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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.
Reviewer: A.Ivić (Beograd)

MSC:
11M41 Other Dirichlet series and zeta functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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