Abstract. February 3, Centre of Excellence in Mathematics (CEM) Department of Mathematics, Mahidol University

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1 February 3, 202 Centre of Excellence in Mathematics (CEM) Department of Mathematics, Mahidol University Transcendental Number Theory: recent results and open problems. Michel Waldschmidt Abstract An algebraic number is a complex number which is a root of a polynomial with rational coe cients. For instance p 2, i = p, the Golden Ratio ( + p 5)/2, the roots of unity e 2i a/b, the roots of the polynomial X 5 6X +3are algebraic numbers. A transcendental number is a complex number which is not algebraic. Institut de Mathématiques de Jussieu miw/ /90 2/90 Abstract (continued) The existence of transcendental numbers was proved in 844 by J. Liouville who gave explicit ad-hoc examples. The transcendence of constants from analysis is harder ; the first result was achieved in 873 by Ch. Hermite who proved the transcendence of e. In 882, the proof by F. Lindemann of the transcendence of gave the final (and negative) answer to the Greek problem of squaring the circle. The transcendence of 2 p2 and e, which was included in Hilbert s seventh problem in 900, was proved by Gel fond and Schneider in 934. During the last century, this theory has been extensively developed, and these developments gave rise to a number of deep applications. In spite of that, most questions are still open. In this lecture we survey the state of the art on know results and open problems. miw/ Rational, algebraic irrational, transcendental Goal : decide upon the arithmetic nature of given numbers: rational, algebraic irrational, transcendental. Rational integers : Z = {0, ±, ±2, ±3,...}. Rational numbers : Q = {p/q p 2 Z, q 2 Z, q > 0, gcd(p, q) =}. Algebraic number : root of a polynomial with rational coe cients. A transcendental number is complex number which is not algebraic. 3/90 4/90

2 Rational, algebraic irrational, transcendental Goal : decide wether a given realnumberisrational, algebraic irrational or else transcendental. Question : what means given? Criteria for irrationality : development in a given basis (e.g. : decimal expansion, binary expansion), continued fraction. Analytic formulae, limits, sums, integrals, infinite products, any limiting process. Algebraic irrational numbers Examples of algebraic irrational numbers : p 2, i = p, the Golden Ratio ( + p 5)/2, p d for d 2 Z not the square of an integer (hence not the square of a rational number), the roots of unity e 2i a/b, for a/b 2 Q, and, of course, any root of an irreducible polynomial with rational coe cients of degree >. 5/90 6/90 Rule and compass ; squaring the circle Construct a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. Any constructible length is an algebraic number, though not every algebraic number is constructible (for example 3 p 2 is not constructible). Quadrature of the circle Marie Jacob La quadrature du cercle Un problème à la mesure des Lumières Fayard (2006). Pierre Laurent Wantzel (84 848) Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas. Journal de Mathématiques Pures et Appliquées (2), (837), /90 8/90

3 Resolution of equations by radicals The roots of the polynomial X 5 6X +3are algebraic numbers, and are not expressible by radicals. Evariste Galois (8 832) Gottfried Wilhelm Leibniz Introduction of the concept of the transcendental in mathematics by Gottfried Wilhelm Leibniz in 684 : Nova methodus pro maximis et minimis itemque tangentibus, qua nec fractas, nec irrationales quantitates moratur,... Born 200 years ago. 9/90 Breger, Herbert. Leibniz Einführung des Transzendenten, 300 Jahre Nova Methodus von G. W. Leibniz ( ), p Franz Steiner Verlag (986). Serfati, Michel. Quadrature du cercle, fractions continues et autres contes, Editions APMEP, Paris (992). 0 / 90 Irrationality Given a basis b 2, a real number x is rational if and only if its expansion in basis b is ultimately periodic. b = 2: binary expansion. b = 0: decimal expansion. For instance the decimal number is rational : 0, = = First decimal digits of p / 90 2 / 90

