Generalized Euler constants

We study the distribution of a family $\{\g(\P)\}$ of generalized Euler constants arising from integers sieved by finite sets of primes $\P$. For $\P=\P_r$, the set of the first r primes, $\g(\P_r) \to \exp(-\g)$ as r → ∞. Calculations suggest that $\g(\P_r)$ is monotonic in r, but we prove it is not. Also, we show a connection between the distribution of $\g(\P_r) - \exp(-\g)$ and the Riemann hypothesis.