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arXiv · 2507.03625

On the dynamics of a semigroup and its relation with the Riemann Hypothesis

Carlos F. Álvarez·Juan Manzur
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Abstract

The semigroup of weighted composition operators $(W_n)_{n\in \mathbb{N}}$, defined by $$W_nf(z)=(1+z+\cdots +z^n)f(z^n),$$ acts on the classical Hardy-Hilbert space $H^{2}(\mathbb{D})$, and exhibits intriguing connections with both the Riemann Hypothesis (RH) and the Invariant Subspace Problem (ISP). In this paper, we prove that the adjoint operators $W^{\ast}_{n}$, for $n\geq 2$, are Devaney chaotic, frequently hypercyclic and mixing. In particular, these operators are hypercyclic and discuss connections with the RH and invariant subspaces.

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Carlos F. Álvarez, Juan Manzur. 2025-07-04. On the dynamics of a semigroup and its relation with the Riemann Hypothesis. https://arxiv.org/abs/2507.03625

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