Let \(p\) be a prime number and let \(\mathbb{F}_q\) be a finite field having \(q=p^\varepsilon\) elements. Let \(T\) be an indeterminate over \(\mathbb{F}_q\) and let \(A= \mathbb{F}_q [T]\), \(k= \mathbb{F}_q (T)\), \(A_+\) be the set of monic polynomials in \(A\) and \(C_\infty\) be a completion of an algebraic closure of \(k_\infty\), where \(k_\infty\) is a completion of \(k\) at \(\frac 1T\).
Let \(\rho\) be the Carlitz module, i.e. \(\rho: A\to\text{End }\mathbb{G}_a\) is the morphism of rings given by \(\rho_T= TX+ X^q\). The theory of Drinfeld modules tells us that there exists a unique lattice \(\Gamma= \pi A\) associated to \(\rho\), where \(\pi\in C_\infty\). Set: \[ e(X)= \pi X\prod_{\gamma\in A\setminus \{0\}} \biggl(1- \frac X\gamma\biggr). \] We fix a monic irreducible polynomial \(P\) in \(A\) of degree \(d\). Set \(\lambda= e(\frac 1P)\), then \(\lambda\) is a root of \(\rho_P\) and \(\lambda\neq 0\). Let \(K= k(\lambda)\) be the \(P\)th cyclotomic function field. Let \(O_K\) be the integral closure of \(A\) in \(K\), then \(\lambda O_K\) is the only prime ideal of \(O_K\) above \(P\). Let \(\widehat{O}_K\) be the \(\lambda\)-adic completion of \(O_K\) and let \(\omega: (A/P)^*\to \widehat{O}_K^*\) be the Teichmüller character. Let \(i\) be an integer, \(1\leq i\leq q^d-1\), then the Goss \(L\)-function attached to \(\omega^i\) at a positive integer \(n\) is given by: \[ L(n,\omega^i)= \sum_{a\in A_+, (a,P)=1} \frac{\omega^i(a)} {a^n}. \] We refer the reader to Goss’ book [D. Goss, Basic structures of function field arithmetic, Springer-Verlag (1996; Zbl 0874.11004)] for more information on cyclotomic function fields and \(L\)-series.
Let \(\theta\in A\) be a fixed generator of \((A/P)^*\). G. Anderson [J. Number Theory 60, 165-209 (1996; Zbl 0868.11031)] obtained a formula for the values at \(s=1\) of Goss \(L\)-functions \(L(s,\omega^i)\) which is an analogue of Kummer’s formula for the Dirichlet \(L\)-functions at \(s=1\). This formula involves Anderson’s root numbers which are defined as follows: let \(U\) be the matrix \(\frac 1P(e(\frac {\theta^j}{P})^i)_{1\leq i,j\leq q^d-1}\) and write \(U^{-1}= (e_j^*(\theta^i))_{1\leq i,j\leq q^d-1}\), then Anderson’s root numbers are: \[ r_{i,j}= \sum_{a\in (A/P)^*} \omega^i(a) e_j^*(a) \qquad \text{for}\quad 1\leq i,j\leq q^d-1. \] If \(i\not\equiv j(q-1)\), Anderson proved that \(r_{i,j}=0\). Recall that the Gauss sums of Thakur are: \[ g_\ell=- \sum_{a\in (A/P)^*} \omega^{-q^\ell} (a)\rho_a(\lambda) \qquad\text{for} \quad 0\leq\ell\leq d-1. \] J. Zhao [J. Number Theory 62, 307-321 (1997; Zbl 0874.11043)] worked out the relationship of these root numbers to Thakur’s Gauss sums. He found a very interesting formula for these root numbers and suggested that, when \(r_{i,j}\neq 0\), \(r_{i,j}\) should be a certain product of Thakur’s Gauss sums up to a simple factor. The main aim of the present article is to affirm the conjecture above, more precisely, the author proves the following Theorem (Theorem 1.2): Let \(1\leq i,j\leq q^d-1\). Write \(q^d-1-i= a_0+ a_1q+\cdots+ a_{d-1} q^{d-1}\), where \(0\leq a_k\leq q-1\) for \(k=0,\dots, d-1\). Then: \[ \frac{r_{i,j}} {g_0^{a_0}\cdots g_{d-1}^{a_{d-1}}}\in A. \] As a consequence of the proof of this theorem, the author obtains several vanishing and nonvanishing results for these root numbers (see especially Corollary 3.7 and Theorem 3.9).