A Universal Space of Arithmetic Functions:The Banach--Hilbert Hybrid Space U

TL;DR

We introduce the Banach--Hilbert hybrid space $\mathbf{U}$ designed to contain all classical arithmetic functions while supporting both a Hilbert-type control of Dirichlet coefficients for every $s>1$ and a Banach-type control of logarithmic averages of partial sums. We establish that $\mathbf{U}$ is a complete normed space in which Dirichlet convolution, pointwise multiplication, and shifts are bounded, and we connect $\mathbf{U}$ to Dirichlet series through absolute convergence, analytic continuation under growth assumptions, and a Dirichlet-algebra structure. The framework shows that fundamental arithmetic functions like $\mu$, $\Lambda$, $\varphi$, $d$, and Dirichlet characters belong to $\mathbf{U}$, enabling unified analysis of $L$-functions and potential spectral approaches to problems such as the Riemann Hypothesis. The paper also discusses applications, including operator theory on $\mathbf{U}$, approximation and sampling, and implications for prime number theory, while outlining several open problems and directions for further development.

Abstract

We introduce a new functional space U designed to contain all classical arithmetic functions (Mobius, von Mangoldt, Euler phi, divisor functions, Dirichlet characters, etc.). The norm of U combines a Hilbert-type component, based on square summability of Dirichlet coefficients for every s > 1, with a Banach component controlling logarithmic averages of partial sums. We prove that U is a complete Banach space which embeds continuously all standard Hilbert spaces of Dirichlet series and allows natural actions of Dirichlet convolution and shift operators. This framework provides a unified analytic setting for classical and modern problems in multiplicative number theory.

Theorems & Definitions (32)

• Definition 2.1: The space $\mathbf{U}$
• Remark 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof