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Functional independence of the singularities of a class of Dirichlet series. (English) Zbl 0905.11036
The authors deal with the algebraic independence and, more generally, with the functional independence of the singularities of $$\log F_j(s)$$, $$j=1,\dots,N$$, and of $${F'_j\over F_j}(s)$$, $$j=1,\dots,N$$, where $$F_j(s)$$ are functions in the Selberg class of axiomatically defined zeta functions. In particular, the following interesting results are proved.
(i) If $$\log F_1(s),\dots,\log F_N(s)$$ are linearly independent over $$\mathbb Q$$, then
$$P(\log F_1(s),\dots,\log F_N(s),s)$$ has infinitely many singularities in the half plane $$\sigma \geq {1\over 2}$$, provided $$P\in \mathbb C[X_1,\dots,X_N,s]$$ with $$\text{deg }P >0$$ as a polynomial in the first $$N$$ variables.
(ii) If $$P\in \mathbb C[X_1,\dots,X_N]$$ with $$\text{deg }P>0$$, then $$P({F'_1\over F_1}(s),\dots,{F'_N\over F_N}(s))$$ is either constant or has infinitely many singularities in the half plane $$\sigma \geq 0$$.
The authors give an arithmetic application of the above results, proving an $$\Omega$$-result for the remainder term of the counting function arising from a factorization problem in algebraic number fields.
 11M41 Other Dirichlet series and zeta functions 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$