Single-valued differentials and special divisors of Prym differentials

The theory of multiplicative functions and Prym differentials on a compact Riemann surface has found numerous applications in function theory, analytic number theory, and equations of mathematical physics. We give a full constructive description for the divisors of elementary abelian differentials of integer order and all three kinds depending holomorphically on the modulus of compact Riemann surfaces $F$. We study the location of zeros of holomorphic Prym differentials on $F$, as well as the structure of the set of (multiplicatively) special divisors on $F$ in the spaces $F_{g-1}$ and $F_{g-2}$.