### 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>