Special values of Green's functions on hyperbolic $3$-space

Gross, Kohnen and Zagier proved an averaged version of the algebraicity conjecture for special values of higher Green’s functions on modular curves. In this work, we study an analogous problem for special values of Green’s functions on hyperbolic $3$-space. We prove that their averages can be computed in terms of logarithms of primes and logarithms of units in real quadratic fields. Moreover, we study twisted averages of special values of Green’s functions, which yield algebraic numbers instead of logarithms.