The impact of the infinite primes on the Riemann hypothesis for characteristic p L-series, by David Goss

In \cite{go2} we proposed an analog of the classical Riemann hypothesis for characteristic $p$ valued $L$-series based on the work of Wan, Diaz-Vargas, Thakur, Poonen, and Sheats for the zeta function $\zeta_{\Fr[\theta]}(s)$. During the writing of \cite{go2}, we made two assumptions that have subsequently proved to be incorrect. The first assumption is that we can ignore the trivial zeroes of characteristic $p$ $L$-series in formulating our conjectures. Instead, we show here how the trivial zeroes influence nearby zeroes and so lead to counter-examples of the original Riemann hypothesis analog. We then sketch an approach to handling such ``near-trivial'' zeroes via Hensel's and Krasner's Lemmas (whereas classically one uses Gamma-factors). Moreover, we show that $\zeta_{\Fr[\theta]}(s)$ is not representative of general $L$-series as, surprisingly, all its zeroes are near-trivial, much as the Artin-Weil zeta-function of $\mathbb{P}^1/\Fr$ is not representative of general complex $L$-functions of curves over finite fields. Consequently, the ``critical zeroes'' (= all zeroes not effected by the trivial zeroes) of characteristic $p$ $L$-series now appear to be quite mysterious. The second assumption made while writing \cite{go2} is that certain Taylor expansions of classical $L$-series of number fields would exhibit complicated behavior with respect to their zeroes. We present a simple argument that this is not so, and, at the same time, give a characterization of functional equations.

David Goss <goss@math.ohio-state.edu>