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

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 3

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

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

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 7

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 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.