Functional independence of the singularities of a class of Dirichlet series
Copyright policy )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 F 1 ( 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 F 1 ( 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" /][ X 1 , . . . , 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" /][ 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 σ ≥ 0.
- University of Genoa Italy
\(\zeta (s)\) and \(L(s, \chi)\), linear independence, Selberg class; L-functions; functional independence, Selberg zeta functions, functional equation, functional independence, Dirichlet series, Other Dirichlet series and zeta functions
\(\zeta (s)\) and \(L(s, \chi)\), linear independence, Selberg class; L-functions; functional independence, Selberg zeta functions, functional equation, functional independence, Dirichlet series, Other Dirichlet series and zeta functions
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