# Generalization of $\gamma$ as limit involving primes

The following limit is utilized in Merten’s theorem.

$$\gamma = \lim\limits_{n\to\infty}\left(-\log\log n - \sum_{p\text{ prime}}^n\log\left(1-\frac{1}{p}\right)\right)$$

I’m interested in the generalization

$$C_x = \lim\limits_{n\to\infty}\left(x\log\log n - \sum_{p\text{ prime}}^n\log\left(1+\frac{x}{p}\right)\right)$$

For $$x\ge 1$$. Obviously we can define it in terms of the first equation. In fact for $$x=1$$ we can say $$C_1 = \log\zeta(2) - \gamma$$, but for $$x\gt 1$$ we are left with an unresolved product of primes.

Can we derive a definition of this limit that’s not in terms of primes? Perhaps we can modify a proof of Mertens.

It may be helpful to note that

$$\sum_{k=1}^{\infty} \frac{x^{\Omega(k)}}{k^s} = \prod_{p\text{ prime}}^\infty\left(1-\frac{x}{p^s}\right)^{-1}$$

• Unclear what you expect. Apr 28, 2022 at 19:43
• @reuns I revised my question Apr 29, 2022 at 16:38

Adding $$x$$ times the limit that gives $$\gamma$$ and the one that gives $$C_x$$ gives
$$C_x+x\gamma=\lim_{n\to\infty}\left(-\sum_{p\leq n}\log\left(1+\frac xp\right)-x\log\left(1-\frac 1p\right)\right)$$
$$=-\log\left(\prod_p \left(1+\frac xp\right)\left(1-\frac 1p\right)^x\right).$$