# Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?

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.

• Apr 23, 2021 at 15:58

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 $$\gamma(i)=\frac{\pi}{4}-\ln\frac{\Gamma(1/4)^2}{\pi\sqrt{2\pi}}+i\ln\frac{8\sqrt\pi}{\Gamma(1/4)^2}.$$

This isn't a full answer but gives another surprising connection between the two constants. One has $$\gamma = \int_1^\infty \frac{1-\{x\}}{x^2} dx$$ and $$\log \frac{4}{\pi} = \int_1^\infty \frac{\Vert x \Vert}{x^2} dx,$$ where $$\{x\} = x-\lfloor x \rfloor$$ denotes the fractional part of $$x$$ and $$\Vert x \Vert$$ denotes the distance from $$x$$ to the nearest integer.

One also has $$\int_1^\infty \frac{1-\{x\}}{x^{s+1}} dx = \frac{\zeta(s)-\frac{1}{s-1}}{s}$$ while \begin{align*} \int_1^\infty \frac{\Vert x\Vert}{x^{s+1}}\, dx = \frac{(4-2^{s})\zeta(s-1)+2^{s}-1}{s(s-1)} \end{align*} for all $$s \in \mathbb{C}\backslash\{1\}$$ with positive real part. These assume the respective limits above at $$s = 1$$. I found the formulas for $$\Vert x \Vert$$ by asking, out of curiosity, what $$\int_1^\infty \frac{\Vert x\Vert}{x^{s+1}}\, dx$$ is. Interestingly, the function $$\int_1^\infty \frac{\Vert x\Vert-\tfrac{1}{4}}{x^{s+1}}\, dx$$ has analytic continuation to all of $$\mathbb{C}$$, where $$\frac{1}{4} = -3\zeta(-1)$$ is the average value of $$\Vert x \Vert$$ over any of its periods, and one has $$\int_1^\infty \frac{\Vert x\Vert-\frac{1}{4}}{x}\, dx = -3(\zeta(-1)+\zeta'(-1)) -\frac{13}{12}\log 2 = 3\log A-\frac{13}{12}\log 2 = -0.00464601\ldots,$$ where $$A$$ is the Glaisher-Kinklein constant. Moreover, the function $$\int_1^\infty \frac{\Vert x\Vert}{x^{s+1}}\, dx$$ has meromorphic continuation to $$\mathbb{C}$$ with a single (simple) pole at $$s = 0$$ with residue $$-3\zeta(-1) = \frac{1}{4}$$,

Also, I stumbled on this on Wikipedia: \begin{aligned}\gamma &=\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{m}}\\&=\log {\frac {4}{\pi }}+\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{2^{m-1}m}}.\end{aligned}

• Are you sure about this formula? I mean, should not it be just $x$, not $x^2$ in the denominator? Nov 7, 2023 at 11:11
• Also, check the sign, please $\gamma=-\underset{x\to \infty }{\text{lim}}\left(\int_1^x \frac{1}{t} \, dt-\sum _{k=1}^x \frac{1}{k}\right)$ Nov 7, 2023 at 11:12
• Yes, I'm sure. The $x^2$ and sign are correct. Nov 7, 2023 at 11:54
• Is not integral without the fractional part the same as the sum? $\sum _{k=1}^{\infty } \frac{1}{k^2}=\frac{\pi ^2}{6}$ Nov 7, 2023 at 12:06
• $\int_1^{\infty } \frac{1}{x^s} \, dx=\frac{1}{s-1}$. How do you get additional $s$ in the denominator? Nov 7, 2023 at 12:11

This is not an answer per se, but some additional insight. It seems that in certain context there is a meaning in a set of integers with period $$2e^{-\gamma}$$.

The Chow's EL-numbers are the numbers that can be obtained from $$0$$ via exponential function, logarithmic function and field operations.

For instance, constants $$e,\pi$$ and $$i$$ are all EL-numbers:

$$\begin{gather*} 1=\exp(0) \\ e=\exp(\exp(0)) \\ i=\exp\left(\frac{\log(-1)}2\right)=\exp\left(\frac{\log(0-\exp(0))}{\exp(0)+\exp(0)}\right) \\ \pi=-i\log(-1)=-\exp\left(\frac{\log(0-\exp(0))}{\exp(0)+\exp(0)}\right)\log(0-\exp(0)). \end{gather*}$$

But what happens if we extend this set with $$\lambda=\log(0)$$? It is a negatively infinite quantity, equal to the harmonic series with negaive sign:$$-\sum_{k=1}^\infty \frac1k$$. Its regularized value $$-\gamma$$.

From the equality $$\lim_{n\to\infty}\left(\sum_{k=1}^n \frac1k-\int_1^n \frac1tdt\right)=\gamma$$ (or via Laplace transform), $$\int_1^\infty\frac1xdx=-\lambda-\gamma.$$ where the integral $$\int_1^\infty\frac1xdx$$ is the germ at infinity of the logarithmic function.
If we introduce a new constant $$\omega$$ as the germ at infinity of the function $$f(x)=x$$, then (taking into account that $$f(x)=\ln x$$ has no infinitesimal part at infinity) $$\int_1^\infty\frac1xdx=\ln\omega$$ and $$e^{-\lambda}=e^\gamma \omega.$$ Now, $$\omega$$ is half the numerocity of integers, it corresponds to the numerocity of integers spaced with distance $$2$$ (like the set of even or odd numbers). So, $$e^{-\lambda}=e^\gamma\omega=\omega/(e^{-\gamma})$$ is $$e^\gamma$$ times greater, which corresponds to the numerocity of a set spaced with $$2e^{-\gamma}$$. Notice in this context that $$2\omega/\pi=\omega/(\pi/2)$$ is the numerocity of a set spaced with step $$\pi$$, such as the numerocity of the roots of sine or cosine.

Thus we see that $$2e^{-\gamma}$$ and $$\pi/2$$ somehow appear as meaningful periods of a lattice of reals.

Another interesting observation: in $$\omega/(\pi/2)$$ the denominator is EL-number but the ratio possibly not. But in $$\omega/(e^{-\gamma})$$ the ratio is EL-number but $$e^{-\gamma}$$ is not. It turns out that $$\omega$$ is EL-number iff $$\gamma$$ is EL-number.

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