# Non-real constants

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.

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A cute question, the best source for checking is Steven R. Finch's book Mathematical Constants (CUP 2003) (unfortunately, I don't have nearby). –  Wadim Zudilin May 15 '10 at 13:03
Why stop with complexes and p-adics? What about quaternions or surreal numbers or even stranger number systems? –  teil May 15 '10 at 13:37
I find it rather odd calling a number a "constant". What precisely makes a number like $\pi$ a "constant"? Surely it's just an interesting number? –  Robin Chapman May 15 '10 at 14:14
I think calling numbers constants is due to the influence of physics. –  teil May 15 '10 at 14:22
Since asking this question I've also come across the term "special values" which is useful because googling "p-adic constants" didn't bring up much but "p-adic" "special values" turns up a lot of things. –  Roy Maclean May 15 '10 at 14:25

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

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More generally, roots of unity are worth mentioning. –  Benoît Kloeckner May 15 '10 at 13:02
i is really shorthand for (0,1) so in complexity is on a par with (1,0) so isn't really as interesting as (e,0) or (pi,0). –  Roy Maclean May 16 '10 at 17:50

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).

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"Carlitz period". I like it. –  Roy Maclean May 15 '10 at 14:21

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.
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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.

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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.

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You can check with Finch's book. –  Wadim Zudilin May 15 '10 at 13:06
A field of functions plus a list of some funtions that stand out within that field is a good answer. Why does the sine function stand out amongst meromorphic functions ? –  Roy Maclean May 15 '10 at 13:21
A variation of the sine function does stand out: $\sin(\pi z)/\pi=z\prod_{n\ne 0}(1-z/n)$ (with the usual interpretation of the product). –  Victor Protsak Jun 8 '10 at 17:19

Initial or terminal objects in categories.

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

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Can you expand the answer? –  Wadim Zudilin May 15 '10 at 13:05
Whenever you have any formula relating objects of a category, the initial or terminal object is going to appear. For instance, characteristic involves the initial object, and exact sequences uses zero object. –  Colin Tan May 15 '10 at 13:17
I am not sure that these can be counted as constants. But this is subjective. –  Wadim Zudilin May 15 '10 at 13:21
Yes it is subjective. But the OP is allows "whole objects... that appear in many formulae that relate such objects." –  Colin Tan May 15 '10 at 14:41

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.

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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.

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Perhaps $\aleph_0$

Perhaps certain complex roots of L-functions.

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"The only p-adic numbers which aren't real are infinite." The $p$-adics and the real numbers do not naturally live inside any common field, so I don't know what it means for a $p$-adic number to be real. –  Pete L. Clark May 15 '10 at 13:56
I thought p-adics were Laurent series in p, so that you could evaluate the series as a real number and identify them that way. –  teil May 15 '10 at 14:04
Every element of $\mathbb{Q}_p$ has a unique representation as a convergent series $\sum_{n = N_0}^{\infty} a_n p^n$ with $a_n \in \{0,\ldots,p-1\}$, yes. But this series is convergent in the $p$-adic topology. If $x$ is not in $\mathbb{Q}$, as a real series the terms do not tend to zero, so of course the series diverges. Maybe this is what you were getting at? –  Pete L. Clark May 15 '10 at 14:43