4 First binary digits of p 2 Computation of decimals of p decimals computed by hand by Horace Uhler in decimals computed in decimals in decimals computed by Yasumasa Kanada and Daisuke Takahashi in 997 with Hitachi SR220 in 7 hours and 3 minutes. Motivation : computation of. 3 / 90 4 / 90 Square root of 2 on the web The first decimal digits of p 2 are available on the web, 4,, 4, 2,, 3, 5, 6, 2, 3, 7, 3, 0, 9, 5, 0, 4, 8, 8, 0,, 6, 8, 8, 7, 2, 4, 2, 0, 9, 6, 9, 8, 0, 7, 8, 5, 6, 9, 6, 7,, 8,... Pythagoras of Samos 569 BC 475 BC a 2 + b 2 = c 2 =(a + b) 2 2ab. The On-Line Encyclopedia of Integer Sequences Neil J. A. Sloane 5 / / 90

5 Irrationality in Greek antiquity Irrationality of p 2 Platon, La République : incommensurable lines, irrational diagonals. Theodorus of Cyrene (about 370 BC.) irrationality of p 3,..., p 7. Theetetes : if an integer n > 0 is the square of a rational number, then it is the square of an integer. Pythagoreas school Hippasus of Metapontum (around 500 BC). Sulba Sutras, Vedic civilization in India, BC. 7 / 90 8 / 90 Émile Borel : 950 Émile Borel (87 956) The sequence of decimal digits of p 2 should behave like a random sequence, each digit should be occurring with the same frequency /0, each sequence of 2 digits occurring with the same frequency /00... I Les probabilités dénombrables et leurs applications arithmétiques, Palermo Rend. 27, (909). Jahrbuch Database JFM I Sur les chi res décimaux de p 2 et divers problèmes de probabilités en chaînes, C. R. Acad. Sci., Paris 230, (950). Zbl / / 90

6 Complexity of the b ary expansion of an irrational algebraic real number Let b 2 be an integer. É. Borel (909 and 950) : the b ary expansion of an algebraic irrational number should satisfy some of the laws shared by almost all numbers (with respect to Lebesgue s measure). Remark : no number satisfies all the laws which are shared by all numbers outside a set of measure zero, because the intersection of all these sets of full measure is empty! \ R \{x} = ;. x2r More precise statements by B. Adamczewski and Y. Bugeaud. Conjecture of Émile Borel Conjecture (É. Borel). Let x be an irrational algebraic real number, b 3 apositiveintegeranda an integer in the range 0 apple a apple b. Then the digit a occurs at least once in the b ary expansion of x. Corollary. Each given sequence of digits should occur infinitely often in the b ary expansion of any real irrational algebraic number. (consider powers of b). An irrational number with a regular expansion in some basis b should be transcendental. 2 / / 90 The state of the art What is known on the decimal expansion of p 2? There is no explicitly known example of a triple (b, a, x), where b 3 is an integer, a is a digit in {0,...,b } and x is an algebraic irrational number, for which one can claim that the digit a occurs infinitely often in the b ary expansion of x. A stronger conjecture, also due to Borel, is that algebraic irrational real numbers are normal : each sequence of n digits in basis b should occur with the frequency /b n, for all b and all n. The sequence of digits (in any basis) of p 2 is not ultimately periodic Among the decimal digits {0,, 2, 3, 4, 5, 6, 7, 8, 9}, at least two of them occur infinitely often. Almost nothing else is known. 23 / / 90

7 Complexity of the expansion in basis b of a real irrational algebraic number 2 Irrationalityoftranscendentalnumbers The number e The number Theorem (B. Adamczewski, Y. Bugeaud 2005 ; conjecture of A. Cobham 968). If the sequence of digits of a real number x is produced by a finite automaton, then x is either rational or else transcendental. 25 / 90 Open problems 26 / 90 Introductio in analysin infinitorum Joseph Fourier Leonhard Euler (737) ( ) Introductio in analysin infinitorum Fourier (85) : proof by means of the series expansion e =+! + 2! + 3! + + N! + r N with r N > 0 and N!r N! 0 as N! +. e is irrational. Continued fraction of e : e = Course of analysis at the École Polytechnique Paris, / / 90

