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On Dirichlet series and holomorphic differential equations. (Über Dirichletsche Reihen und holomorphe Differentialgleichungen.) (German) Zbl 0556.10027

Let \({\mathcal D}\) be the class of all ordinary Dirichlet series \(f(s)=\sum a_ nn^{-s}\) with finite abscissa of absolute convergence such that the set of all indices \(n\) with \(a_ n\neq 0\) has infinitely many prime divisors. For real numbers \(h_ 0<h_ 1<...<h_{\mu}\) and integers \(\nu_ 0,\nu_ 1,...,\nu_{\mu}\geq 0\) the following theorem is proved: If for holomorphic \(\Phi\) the differential-difference equation \[ \Phi (f(s+h_ 0),...,f^{(\nu_ 0)} (s+h_ 0),...,f^{(\nu_{\mu})} (s+h_{\mu}))=0 \] holds in a right half-plane, then \(\Phi =0\). This generalizes the result of A. Ostrowski [Math. Z. 8, 241–298 (1920; JFM 47.0292.01)], who proved the result above for exactly the same class \({\mathcal D}\) of Dirichlet series in the case \(\Phi\) polynomial.

MSC:

11M41 Other Dirichlet series and zeta functions
30B50 Dirichlet series, exponential series and other series in one complex variable
34K05 General theory of functional-differential equations
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

Citations:

JFM 47.0292.01
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