Euler constants for the ring of \(S\)-integers of a function field (Q2456037)

Summary: The Euler constant \(\gamma\) may be defined as the limit for \(n\) tending to \(+\infty\), of the difference \(\sum_{j=1}^n \frac1j-\log n\). Alternatively, it may be defined as the limit at 1 of the difference \(\sum_{n=1}^\infty \frac1{j^s}-\frac1{s-1}\), \(s\) being a complex number in the half-plane \(\text{Re}(s)>1\). Mertens' theorem states that for \(x\) a real number tending to \(+\infty\), \(\prod_{p\leq x}(1-\frac1p)\sim \frac{e^\gamma}{\log x}\), the product being over prime numbers \(\leq x\). We prove analogous results for the ring of \(S\)-integers of a function field. However, in the function field case, the three approaches lead to different constants.