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Transcendental Numbers and Periods TRANSCENDENTAL NUMBERS AND PERIODS JAMES CARLSON Contents 1. Introduction 1 1.1. Diophantine approximation I: upper bounds 2 1.2. Diophantine approximation II: lower bounds 4 1.3. Proof of the lower bound 5 2. Periods 6 2.1. Periods 6 3. Computable numbers 8 4. References 11 1. Introduction These are notes for a talk given at Colorado State University on February 12, 2015. The numbers we are familiar with fall into a hierarchy of sets of ever greater scope: N ⊂ Z ⊂ Q ⊂ Q¯ ⊂ C These are the natural numbers, the integers, the rationals, the field of algebraic numbers, and complex numbers, respectively. The field Q¯ is the set of all possible solutions of polynomial equations with rational coefficients. Such equations can be enumerated, so Q¯ , like the field of rational numbers itself, is countable. Nonetheless, the Q¯ is considerably larger than Q in the sense that it infinite dimensional rational vector space. For example, Date: 02-19-2014. 1 2 JAMES CARLSON p the numbers p, where p is prime, are linearly independent. These reults have their roots in the Greek mathematics of around 500 BC. Consider now the field of complex numbers. By the work of Georg Cantor, it is anun- countable field—a remarkable fact that rests on the Cantor’s famous diagonal argument. It follows that ”most” numbers are transcendental. That is, they satisfy no polynomial equation with rational coefficients. Consequently, there is a decomposition C = Q¯ [ (C − Q¯ ) where the first term is countable and the second is uncountable. Note also that thefirstis of measure zero while the second is of infinite measure. The same considerations apply to the deomposition [0; 1] = (Q¯ \ [0; 1]) [ (C − Q¯ ) \ [0; 1]; except that [0; 1] and the second term on the right have measure one. We are now faced with a paradox. The set of algebraic numbers is small both in cardinality and measure relative to the set of all complex numbers. A number chosen at random is transcendental with probability one. Despite this fact, finding explicit transcendental numbers is quite hard. Specific transcendental numbers were exhibted first exhibited in 1844 by Liouville. Theorem 1. The ”Liouville” number 1 X 1 λ = : 10n! n=0 is transcendental. In what follows we discuss (1) The proof of Theorem 1 (2) The definition and properties of a countable class of transcendental numbers, the periods (3) The Konsevich-Zagier philosophy that almost all known transcendental numbers are periods. (4) Yoshinaga’s solution of the problem of exhibiting an explicit non-period, analogous to Liouville’s construction of λ. (5) The K-Z conjecture that e is not a period. The first item depends on arguments in Diophantine approximation,which we take upnow. TRANSCENDENTAL NUMBERS AND PERIODS 3 1.1. Diophantine approximation I: upper bounds. The first numbers known to be transcendental are the so-called Liouville numbers. To understand them, we must consider the task of approximating an arbitrary real number by rational numbers. If we divide the number line at the rational numbers with denominator q, we divide it into intervals of length 1/q. Consequently, for any real number α and any integer q > 0, there is an integer p such that p 1 α − < : q 2q It turns out, however, that there are specially good ways of approximating a real number by a rational number. These are the continued fractions. Suppose given a real number α > 0. Write it as the sum of an integer and a fractional part, α = [α] + f α g; where 0 < α < 1. Thus for α = π we have π = 3 + 0:14159265:: Let α1 = 1/f α g, and treat α1 in the same way as we treated α: α1 = [α1] + f α1 g; In the case of π, α1 = 7 + 0:062513484986686 Putting together what we have done s far gives us 1 π = 3 + 7 + 0:062513484986686 ::: This is the beginning of the continued fraction expansion of π. You are urged to work out more terms. The next step is + 1 π = 3 + 1 7 + 15+ϵ where ϵ = 0:9 :::. 