Abstract

We deal with the algebraic independence and, more generally, with the functional independence of the singularities of log Fj(s), j = 1, . . . , N, and of F'j/Fj(s), j = 1, . . . , N, where Fj(s) are functions in the Selberg class. In particular, we prove the following results: (i) If log F1(s), . . . , log FN(s) are linearly independent over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], then P(log F1(s), . . . , log FN(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 , . . . , XN+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 , . . . , XN] with deg P > 0, then P(F'1/F1(s), . . . , F'N/FN (s)) is either constant or has infinitely many singularities in the half plane σ ≥ 0.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 289-303
Launched on MUSE
1998-04-01
Open Access
No
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