What are some large or small mathematical constants?

Question

I understand this question is a bit vague, but I would like to know about notable mathematical constants that are large or small, and I clarify what I mean now:

• Notable, as in not a product, sum, exponentiation, or other operation on other constants simply for the sake that they are a product, sum, exponentiation, or operation on other constants. So, not $e^{e^2}$ or $\pi^{50\gamma}$. Also, not large or small for the sake of being large or small (no googolplex.)
• Large or small, as in larger than 100 or less than 0.1.

Motivation: there is one number on Wikipedia's list of mathematical constants page larger than 100, and three less than 0.1, and I am stunned by this normality! We need better representation of large and small constants.

Answer 1

I'll start. Ramanujan's constant (sometimes called the Hermite-Ramanujan constant) is a large transcendental number that is very close to an integer. It is expressible as $$R = e^{\pi\sqrt{163}} = 262537412640768743.99999999999925\dots \approx 640320^3+744$$

Answer 2

Some are very arbitrary, like $\pi^\pi$. If that number is considered important enough for inclusion in the wiki list, then why not $\pi^{\pi^{\pi}}$, $e^e$, $e^{e^e}$, and many other tetrations?

Answer 3

As to why notable constants are generally small, I think it is best to think about what they mean and where they come from.

For example, pi is the ratio between a circles radius and its circumference. These are 2 relatively "similar" (as in the ratio between the 2 is relatively small) distances, so pi is a relatively small number.

Gamma is approximately the difference between the harmonic series in respect to n and ln(n). Both ln and the harmonic series grow very slowly and close together (about a gamma distance apart!) but both diverge.

Similar philosophies can be applied to most commonly used numbers like these.

Answer 4

Wyler's constant is defined as \begin{align}\alpha_{\small{W}}&=\frac{9}{8\pi^4}\left(\frac{\pi^5}{2^45!}\right)^{1/4}\\&=0.0072973481300\dots\\&= \frac{1}{137.0360824\dots} \end{align}

(Wyler 1969; OEIS A180872), which at the time it was proposed, agreed with experiment to within $\pm1.5$ ppm for the value of the fine structure constant $\alpha$ in physics.

The constant is a transcendental number.

Answer 5

• Constant $\sigma_3$: The non-zero constant with smallest absolute value stated in Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94 is \begin{align*} \color{blue}{\sigma_3=-0.000\,111\,158\,2...} \end{align*}

We can find in section 2.21 Stieltjes Constants: \begin{align*} \sigma_n=\sum_{\rho}\frac{1}{\rho^n}= \begin{cases} -\frac{1}{2}\ln(4\pi)+\frac{\gamma_0}{2}+1=0.023\,095\,708\,9\ldots&n=1,\\ -\frac{\pi^2}{8}+\gamma_0^2+2\gamma_1+1=-0.046\,154\,317\,2\ldots&n=2,\\ -\frac{7\zeta{3}}{8}+\gamma_0^3+3\gamma_0\gamma_1+\frac{3\gamma_2}{2}+1=\color{blue}{-0.000\,111\,158\,2\ldots}&n=3, \end{cases} \end{align*} where each sum is over all nontrivial zeros $\rho$ of the Riemann zeta function $\zeta(z)$. The constants $\gamma_n$ come from the Laurent expansion in a neighborhood of its simple pole at $z=1$: \begin{align*} \zeta(z)=\frac{1}{z-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!}\gamma_n(z-1)^n. \end{align*}

• Stieltjes Constant $\gamma_{100\,000}$:

  The coefficients $\gamma_n$ can be proved to satisfy \begin{align*} \gamma_n=\lim_{m\to\infty}\left(\sum_{k=1}^m\frac{\ln(k)^n}{k}-\frac{\ln(m)^{n+1}}{n+1}\right) \end{align*} In particular $\gamma_0=\gamma=0.577\,215\,664\,9\ldots$ is the Euler-Mascheroni constant. A somewhat larger constant of this family is the constant \begin{align*} \color{blue}{\gamma_{100\,000}=1.991\,927\,306\,3\ldots\times10^{83\,432}} \end{align*} which is stated here.