Lerch’s formulae for Euler quotients in the rings Z and Fq[t] have already been studied. In this paper, we extend the study of these quotients to number fields and the Carlitz module. In the number fields case, we prove a version of Lerch’s formula for OKHil , the ring of integers of the Hilbert class field of a number field K. In the Fq[t] case, we replace the usual multiplication in Fq[t] with the Carlitz module action ? and prove two new versions of this formula. In addition, we relate these congruences to Carlitz Wieferich primes in Fq[t]. All our proofs use properties of Carlitz polynomials.
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