## 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.

Zbl 0293.10001
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### References:

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