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Transcendence of the values of the Carlitz zeta function by Wade’s method. (English) Zbl 0743.11070

Let \(i>1\) be a positive integer. The zeta-values, \(\zeta(i)=\sum^ \infty_{n=1}n^{-i}\), have long been of interest to mathematicians. This interest stems from the elegant formula of Euler which states that for \(i\equiv 0\bmod 2\) one has \(\zeta(i)/\pi^ i\in\mathbb{Q}\). These zeta- values were then later subsumed into the complex analytic function \(\zeta(s)\) for \(s\in\mathbb{C}-\{1\}\) by Riemann. (For a very enjoyable introduction to Euler’s approach to \(\zeta(i)\) for both positive and negative \(i\), see R. Ayoub, Euler and the zeta-function, Am. Math. Mon. 81, 1067-1086 (1974; Zbl 0293.10001).)
Now let \({\mathbb{A}}=\mathbb{F}_ r[t]\) be the polynomial ring over the finite field with \(r\) elements. Let \({\mathbb{K}}=\mathbb{F}_ r\left(\left({1\over t}\right)\right)\) be the completion of \(\mathbb{F}_ r(t)\) at the infinite prime. In 1935, L. Carlitz began the study of the “zeta-values” \(\zeta(i)=\sum_{n\in{\mathbf A}, n\text{ monic}}n^{-i}\in{\mathbb{K}}\) for \(i\) a positive integer. The analogy with the above special values studied by Euler is clear. Indeed, let \(\xi\) be the “period” of the Carlitz exponential (i.e., the \({\mathbb{A}}\)-analog of \(2\pi i)\); then one knows that for \(i\equiv 0 \pmod {r-1}\), one has \(\zeta(i)/\xi^ i\in\mathbb{F}_ r(t)\). As Wade has established that \(\xi\) is transcendental over \(\mathbb{F}_ r(t)\), we deduce the same for \(\zeta(i)\) for the above values of \(i\). These zeta-values were also later put into an “analytic” function (which is defined on the space \(S_ \infty=\overline{\mathbb{K}}^*\times\mathbb{Z}_ p)\).
These zeta values have recently attracted a great deal of interest. This has culminated in the theorem of Jing Yu which states: 1) \(\zeta(i)\) is transcendental for all positive \(i\); 2) Let \(i\not\equiv 0 \pmod {r-1}\), then \(\zeta(i)/\xi^ i\) is also transcendental. In this paper, the authors give a proof of the first part of Yu’s results which is based on Wade’s methodology.
Reviewer: D.Goss (Columbus)

MSC:

11T55 Arithmetic theory of polynomial rings over finite fields
11R58 Arithmetic theory of algebraic function fields
11G09 Drinfel’d modules; higher-dimensional motives, etc.

Citations:

Zbl 0293.10001
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References:

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