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The Euler constant may be defined as the limit for tending to , of the difference . Alternatively, it may be defined as the limit at 1 of the difference , being a complex number in the half-plane . Mertens theorem states that for real number tending to +, , the product being over prime numbers . We prove analog results for the ring of -integers of a function field. However, in the function field case, the three approaches lead to different constants.