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Approximation by translates of Taylor polynomials of the Riemann zeta function. (English) Zbl 1221.30084

From the authors’ summary: “We show that each function holomorphic on a compact set with connected complement can be approximated by translates of Taylor polynomials of the Riemann zeta function. From this it can be concluded that each entire function can be approximated by translates of Taylor polynomials of the \(\zeta\)-function.”
For the proof the author uses a result of V. Nestoridis in [Ann. Inst. Fourier 46, No. 4, 1293–1306 (1996; Zbl 0865.30001)]. The paper ends with the remark that the universality property proved for the Riemann zeta function is shared by almost all entire functions. But one knows explicitly not one other entire function aside the Riemann zeta function (and its close cousins, e.g., other zeta functions) having such a universality property.

MSC:

30E10 Approximation in the complex plane
30B10 Power series (including lacunary series) in one complex variable
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
30B99 Series expansions of functions of one complex variable

Citations:

Zbl 0865.30001
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Full Text: DOI

References:

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