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Phase variation and angular momentum of the Riemann, and, Dirichlet Xi functions

Giovanni Lodone

TL;DR

This paper introduces an angular momentum concept, encoded as the phase-variation rate of the Riemann and Dirichlet xi functions, to derive RH and GRH equivalences. By relating the phase derivative to a determinant-based angular momentum $\mathcal{L}[\cdot]$, it provides lemmas showing how multiplicative factors and Hadamard product decompositions influence $\mathcal{L}$, and extends these ideas from $\xi(s)$ to $\xi(s,\chi)$ for primitive Dirichlet characters. The core results establish that, under certain conditions, the sign of $\mathcal{L}$ tracks the distance from the critical line $\epsilon$, leading to equivalences: (i) if all zeros lie on the critical line, then $\mathcal{L}$ mirrors the sign of $\epsilon$; (ii) conversely, if $\mathcal{L}$ maintains this sign, then all zeros lie on the critical line. The work also connects these ideas to a whole-strip derivative relation and to a practical, line-based criterion via a modified function $\eta$, offering a Bombieri-style route to RH/GRH from critical-line data.

Abstract

The concept of angular momentum is used to find new RH equivalence statements, and, generalize some known results from Riemann to Dirichlet primitive Xi functions

Phase variation and angular momentum of the Riemann, and, Dirichlet Xi functions

TL;DR

This paper introduces an angular momentum concept, encoded as the phase-variation rate of the Riemann and Dirichlet xi functions, to derive RH and GRH equivalences. By relating the phase derivative to a determinant-based angular momentum , it provides lemmas showing how multiplicative factors and Hadamard product decompositions influence , and extends these ideas from to for primitive Dirichlet characters. The core results establish that, under certain conditions, the sign of tracks the distance from the critical line , leading to equivalences: (i) if all zeros lie on the critical line, then mirrors the sign of ; (ii) conversely, if maintains this sign, then all zeros lie on the critical line. The work also connects these ideas to a whole-strip derivative relation and to a practical, line-based criterion via a modified function , offering a Bombieri-style route to RH/GRH from critical-line data.

Abstract

The concept of angular momentum is used to find new RH equivalence statements, and, generalize some known results from Riemann to Dirichlet primitive Xi functions
Paper Structure (11 sections, 38 equations, 1 figure)

This paper contains 11 sections, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Angular Momenta
  3. Phase variations and Angular Momentum of the $\xi$ function
  4. First lemma on angular momentum
  5. Second lemma on angular momentum
  6. Some RH-GRH equivalences
  7. Third lemma on angular momentum
  8. Fourth lemma on angular momentum
  9. Another equivalence linked to sign of angular momentum on whole critical strip-width.
  10. A practical equivalence based ONLY on critical line points
  11. Conclusions

Figures (1)

  • Figure 1: Square module of $Z(t,\epsilon) = \frac{-\xi(t,\epsilon)}{F(t) e^{i\epsilon\frac{\pi}{4}}}$ ( \ref{['ZSinhECosh1']} ) , in the interval $0< \epsilon < 0.5$, for different $t$ values.