Non-real constants

Question

Constants are usually real numbers e.g. e, pi, gamma etc. Can you give examples of special constants that are not real? e.g. complex or p-adic constants.

A real number in base10 can be viewed as the coefficients of a power series evaluated at x=1/10, so I suppose a constant in another context such as a complete ring could just be some value of a function evaluated at some point. Can you give examples of such a value that could be considered as a special mathematical constant.

More generally a whole object such as a set or group could be thought of as a constant if it appeared in many formulae relating such objects.

Answer 1

I have a curious example of $p$-adic constant which can be called Kurepa--Vladimirov constant, although it's very close to one of Euler's constants, $-1=\sum_{n=0}^\infty n\cdot n!$.

In 1971, D. Kurepa [Math. Balkanica 1 (1971) 147--153] introduced the left factorial $!n=\sum_{k=0}^{n-1}k!$ and investigated its divisibility properties. One of his conjectures, finally proved by D. Barsky and B. Benzaghou [J. Th\'eor. Nombres Bordeaux 16 (2004) 1--17] asserts that for any odd prime $p$ the left factorial $!p$ is never divisible by $p$. This is equivalent to saying that the corresponding $p$-adic number $$ \xi=\xi_p=\sum_{n=0}^\infty n! $$ is a $p$-adic unit, $|\xi|_p=1$ (cf. [V.S. Vladimirov, Publ. Inst. Math. (Beograd) (N.S.) 72(86) (2002) 11--22]). On the other hand, a folklore conjecture says that the number $\xi_p$ is irrational for any prime $p$. The problem is considered to be hard, since a very similar number $\sum_{n=0}^\infty n\cdot n!$ is $-1$ in any $p$-adic valuation, the fact already known to L. Euler. Note that $\xi$ is one of the simplest constants (from the definining series point of view), for which the expected irrationality was not yet shown.

Answer 2

Well for me the imaginary unit $i = \sqrt{-1}$ is a very natural yet non-real mathematical constant.

Answer 3

Alongside the ubiquitous complex period $2\pi i$, there's also the p-adic analogue $t$ which is a uniformiser in Fontaine's period ring $B_{dR}^+$ (see for instance anything written by Colmez e.g. this) as well as characteristic $p$ analogues such as the Carlitz period (see e.g. notes of Brownawell and Papanikolas).

Answer 4

We often encounter interesting constants as:

1. solutions to interesting equations
2. values of special functions at special inputs
3. integrals of differential forms on geometric objects

The third case is possibly viewable as a special case of the first. None of these are necessarily constrained to be real. Here are some examples:

1. Small-order roots of unity (complex or p-adic) have plenty of number-theoretic utility, along with roots of interesting non-cyclotomic polynomials (e.g., $x^p-x-1/p$ in the p-adic world). We get $2\pi i$ by choosing a generator of the kernel of the exponential map $\mathbb{C} \to \mathbb{C}$, i.e., a special solution to the equation $e^z = 1$. Integers in imaginary quadratic fields can be viewed as locations of poles of special Weierstrass functions, and the zeroes of the Riemann zeta function are reasonably interesting, although perhaps not as isolated examples.
2. Constants like $e$ and $\gamma$ arise as values of special functions, namely $e^x$ and $-\Gamma'(x)$ at $x=1$. We can get similar constants in other rings by evaluating special functions in those domains, or taking residues at poles. For example, certain modular functions evaluated at imaginary quadratic integers yield algebraic integers whose degree depends on class number.
3. We can also think of $2 \pi i$ as the integral of $dz/z$ along a 1-cycle in $\mathbb{C}^\times$. More complicated constants arise from integrals on cycles in more complicated varieties. These are known as periods, and they exist in both complex and nonarchimedean worlds.

Answer 5

Initial or terminal objects in categories.

Eg: The integers ${\mathbb{Z}}$ are initial in the category of rings.