8 Variant of Fourier s proof : e is irrational F. Beukers : alternating series For odd N,! + 2! N! < e <! + 2! a N N! < e < a N N! + (N +)!, a N 2 Z Hence N!e is not an integer. a N < N!e < a N +. + (N +)! Irrationality of Āryabhaṭa, born 476 AD : Nīlakaṇṭha Somayājī, born 444 AD : Why then has an approximate value been mentioned here leaving behind the actual value? Because it (exact value) cannot be expressed. K. Ramasubramanian, The Notion of Proof in Indian Science, 3th World Sanskrit Conference, / / 90 Irrationality of Lambert and Frederick II, King of Prussia Johann Heinrich Lambert ( ) Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques, Mémoires de l Académie des Sciences de Berlin, 7 (76), p ; lu en 767 ; Math. Werke, t. II. Quesavezvous, Lambert? Tout,Sire. Etdequile tenez vous? Demoi-même! tan(v) is irrational when v 6= 0is rational. As a consequence, is irrational, since tan( /4) =. 3 / / 90

9 Known and unknown transcendence results Catalan s constant Known : e,, log 2, e p2, e, 2 p2, (/4). Not known : e +, e, log, e, (/5), (3), Euler constant Is Catalan s constant X ( ) n (2n +) 2 n = an irrational number? Why is e known to be transcendental while e is not known to be irrational? Answer : e =( ) i. 33 / / 90 Catalan s constant, Dirichlet and Kronecker Catalan s constant is the value at s =2of the Dirichlet L function L(s, 4) associated with the Kronecker character 8 n >< 0 if n is even, 4(n) = = if n (mod 4), 4 >: if n (mod 4). Catalan s constant, Dedekind and Riemann The Dirichlet L function L(s, 4) associated with the Kronecker character 4 is the quotient of the Dedekind zeta function of Q(i) and the Riemann zeta function : Q(i) (s) =L(s, 4) (s) Johann Peter Gustav Lejeune Dirichlet Leopold Kronecker / 90 Julius Wilhelm Richard Dedekind Georg Friedrich Bernhard Riemann / 90

10 Riemann zeta function The function (s) = X n s n was studied by Euler ( ) for integer values of s and by Riemann (859) for complex values of s. Euler : for any even integer value of s arationalmultipleof s. 2, the number (s) is Examples : (2) = 2 /6, (4) = 4 /90, (6) = 6 /945, (8) = 8 /9450 Coe cients : Bernoulli numbers. Riemann zeta function The number (3) = X n =, n3 is irrational (Apéry 978). Recall that (s)/ s is rational for any even value of s 2. Open question : Is the number (3)/ 3 irrational? 37 / / 90 Riemann zeta function Infinitely many odd zeta values are irrational Tanguy Rivoal (2000) Is the number (5) = X n irrational? = n5 T. Rivoal (2000) : infinitely many (2n +)are irrational. Let > 0. For any su ciently large odd integer a, the dimension of the Q vector space spanned by the numbers, (3), (5),, (a) is at least log a. +log2 39 / / 90

11 Euler Mascheroni constant Euler s constant Euler s Constant is = lim + n! n Lorenzo Mascheroni ( ) log n = Recent work by J. Sondow inspired by the work of F. Beukers on Apéry s proof. Is it a rational number? X = log + = k k = k= Z Z 0 0 ( x)dxdy ( xy)log(xy) Z [x] dx x F. Beukers Jonathan Sondow jsondow/ 4 / / 90 Jonathan Sondow jsondow/ Euler Gamma function Is the number (/5) = = Z 2t(t +) F = Z 0 = lim s!+ X k 2 k=2 X n=, 2, 2 dt. 3, t +2 t+k k dt n s s n irrational? (z) =e z z Y n= + z n e z/n = Z 0 e t t z dt t Here is the set of rational values for z 2 (0, ) for which the answer is known (and, for these arguments, the Gamma value is a transcendental number) : r 2 6, 4, 3, 2, 2 3, 3 4, 5 (mod ) / / 90

12 Georg Cantor (845-98) Henri Léon Lebesgue (875 94) The set of algebraic numbers is countable, not the set of real (or complex) numbers. Almost all numbers for Lebesgue measure are transcendental numbers. Cantor (874 and 89). 45 / / 90 Most numbers are transcendental Special values of hypergeometric series Meta conjecture : any number given by some kind of limit, which is not obviously rational (resp. algebraic), is irrational (resp. transcendental). Goro Shimura Jürgen Wolfart Frits Beukers 47 / / 90

13 Sum of values of a rational function Work by S.D. Adhikari, N. Saradha, T.N. Shorey and R. Tijdeman (200), Let P and Q be non-zero polynomials having rational coe cients and deg Q 2+degP. Consider X n 0 Q(n)6=0 P(n) Q(n) Robert Tijdeman Sukumar Das Adhikari N. Saradha Telescoping series Examples X n=0 X n= 4n + X n=0 n(n +) =, 5n +2 X n=0 n 2 = 3 4, 3 4n n n n +7 + = 5 5n 3 6 =0 49 / / 90 Transcendental values Transcendental values X n=0 X n=0 are transcendental. (2n +)(2n +2) =log2, X n= n 2 = 2 6, (n +)(2n +)(4n +) = 3 X n=0 (6n +)(6n +2)(6n +3)(6n +4)(6n +5)(6n +6) n=0 = 4320 (92 log 2 8 log 3 7 p 3) X n 2 + = e + e = e e X n=0 ( ) n n 2 + = 2 e e = / / 90

14 Leonardo Pisano (Fibonacci) The Fibonacci sequence (F n ) n 0 : 0,,, 2, 3, 5, 8, 3, 2, 34, 55, 89, 44, is defined by Leonardo Pisano (Fibonacci) (70 250) Encyclopedia of integer sequences (again) 0,,, 2, 3, 5, 8, 3, 2, 34, 55, 89, 44, 233, 377, 60, 987, 597, 2584, 48, 6765, 0946, 77, 28657, 46368, 75025, 2393, 9648, 378, 54229, , , , , , ,... The Fibonacci sequence is available online The On-Line Encyclopedia of Integer Sequences Neil J. A. Sloane F 0 =0, F =, F n = F n + F n 2 (n 2) / / 90 Series involving Fibonacci numbers Series involving Fibonacci numbers The number is rational, while and X F n=0 2 n X n= = 7 p 5, 2 X n= are irrational algebraic numbers. F n F n+2 = X n= p 5 F 2n + = 2 ( ) n = F n F n+ 2 p 5 The numbers X n= X F 2 n= n, F 2n, n= are all transcendental X F 4 n= n, X ( ) n, n= F 2 n X, F 2 n + F 2 n + X F 6 n= n X n= X, n F 2n, F n= 2 n + 55 / / 90

15 Series involving Fibonacci numbers Each of the numbers X n= F n, X n X n= F n + F n+2 F F 2 F n is irrational, but it is not known whether they are algebraic or transcendental. The first challenge here is to formulate a conjectural statement which would give a satisfactory description of the situation. The Fibonacci zeta function For <e(s) > 0, F (s) = X n F s n F (2), F (4), F (6) are algebraically independent. Iekata Shiokawa, Carsten Elsner and Shun Shimomura (2006) Iekata Shiokawa 57 / / 90 3 Transcendentalnumbers Liouville (844) Hermite (873) Lindemann (882) Hilbert s Problem 7th (900) Gel fond Schneider (934) Baker (968) Existence of transcendental numbers (844) J. Liouville ( ) gave the first examples of transcendental numbers. For instance X n = n! is a transcendental number. Nesterenko (995) 59 / / 90

