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On theorems of Gelfond and Selberg concerning integral-valued entire functions. (English) Zbl 1081.11051
For each positive integer \(s\), the authors denote by \(\theta_{s}\) the positive number having the following property. On the one hand, any entire function \(g\) of exponential type \(<\theta_{s}\) satisfying \((d/dz)^{\sigma}g(m)\in\mathbb{Z}\) for any \(\sigma=0,\ldots , s-1\) and any \(m=0,1,2,\dots\) is a polynomial. On the other hand, conversely, for any \(\delta>0\) there exists an entire function \(g\), which is not a polynomial, whose first \(s\) derivatives at all the nonnegative integers are rational integers, and \(g\) has exponential type \(\leq \theta_{s}+\delta\). A well-known theorem of Pólya states that \(\theta_{2}=\log 2\). The authors point out that lower bounds for \(\theta_{s}\) have been obtained, especially by A. O. Gel’fond and A. Selberg, and that no reasonable upper bound has been established so far. Their aim is twofold. On the one hand they establish upper bounds for \(\theta_{s}\), including the estimate \(\theta_{s}\leq s\pi/3\). More precisely, a consequence of their results is that for any \(s\geq 1\), there exists a transcendental entire function, having an exponential type bounded by \(s\pi/3+\epsilon\) for any \(\epsilon>0\), whose \(s\) first derivatives at all integers (positive as well as negative) are integers. On the other hand they improve previously known lower bounds for \(\theta_{s}\). An interesting feature of their proof of these refined lower bounds is the use of the Rhin-Viola group-structure arithmetic method [G. Rhin and C. Viola, Acta Arith. 77, No. 1, 23–56 (1996; Zbl 0864.11037); ibid. 97, No. 3, 269–293 (2001; Zbl 1004.11042)], which enables them to get rid of the Selberg integral.

MSC:
11J81 Transcendence (general theory)
30D15 Special classes of entire functions of one complex variable and growth estimates
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References:
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