Answer 6

Since all the known non-trivial zeroes of Riemann Zeta function are on the Re(z)=1/2 line we only give their imaginary parts, but in fact their are complex,

$$1/2+ i*14.1347251417346937904572519835624702707842571156992431756855674601499...$$

being the first above the real line. If Riemann Hypothesis is true we will never have to mention a different real part.

Answer 7

What do you mean by a "constant"? The sine function is a very interesting constant in the field of meromorphic functions on the complex plane.

Answer 8

Since we do not want to restrict ourselves to values in traditional number systems...

• Each remarquable/exceptional finite algebraic structure, graph, can be described/encoded as a specific value in a numbering system. For instance as a series of generating matrices, multiplication table, representation tables, etc.

• We can consider each sporadic group as a remarquable constant. To me the Monster Group is a mathematical attraction point that can be compared to $\pi$ or $e$. And it is well hidden in the armies of soluble groups around it.

Answer 9

Perhaps $\aleph_0$

Perhaps certain complex roots of L-functions.

Answer 10

Good question. I will list some constants groupped mainly into 4 categories: 1) hypercomplex numbers, 2) umbral calculus, 3) formal power series, infinite germs and divergent integrals and series, 4) linear operators (infinite matrices).

1. Hypercomplex numbers

• Split-complex (tessarine) unity $j$. Logarithm: $\ln j=\frac{i \pi }{2}-\frac{i j\pi}{2}$. Has property $i^j=ij$, where $i$ is complex unity.

• Split-complex idempotents and zero divisors $1/2+j/2$ and $1/2-j/2$. Have properies $0^{1/2+j/2}=1/2-j/2$ and $0^{1/2-j/2}=1/2+j/2$

• Dual unity $\varepsilon$ and its roots in higher-dimension algebras (e.g. $\varepsilon_1^2=\varepsilon$), all nilpotents. Has property $f(\varepsilon)=f(0)+\varepsilon f'(0)$

• Grassmann algebra generators $\theta_i$ and their products, also all nilpotents.

• Triplex numbers unities $j_1$ and $j_2$. As well as $c=i^{j_1+j_2}=\frac{1}{3} \left(-\left(\left(\sqrt{3}+1\right) j_1\right)+\left(\sqrt{3}-1\right) j_2-1\right)$ (here $i$ is usual complex unity), satisfying $c^3+c^2+c+1=0$. Logarithm: $\ln j_1=\frac{2 \pi \left(j_1-j_2\right)}{3 \sqrt{3}}$, $\ln c=\frac{1}{3} i \pi j_1-\frac{\pi j_1}{2 \sqrt{3}}+\frac{1}{3} i \pi j_2+\frac{\pi j_2}{2 \sqrt{3}}+\frac{i \pi }{3}$. $j_1^2=j_2$, $j_2^2=j_1$, $j_1j_2=1$.

2. Umbral calculus

• The most important is Bernoulli's umbra $B^-$ and $B^+=B^-+1$. $\operatorname{eval}(B^-)=-1/2$, $\operatorname{eval}(B^+)=1/2$ (here "eval" is evaluation, akin to taking real part). Logarithm of $B^-$ is undefined, but $\operatorname{eval}(\ln B^+)=-\gamma$. Links logarithms with trigonometric functions: $\operatorname{eval}\frac1{\pi }\ln \left(\frac{B^+-\frac{x}{\pi }}{B^-+\frac{x}{\pi }}\right)=\cot x$.

• $e^{B^-}$ and $e^{B^+}$ respectively have evaluations $\frac 1{e-1}$ and $\frac1{1-1/e}$. $(-1)^{B^-+1/2}$ has evaluation $\frac{\pi}2$.

