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Fermat versus Wilson congruences, arithmetic derivatives and zeta values. (English) Zbl 1378.11102

Summary: We look at two analogs each for the well-known congruences of Fermat and Wilson in the case of polynomials over finite fields. When we look at them modulo higher powers of primes, we find interesting relations linking them together, as well as linking them with derivatives and zeta values. The link with the zeta value carries over to the number field case, with the zeta value at 1 being replaced by Euler’s constant.

MSC:

11T55 Arithmetic theory of polynomial rings over finite fields
11R58 Arithmetic theory of algebraic function fields
11G09 Drinfel’d modules; higher-dimensional motives, etc.
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