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arXiv · 2510.00008

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

Es-said En-naoui
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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.

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Es-said En-naoui. 2025-09-14. A Universal Space of Arithmetic Functions:The Banach--Hilbert Hybrid Space U. https://arxiv.org/abs/2510.00008

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