16 Charles Hermite and Ferdinand Lindemann Hermite Lindemann Theorem For any non-zero complex number z, one at least of the two numbers z and e z is transcendental. Corollaries : Transcendence of log and of e for and non-zero algebraic complex numbers, provided log 6= 0. Hermite (873) : Transcendence of e e = Lindemann (882) : Transcendence of = / / 90 Transcendental functions A complex function is called transcendental if it is transcendental over the field C(z), which means that the functions z and f (z) are algebraically independent : if P 2 C[X, Y ] is a non-zero polynomial, then the function P z, f (z) is not 0. Exercise. An entire function (analytic in C) is transcendental if and only if it is not a polynomial. Example. The transcendental entire function e z takes an algebraic value at an algebraic argument z only for z =0. Weierstrass question Is it true that a transcendental entire function f takes usually transcendental values at algebraic arguments? Examples : for f (z) =e z, there is a single exceptional point algebraic with e also algebraic, namely =0. For f (z) =e P(z) where P 2 Z[z] is a non constant polynomial, there are finitely many exceptional points, namely the roots of P. The exceptional set of e z + e +z is empty (Lindemann Weierstrass). The exceptional set of functions like 2 z or e i z is Q, (Gel fond and Schneider). 63 / / 90

17 Exceptional sets Answers by Weierstrass (letter to Strauss in 886), Strauss, Stäckel, Faber, van der Poorten, Gramain... If S is a countable subset of C and T is a dense subset of C, there exist transcendental entire functions f mapping S into T, as well as all its derivatives. Any set of algebraic numbers is the exceptional set of some transcendental entire function. Also multiplicities can be included. van der Poorten : there are transcendental entire functions f such that D k f ( ) 2 Q( ) for all k 0 and all algebraic. Integer valued entire functions An integer valued entire function is a function f, which is analytic in C, and maps N into Z. Example : 2 z is an integer valued entire function, not a polynomial. Question : Are there integer valued entire function growing slower than 2 z without being a polynomial? Let f be a transcendental entire function in C. ForR > 0 set f R =sup f (z). z =R 65 / / 90 Integer valued entire functions G. Pólya (94) : if f is not a polynomial and f (n) 2 Z for n 2 Z 0, then lim sup 2 R f R. R! Further works on this topic by G.H. Hardy, G. Pólya, D. Sato, E.G. Straus, A. Selberg, Ch. Pisot, F. Carlson, F. Gross,... Integer valued entire function on Z[i] A.O. Gel fond (929) : growth of entire functions mapping the Gaussian integers into themselves. Newton interpolation series at the points in Z[i]. An entire function f which is not a polynomial and satisfies f (a + ib) 2 Z[i] for all a + ib 2 Z[i] satisfies lim sup R! R 2 log f R. F. Gramain (98) : = /(2e) = This is best possible : D.W. Masser (980). 67 / / 90

18 Transcendence of e Hilbert s Problems A.O. Gel fond (929). If e = is rational, then the function e z takes values in Q(i) when the argument z is in Z[i]. Expand e z into an interpolation series at the Gaussian integers. August 8, 900 David Hilbert ( ) Second International Congress of Mathematicians in Paris. Twin primes, Goldbach s Conjecture, Riemann Hypothesis Transcendence of e and 2 p 2 69 / / 90 A.O. Gel fond and Th. Schneider Solution of Hilbert s seventh problem (934) : Transcendence of and of (log )/(log 2 ) for algebraic,, and 2. Transcendence of and log / log 2 :examples The following numbers are transcendental : 2 p2 = log 2 = log 3 e = (e =( ) i ) e p 63 = / / 90

19 e =( ) i Example : Transcendence of the number e p63 = Remark. For Beta values : Th. Schneider 948 Euler Gamma and Beta functions B(a, b) = (z) = Z x a ( x) b dx. 0 Z 0 e t t z dt t we have j( ) = = +ip 63, q = e 2i = e p and B(a, b) = (a) (b) (a + b) j( ) q 744 < / / 90 Algebraic independence : A.O. Gel fond 948 The two numbers 2 3p2 and 2 3p4 are algebraically independent. Alan Baker 968 Transcendence of numbers like log + + n log n More generally, if is an algebraic number, 6= 0, 6= and if is an algebraic number of degree d 3, then two at least of the numbers or e 0 for algebraic i s and j s., 2,..., d are algebraically independent. 75 / 90 Example (Siegel) : Z 0 dx +x 3 = 3 is transcendental. log 2 + p3 = / 90

