TITLE: Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?
QUESTION [55 upvotes]: Here is a couple of examples of the similarity from Wikipedia, in which the expressions differ only in signs.
I encountered other analogies as well.
$${\begin{aligned}\gamma &=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1-xy)\ln xy}}\,dx\,dy\\&=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right).\end{aligned}}$$
$${\begin{aligned}\ln {\frac {4}{\pi }}&=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1+xy)\ln xy}}\,dx\,dy\\&=\sum _{n=1}^{\infty }\left((-1)^{n-1}\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right)\right).\end{aligned}}$$
$${\begin{aligned}\gamma &=\sum _{n=1}^{\infty }{\frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}\\\ln {\frac {4}{\pi }}&=\sum _{n=1}^{\infty }{\frac {N_{1}(n)-N_{0}(n)}{2n(2n+1)}},\end{aligned}}$$
(where $N_1(n)$ and $N_0(n)$ are the number of 1's and 0's, respectively, in the binary expansion of $n$).
I wonder whether is there any algebraic system where $4e^{-\gamma}$ would play a role similar to what $\pi$ plays, say in complex numbers, or a geometric system where $4e^{-\gamma}$ would play some special role, like $\pi$ in Euclidean and Riemannian geometries.

REPLY [55 votes]: The intuition may be helped by considering the generalized Euler constant function 
$$\gamma(z)=\sum_{n=1}^\infty z^{n-1}\left(\frac{1}{n}-\ln\frac{n+1}{n}\right),\;\;|z|\leq 1.$$
Its values include the Euler constant $\gamma=\gamma(1)$ and the "alternating Euler constant" $\ln 4/\pi=\gamma(-1)$. So any general integral formula or recursion relation for $\gamma(z)$ will establish a connection of the type noted in the OP.   
The properties of the function $\gamma(z)$ have been studied in The generalized-Euler-constant function 
 and a generalization of Somos's quadratic recurrence constant (2007). Somos's constant $\sigma=\sqrt{1\sqrt{2\sqrt{3\cdots}}}$ is obtained as
$\gamma(1/2)=2\ln(2/\sigma)$.
Another special value
$$\gamma(i)=\frac{\pi}{4}-\ln\frac{\Gamma(1/4)^2}{\pi\sqrt{2\pi}}+i\ln\frac{8\sqrt\pi}{\Gamma(1/4)^2}.$$