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The Generalized Riemann Zeta heat flow

Víctor Castillo, Claudio Muñoz, Felipe Poblete, Vicente Salinas

Abstract

We consider the PDE flow associated to Riemann zeta and general Dirichlet $L$-functions. These are models characterized by nonlinearities appearing in classical number theory problems, and generalizing the classical holomorphic Riemann flow studied by Broughan and Barnett. Each zero of a Dirichlet $L$-function is an exact solution of the model. In this paper, we first show local existence of bounded continuous solutions in the Duhamel sense to any Dirichlet $L$-function flow with initial condition far from the pole (as long as this exists). In a second result, we prove global existence in the case of nonlinearities of the form Dirichlet $L$-functions and data initially on the right of a possible pole. Additional global well-posedness and convergence results are proved in the case of the defocusing Riemann zeta nonlinearity and initial data located on the real line and close to the trivial zeros of the zeta. The asymptotic stability of any stable zero is also proved. Finally, in the Riemann zeta case, we consider the ``focusing'' model, and prove blow-up of solutions near the pole $s=1$.

The Generalized Riemann Zeta heat flow

Abstract

We consider the PDE flow associated to Riemann zeta and general Dirichlet -functions. These are models characterized by nonlinearities appearing in classical number theory problems, and generalizing the classical holomorphic Riemann flow studied by Broughan and Barnett. Each zero of a Dirichlet -function is an exact solution of the model. In this paper, we first show local existence of bounded continuous solutions in the Duhamel sense to any Dirichlet -function flow with initial condition far from the pole (as long as this exists). In a second result, we prove global existence in the case of nonlinearities of the form Dirichlet -functions and data initially on the right of a possible pole. Additional global well-posedness and convergence results are proved in the case of the defocusing Riemann zeta nonlinearity and initial data located on the real line and close to the trivial zeros of the zeta. The asymptotic stability of any stable zero is also proved. Finally, in the Riemann zeta case, we consider the ``focusing'' model, and prove blow-up of solutions near the pole .
Paper Structure (24 sections, 28 theorems, 193 equations, 6 figures)

This paper contains 24 sections, 28 theorems, 193 equations, 6 figures.

Table of Contents

  1. Introduction and Main Results
  2. Setting
  3. Dirichlet $L$-functions
  4. Main results
  5. Preliminaries
  6. Banach spaces
  7. General Hurwitz zeta functions
  8. Dirichlet characters and Dirichlet $L$-functions
  9. Comparison principles
  10. Invariant domains
  11. Comparison Principles
  12. Uniform bounds
  13. The Hurwitz zeta case
  14. The general Dirichlet $L$-function case
  15. Local well-posedness
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\lambda=\pm1,$ and let $L_m$ be a Dirichlet $L$-function obtained from a principal character $m$. Consider an initial datum $g\in L^{\infty}(\mathbb{R}^d;\mathbb{C})\cap C(\mathbb{R}^d;\mathbb{C})$, $g=g_1+ig_2$, and $g_i\in\mathbb{R}$ be such that the uniform condition is satisfied. Then there exists a time $T>0$ such that the problem has a unique solution $u\in C([0,T); L^{\infty}(\mathbb

Figures (6)

  • Figure 1.1: Left: The holomorphic flow BB around the pole $s=1$. Right: The same flow around the first Riemann zeta nontrivial zero $s_1\sim \frac{1}{2}+14.13 i$ (unstable source).
  • Figure 1.2: Left: Sets considered in Theorem \ref{['Domaint']}$(ii)$. Red points are sinks, while black ones sources. Right: A scheme of the long time behavior of $u$ in Theorem \ref{['Domaint']}$(ii)$ in the case $g(x)\in (-10,-2)$.
  • Figure 1.3: Left: Example of a real-valued initial condition $g$ is located on the region determined by $I$ and $S$, in this case, above -8 and below -2. Right: Evolution of the solution $u$ in the previous setting. It is shown in Theorem \ref{['teo:globalReal']}$(iii)$ that $\lim_{t\to+\infty} u$ will stay between $-6$ by below and $-2$ by above.
  • Figure 1.4: Left: Distribution graph depicting the proportion sinks vs. nontrivial zeros among the first 10000 nontrivial zeros on the critical line. The $y$-axis describes the proportion $P_n = \dfrac{\# \text{sinks of the first } n \text{ nontrivial zeros}}{n},$where $n$ represents the first $n$ nontrivial zeros (on the $x$-axis). Right: The proportion sources vs. nontrivial zeros.
  • Figure 1.5: Blow up (a.k.a. quenching) gathered by the pole $s=1$ as the attractor of real-valued trajectories placed in its vicinity.
  • ...and 1 more figures

Theorems & Definitions (69)

  • Theorem 1.1: Local well-posedness
  • Remark 1.1
  • Corollary 1.2: Non principal character case
  • Definition 1.3: van de Lune VanLune
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Global well-posedness
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • ...and 59 more