FUNCTIONAL INDEPENDENCE OF THE SINGULARITIES OF A CLASS OF DIRICHLET SERIES

We deal with the algebraic independence and, more generally, with the functional independence of the singularities of log F-j(s), j = 1,.... N, and of F-j'/F-j(s), j = 1,...,N, where F-j(s) are functions in the Selberg class. In particular, we prove the following results: (i) If l og F-1(s),...,log F-N(s) are linearly independent over Q, then P(log F -1(s),...,log F-N(s),s) has infinitely many singularities in the half plane a i 1, provided P is an element of C[X-1,...,XN+1] with deg P > 0. as a polynomial in the first N variables; and (ii) If P is an eleme nt of C[X-1,...,X-N] with deg P > 0, then P(F-1'/F-1(s),...,F-N'/F-N(s )) is either constant or has infinitely many singularities in the half plane sigma greater than or equal to 0.