Polylogarithmic Analogues of Euler's Constant

We introduce a family of constants $$C_m:=\underset{n\to \mathrm{\infty }}{\lim }(\sum _{k=1}^n{\mathrm{Li}}_m\text{\hspace{0.17em}⁣}\left(\frac{1}{k}\right)-\log n),$$ which may be regarded as polylogarithmic analogues of Euler’s constant. We study their basic properties and derive representations in terms of iterated logarithmic integral structures associated with the gamma function. We further introduce associated polylogarithmic zeta potentials and polylogarithmic gamma functions, establish differential relations and integral representations, and describe logarithmic branch asymptotics near the singular points. As an application, we relate the constants $C_m$ to special values of certain Dirichlet series involving the Riemann zeta function.