A formal Mellin transform in the arithmetic of function fields

Goss, David

doi:10.1090/S0002-9947-1991-1041048-5

A formal Mellin transform in the arithmetic of function fields
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Trans. Amer. Math. Soc. 327 (1991), 567-582 Request permission

Abstract:

The Mellin transform is a fundamental tool of classical arithmetic. We would also like such a tool in the arithmetic of function fields based on Drinfeld modules, although a construction has not yet been found. One formal approach to finding Mellin transforms in classical theory is through $p$-adic measures. It turns out that this approach also works for function fields. Thus this paper is devoted to exploring what can be learned this way. We will establish some very enticing connections with gamma functions and the Kummer-Vandiver conjecture for function fields.

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