The Impact of the Infinite Primes on the Riemann Hypothesis for Characteristic p Valued L-series

In [12] we proposed an analog of the classical Riemann hypothesis for characteristic p valued L-series based on known results for $$ \zeta _{\mathbb{F}_r [\theta ]} (s) $$ and 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. Moreover, we show that $$ \zeta _{\mathbb{F}_r [\theta ]} (s) $$ is not representative of general L-series as, surprisingly, all its zeroes are near-trivial, much as the Artin-Weil zeta-function of $$ \mathbb{P}^1 /\mathbb{F}_r $$ is not representative of general complex L-functions of curves. The second assumption in [12] is that certain Taylor expansions associated to 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, characterize functional equations.