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Values of certain \(L\)-series in positive characteristic. (English) Zbl 1336.11064

Summary: 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.

MSC:

11M38 Zeta and \(L\)-functions in characteristic \(p\)
11R58 Arithmetic theory of algebraic function fields
11F52 Modular forms associated to Drinfel’d modules
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