Let \({\mathbb{F}}_ q\) denote the finite field of q elements, \(A={\mathbb{F}}_ q[T]\), \(k={\mathbb{F}}_ q(T)\) and denote by \(k_ w\) the completion of k with respect to a valuation w of k. (Thus w is associated with a monic prime of A or, when \(w=\infty\), with 1/T.) The Carlitz zeta function of A is defined by \(\zeta_ A(s)=\sum f^{-s}, \) where f runs through all the monic polynomials of A and where s is a natural number. The function \(\zeta_ A(s)\) takes values in \(k_{\infty}\) and is not to be confused with the congruence zeta function, which takes values in \({\mathbb{C}}\). The values of the Riemann zeta function at the even natural numbers are related to the Bernoulli numbers by the relation \(\zeta (2k)=(- 1)^{k+1}(2\pi)^{2k}B_{2k}/(2-(2k)!)\). L. Carlitz [Duke Math. J. 1, 137-168 (1935; Zbl 0012.04904); cf. the author, ibid. 45, 885-910 (1978; Zbl 0404.12013)], related the value of \(\zeta_ A((q-1)m)\) to an expression \(B_{(q-1)m}\pi^{(q-1)m}/((q-1)m)!\), where \(\pi\) is transcendental over k (and is analogous to \(2\pi\) i) and the factorial ((q-1)m)!, defined in terms of the Carlitz module, is in A.
In this paper, the author begins by recalling and recasting the theory of the classical gamma function, \(\Gamma\) (s), in a form which is adapted to his treatment of the function field analogue, which follows. He defines the Carlitz factorial in terms of the Carlitz module and shows how it may be interpolated by a gamma function \(\Gamma_ w\) for \(k_ w\), where \(\Gamma_ w\) is a continuous function from \({\mathbb{Z}}_ p\) into \(k_ w\). In the classical case P. Deligne [in “Hodge cycles, motives, and Shimura varieties”, Lecture Notes Math. 900, 9-100 (1982; Zbl 0537.14006)], has shown that products of special values of \(\Gamma\) (s) generate an Abelian extension of \({\mathbb{Q}}(\zeta_ p)\), whilst Morita has constructed a p-adic gamma function related to Gauss sums (the theorem of Gross-Koblitz). The author obtains the function field analogues of those results and poses some further problems.
He describes the ‘magic numbers’, which constitute the evidence for a relationship between Carlitz factorials and the zeta functions of certain function fields arising in cyclotomic theory, and he derives the ‘rigid gamma function’, also denoted by \(\Gamma\) (s), which is a meromorphic function on \(k^*_{\infty}\times {\mathbb{Z}}_ p\), on which the L- functions are also defined. The rigid gamma function has properties analogous to those of the classical gamma function, such as a multiplication formula and a functional equation, in connection with which the author defines the analogue of sin(\(\pi\) s).
Finally he draws attention to the work of Thakur on Drinfeld modules, which is related to Gauss sums and the analogues of many other classical results.