mathbb{N}}$.
  8. Conclusion of the argument
  9. Lemmas for the proof of Theorem \ref{['theorem 1.2']}
  10. Proof of Theorem \ref{['theorem 1.2']}
  11. Concluding Remarks

Key Result

Theorem 1.1

Let $(c_{j})_{j\in\mathbb{N}}$ be a sequence of real numbers such that $\sum_{j=1}^{\infty}|c_{j}|<\infty$ and $(\lambda_{j})_{j\in\mathbb{N}}$ be a bounded sequence of real numbers attaining its bounds. Suppose that $(c_{j})_{j\in\mathbb{N}}$ and $(\lambda_{j})_{j\in\mathbb{N}}$ can be extended to Moreover, let $N_{\alpha,z}(T)$ be the number of zeros written in the form $s=\frac{\alpha}{4}+it,\

Figures (1)

  • Figure 1: The partition of the unit circle into $\mathcal{A}^{+}$ and $\mathcal{A}^{-}$ and the representations of the subsets $\mathcal{A}_{1}^{\pm}$, $\mathcal{A}_{2}^{\pm}$ and $\mathcal{I}^{\pm}$. The line dividing the unit circle is defined by the equation $Y=\arctan\left(\cot\left(\frac{\pi\alpha}{8}\right)\coth\left(\frac{\pi|\lambda_{M}|}{2}\right)\right)\,X$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1: RYCE, Lemma 3.3
  • ...and 24 more