• Euler's umbra $E$. $\operatorname{eval}E=0;$ $\operatorname{eval}\ln E=-\pi/2$

3. Infinite germs and divergent integrals/series.

• The most simple infinite number $\omega$ is the germ at infinity of the function $f(x)=x$. Denoted in Levi-Civita field as $\varepsilon^{-1}$ (so has inverse $\varepsilon$ in that field, as well as in Hardy field). Can be represented as divergent integrals $\int_0^\infty dx$ and $\int_0^\infty\frac1{x^2}dx$ (via Laplace transform). Half the numerocity of integers, equal to numerocity of even or odd numbers. Divergent sum representation: $1/2+\sum_{k=1}^\infty 1$. Has finite part (regularized value) $0$. Also, the germ of the function $\frac1x$ at zero (from above). If we modify Delta distribution to behave like a function, $\omega=\pi\tilde{\delta}(0)$ (via Fourier transform).

• $\lambda=-\int_0^1 \frac1x dx$. This is negatively infinite constant. Via its definition can be used to extend logarithmic function to zero (thus $\lambda=\ln 0$). Has finite part (regularized value) $-\gamma$. Since Harmonic series is the Riemann sum of this integral, $\lambda$ is the negative of the sum of Harmonic series: $\lambda=-\sum_{k=1}^\infty \frac1k$. From this equality (or via Laplace transform), $\int_1^\infty\frac1xdx=-\lambda-\gamma$. Thus, $\int_0^\infty \frac1x dx=-2\lambda-\gamma$. Other integral representations include: $\int_0^\infty\frac{e^{-x}-1}{x}dx,-\int_0^\infty \frac{1}{x^2+x}dx$. Can be expressed via $\omega$: $\lambda=-\ln\omega-\gamma$. Some divergent integrals can be expressed in terms of $\lambda$: $\int_0^\infty \frac{\ln t}{t} \, dt=\gamma\lambda+\gamma^2/2,$ $\int_0^{\infty } | \tan x| \, dx=-\lambda\omega$. If we generalize the notions of periods and Chow's $EL$-numbers to our extended set, then $\lambda$ would be both, because it can be represented as $\int_0^1 \frac{-1}t dt$ (integral of an algebraic function over algebraic domain) and $\ln 0$ respectively. Logarithms of zero divisors can be expressed via $\lambda$, in split-complex numbers: $\ln \left(\frac{a j}{2}+\frac{a}{2}\right)=\frac{j}{2} (\ln a-\lambda)+\frac{1}{2} (\ln a+\lambda)$, in dual numbers $\ln \varepsilon=\lambda+\varepsilon \omega$, $\varepsilon^\varepsilon=1+\varepsilon(1+\lambda)$.

• $-\lambda-\gamma$. Equal to $\ln \omega$ and germ at infinity of logarithmic function $f(x)=\ln x$. Has finite part $0$. Unlike $\lambda$, is neither generalized period, nor EL-number. Has representations $\int_1^\infty\frac1xdx,$ $\int_0^\infty \frac{e^{-x}}{x}dx,$ $\int_0^\infty \frac{x-\log (x)-1}{(x-1)^2}dx$.

• $\int_{-\infty}^\infty e^x dx$. Equal to $e^\omega$. Has finite part $0$. Germ of the function $f(x)=e^x$ at infinity.

• $\sum_{k=1}^\infty k=\frac{\omega^2}2-\frac1{12}$ Has finite part $-1/12$.

4. Among infinite matrices one constant is the matrix of derivative operator $D$. Many other operators can be represented via $D$: finite difference $\Delta=e^D-1$, Fourier transform $\mathcal{F}=e^{\frac{\pi i}{4}(D^2-x^2+1)}$, scaling $f(x)\to f(ax)=a^{xD}$, etc.

Answer 11

Special values of $p$-adic zeta functions, or, more generally, $p$-adic $L$-functions. Start with the most basic fact that $\sum_{n = 0}^\infty p^n = 1/(1-p)$ in the ring of $p$-adic integers.

https://en.wikipedia.org/wiki/P-adic_L-function