Values of certain $L$-series in positive characteristic

Abstract

We introduce a class of deformations of the values of the Goss zeta function. We prove, with the use of the theory of deformations of vectorial modular forms as well as with other techniques, a formula for their value at $1$, and some arithmetic properties of values at other positive integers. Our formulas involve Anderson and Thakur’s function $\omega$. We discuss how our formulas may be used to investigate the existence of a kind of functional equation for the Goss zeta function.