Natural dual space U* capturing prime-counting distributions

Determine whether there exists a natural dual space U* for the Banach space U of arithmetic functions, equipped with the norm ||f||_U = sup_{n in N} |f(n)| / log(2+n), such that U* captures distributional limits of prime counting functions.

Background

The paper introduces a Banach space U designed to host classical arithmetic functions and to support natural operations such as Dirichlet convolution and shifts. Establishing an appropriate dual space U* would enable distributional viewpoints on arithmetic phenomena, particularly those tied to prime counting.

The authors explicitly ask whether a natural dual space U* can capture distributional limits related to prime counting functions, reflecting the goal of linking functional-analytic structures with asymptotic prime distribution behavior.