20 Gregory V. Chudnovsky Yuri V. Nesterenko Yu.V.Nesterenko (996) Algebraic independence of (/4), and e. Also : Algebraic independence of p (/3), and e 3. G.V. Chudnovsky (976) Algebraic independence of the numbers and (/4). Also : algebraic independence of the numbers and (/3). Corollary : The numbers = and e = are algebraically independent. Corollaries : Transcendence of (/4) = and (/3) = Transcendence of values of Dirichlet s L functions : Sanoli Gun, Ram Murty and Purusottam Rath (2009). 77 / / 90 4 : Conjectures Weierstraß sigma function Borel 909, 950 Let = Z! + Z!2 be a lattice in C. The canonical product attached to is the Weierstraß sigma function Y z (z/!)+(z 2 /2!2 ) (z) = (z) = z e.! Schanuel 964 Grothendieck 960 s!2 \{0} Rohrlich and Lang 970 s The number Andre 990 s Z[i] (/2) = 25/4 /2 e /8 (/4) 2 Kontsevich and Zagier 200. is transcendental. 79 / / 90

21 Periods : Maxime Kontsevich and Don Zagier Periods, Mathematics unlimited 200 and beyond, Springer 200, A period is a complex number whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coe cients, over domains in R n given by polynomial inequalities with rational coe cients. The number Basic example of a period : = = Z Z e z+2i = e z Z 2i = Z x 2 +y 2 apple dx p x 2 z = dz z dxdy =2 = Z Z p dx x 2 x 2 dx 8 / / 90 Further examples of periods p 2= Z and all algebraic numbers. Z log 2 = 2x 2 apple <x<2 dx dx x and all logarithms of algebraic numbers. Z = dxdy, x 2 +y 2 apple A product of periods is a period (subalgebra of C), but / is expected not to be a period. Relations among periods Additivity (in the integrand and in the domain of integration) Z b a f (x)+g(x) dx = Z b a f (x)dx = Z c a Z b a f (x)dx + f (x)dx + Z b c Z b a f (x)dx. g(x)dx, 2 Change of variables : if y = f (x) is an invertible change of variables, then Z f (b) f (a) F (y)dy = Z b a F f (x) f 0 (x)dx. 83 / / 90

22 Relations among periods (continued) Conjecture of Kontsevich and Zagier A widely-held belief, based on a judicious combination of experience, analogy, and wishful thinking, is the following 3 Newton Leibniz Stokes Formula Z b a f 0 (x)dx = f (b) f (a). Conjecture (Kontsevich Zagier). If a period has two integral representations, then one can pass from one formula to another by using only rules, 2, 3 in which all functions and domains of integration are algebraic with algebraic coe cients. 85 / / 90 Conjecture of Kontsevich and Zagier (continued) In other words, we do not expect any miraculous coïncidence of two integrals of algebraic functions which will not be possible to prove using three simple rules. This conjecture, which is similar in spirit to the Hodge conjecture, is one of the central conjectures about algebraic independence and transcendental numbers, and is related to many of the results and ideas of modern arithmetic algebraic geometry and the theory of motives. Advice : if you wish to prove a number is transcendental, first prove it is a period. Conjectures by S. Schanuel and A. Grothendieck Schanuel : if x,...,x n are Q linearly independent complex numbers, then n at least of the 2n numbers x,...,x n, e x,...,e xn are algebraically independent. Periods conjecture by Grothendieck : Dimension of the Mumford Tate group of a smooth projective variety. 87 / / 90

23 Motives February 3, 202 Y. André : generalization of Grothendieck s conjecture to motives. Centre of Excellence in Mathematics (CEM) Department of Mathematics, Mahidol University Case of motives : Elliptico-Toric Conjecture of C. Bertolin. Transcendental Number Theory: recent results and open problems. Michel Waldschmidt Institut de Mathématiques de Jussieu miw/ 89 / / 90