4 JAMES CARLSON Convergents. Truncate the continued fraction expansion at any stage, discarding the fractional part if necessary. The resulting compound fractions are called convergents, and can be notated as c0, c1, etc. They satisfy lim cn = α n!1 In the case of π, c0 = 3, and 22 c = : 1 7 For the second convergent, set ϵ = 0 to get 1 c2 = 3 + 1 ; 7 + 15 then 1 c = 3 + ; 1 7:06666666666666667 and finally c1 = 3:14150943396226415 + −5 −2 Thus jπ − c2j < 8:4 × 10 . By contrast, 1/15 < 7 × 10 . These popular ancient values of π give especially good approximations, e.g., 22 1 π − = jπ − 3:142857142857144::j < 0:001264493 << 7 2 · 7 Indeed, we have the following general result: The convergents p/q of a real number α satisfy p 1 α − < : q q2 TRANSCENDENTAL NUMBERS AND PERIODS 5 1.2. Diophantine approximation II: lower bounds. The key to showing that λ is transcendental is the following lower bound on approximation of algebraic numbers by rational numbers: Theorem 2. Let α is an irrational number which is a root of an a polynomial of degree n with integer coefficients. Then there is a constant c(α) such that p c(α) α − > q qn Since the partial sums of the series in Theorem 1 converge exponentailly fast, we conclude the following: Corollary 1. Liouville’s number λ is transcendental. Much more sophisticated arguments in Diophantine approximation soon came. Theorem 3. (Hermite 1873) e is transcendental. Theorem 4. (Lindemann 1882) π is transcendental. Theorem 5. (Weierstrass-Lindemann) The exponential of a nonzero algebraic number is transcendental. The transcendence of e is an immediate consequence of the Weierstrass-Lindemann theo- rem. That theorem also implies the transcendence of π: observe that p e2π −1 = 1 p If π is algebraic, then so is 2π −1. Therefore the expression above is transcendental, a contradiction. 1.3. Proof of the lower bound. Let α be a simple root of a polynomial f(x) with integer coefficients. By the mean value theorem, ( ) ( ) ( ) p p 0 p p f(α) − f = f = f a − q q q q We can rewrite this as ( ) ( ) −1 p p 0 p a − = f f q q q 6 JAMES CARLSON j − p j The interval a q is bounded. Since α is a simple root of f, the derivative of f is nonzero near α and so is bounded below by some non-zero constant c(α)−1. Thus we have ( ) p p a − ≥ c(α) f q q Now write ( ) p a f = q qn where ( ) p a = qnf q is a a polynomial with integer coefficients evaluated on an integer. Since a =6 0, jaj ≥ 1. This observation yields the result. 2. Periods Kontsevich and Zagier posit that almost all transcendental numbers we know are in fact members of a countable set P, the periods. Let us take as a provisional definition of period the length, area, or n-dimensional volume of a set defined by polynomial inequalities with rational coefficients. This set is clearly countable, and it includes Z p p/q = dx; q 0 where p and q are integers, the roots Z p n a = dx; f 0 ≤ xn≤ a g and all algebraic numbers. It also includes the special transcendental numbers Z p p/q dx log = q 1 x TRANSCENDENTAL NUMBERS AND PERIODS 7 Z π = dx ^ dy x2+y2≤1 Thus our hierarchy of numbers now reads N ⊂ Z ⊂ Q ⊂ Q¯ ⊂ P ⊂ C Remark. Let a be real number in the interval (0; 1) that has infinitiely many nonzero binary digits ai. The Liouville number 1 X a λ(a) = n 10n! n=1 is transcendental. Thus there is an uncountable set of ”explicit” transcendental numbers which are not periods. 2.1. Periods. in algebraic geometry A seemingly much larger class of interesting numbers comes from algebraic geometry. Let X be an algebraic variety defined over the rational numbers, and let ! be any rational differential k-form on X with rational coefficients. Let γ be a homology cycle of dimension k. Then we an form the integral Z α = !: γ This gives a seemingly much more general class of periods. However, Kontsevich and Zagier remark that γ can be approximated by a homologous cycle whose pieces are of the first kind in suitable coordinates. Thus there is no real gain in generality. As an example, consider the form ! = dz/z on the punctured complex line C∗ = f z 2 C j z =6 0 g. Let γ be the unit circle centered a the origin taken with some parameterization. Then Z dz p γ = 2π −1 z is a period in the second sense. Replacing γ by a polygon which encircles the origin, we see that it is a period in the seemingly more restricted sense. 8 JAMES CARLSON For another example, consider the elliptic curve E defined by the equation y2 = x(x − 1)(x − a) A homology basis f δ; γ g for E is given in Homology basis A. Figure 1. Homology basis A The cycle δ encircles the branch points 0 and 1, winding once around the branch cut from 0 to 1. The cycle γ starts from a point p on a branch cut from a to infinity, travels in one sheet of E to a point q on the he branch cut, and from there back p in the other sheet. By deforming contours, one sees that Z Z a dx ! = : γ 1 y Such integrals, classically known as elliptic integral, are transcendental—the result of deep and difficult theorems of Siegel, Schneider, and Gelfond.
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