On Certain Arithmetic Functions $\tilde{M}(s;z_1,z_2)$ Associated with Global Fields: Analytic Properties

Abstract

The arithmetic functions in the title arose from value-distribution theories related to $L$-functions of global fields. They are “complexifications” of the Fourier duals of the corresponding density functions. We shall study their complex analytic properties including analytic continuations and the limit behaviors at the critical point $s=1/2$.