Residues and Elementary Prym Differentials on the Compact Riemann Surface

The paper are obtained new results on the theory of multiplicative functions and Prym differentials on a variable compact Riemann surfaces of genus $g>1$. For the first time full sum residues theorems are obtained for Prym differentials for every integer order. As a corollary reciprocity laws and existence theorems for Prym differentials with given poles and residues are proven. All kinds of elementary Prym differentials, which holomorphically depend on modules of surfaces and characters are constructed. Analogues of Appell's decomposition formula for functions with characters are proven.