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 log F₁(s), . . . , log F_N(s) are linearly independent over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], then P(log F₁(s), . . . , log F_N(s), s) has infinitely many singularities in the half plane σ ≥ ½, provided P ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /][X¹ , . . . , X_N+1] with deg P > 0 as a polynomial in the first N variables; and (ii) If P ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /][X₁ , . . . , X_N] with deg P > 0, then P(F'₁/F₁(s), . . . , F'_N/F_N (s)) is either constant or has infinitely many singularities in the half plane σ ≥ 0.