Numb er Theo ry Da ys in KKU

Novemb er 2, 2009 Khon Ka en Univers it y.

Numb er Theo ry Da ys in KKU

http://202.28.94.202/math/thai/

H isto ry of irrational and transcendental numb ers

Michel W alds chmidt

Institut de Math ´ematiques de Jussieu & CIMP A

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Abs tract

The transcendence pro ofs fo r constants of analysis are essentially all based on the seminal w ork by Ch. Hermite : his pro of of the transcendence of the numb er e in 1873 is the protot yp e of the metho ds which have b een subsequently develop ed. W e first sho w ho w the founding pap er by Hermite w as influenced by ea rlier autho rs (Lamb ert, Euler, F ourier, Liouville), next w e explain ho w his arguments have b een expanded in several directions : P ad ´e app ro ximants, interp olation series, auxilia ry functions.

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Numb ers : rational, irrational

Numb ers = real or complex numb ers R , C .

Natural integers : N = { 0 , 1 , 2 ,. .. } .

Rational integers : Z = { 0 , ± 1 , ± 2 ,. .. } .

Rational numb ers : a / b with a and b rational integers, b > 0 .

Irreducible rep resentation : p / q with p and q in Z , q > 0 and gcd( p , q ) = 1 .

Irrational numb er : a real (o r complex) numb er which is not rational.

Numb ers : algeb raic, trans cendental

Algeb raic numb er : a complex numb er which is ro ot of a non-zero p olynomia l with rational co e ffi cients.

Examples : rational numb ers : a / b , ro ot of bX − a . √ 2 , ro ot of X

− 2 . i , ro ot of X

+1 .

The sum and the pro duct of algeb raic nu mb ers are algeb raic numb ers. The set of complex algeb raic numb ers is a field, the algeb raic closure of Q in C .

A transcendental numb er is a complex numb er which is not algeb raic.

Irrationalit y of √ 2

Pythago reas scho ol

Hippasus of Metap ontum (a round 500 BC).

Sulba Sutras, V edic civilization in India, ∼ 800-500 BC.

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Irrationalit y of √ 2 : geometric pro of

• Sta rt with a rectangle have side length 1 and 1+ √ 2 . • Decomp ose it into tw o squa res with sides 1 and a smaller rectangle of sides 1+ √ 2 − 2= √ 2 − 1 and 1 . • This second small rectangle has side lenghts in the prop ortion 1 √ 2 − 1 = 1 + √ 2 ,

which is the same as fo r the la rge one. • Hence the second small recta ngle can b e split into tw o squa res and a third smaller rectangle, the sides of which are again in the same prop ortion. • This pro cess do es not end.

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Rectangles with prop orti on 1+ √ 2

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Irrationalit y of √ 2 : geometric pro of

If w e sta rt with a recta ngle having integer side lengths, then this pro cess stops after finitely ma y steps (the side lengths are p ositive decreasing integers).

Also fo r a rectangle with side lengths in a rational prop ortion, this pro cess stops after finitely ma y steps (reduce to a common denominato r and scale).

Hence 1+ √ 2 is an irrational numb er, and √ 2 also.

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The fabulous des tiny of √ 2

• Beno ˆıt Rittaud, ´Editions Le P ommier (2006).

http://www.math.univ-paris13.fr/∼rittaud/RacineDeDeux

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Continued fraction The numb er √ 2=1 . 414 213 562 373 095 048 801 688 724 209 .. .

satisfies √ 2 = 1 + 1 √ 2+1 · Hence √ 2 = 1 + 1

2+ 1 √ 2+1 = 1 + 1

2+ 1

2+ 1 .. .

W e write the continued fraction expansion of √ 2 using the sho rter notation √ 2 = [1; 2 , 2 , 2 , 2 , 2 , .. . ] = [1; 2] .

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Continued fractions

• H.W. Lenstra Jr, Solving the P el l Equation , Notices of the A.M.S. 49 (2) (2002) 182–192.

Irrationalit y criteria

A real numb er is rational if and only if its continued fraction expansion is finite.

A real numb er is rational if and only if its bina ry (o r decimal, or in any basis b ≥ 2 ) expansion is ultimately p erio dic .

Consequence : it should not b e so di ffi cult to decide whether a given numb er is rational or not.

T o prove that certain numb ers (o ccurring as constants in analysis) are irrational is most often an imp ossible chall en ge. Ho w ever to construct irrational (even transcendental) numb ers is eas y.

Euler–Mascheroni cons tant

Euler ’s Constant is

γ = lim

! 1+ 1 2 + 1 3 + ·· · + 1 n − log n "

=0 . 577 215 664 901 532 860 606 512 090 082 .. .

Is–it a rational numb er ?

γ =

#

k=1

! 1 k − log ! 1+ 1 k "" = $

! 1 [ x ] − 1 x " dx

= − $

0

$

0

(1 − x ) dxdy (1 − xy ) log ( xy ) ·

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Euler’s constant

Recent w ork by J. Sondo w inspired by the w ork of F. Beuk ers on Ap ´ery ’s pro of.

F. Beuk ers Jonathan Sondo w

http://home.earthlink.net/∼jsondow/

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Jonathan Sondo w

γ = $

0 ∞

#

k=2

1 k

%

& dt

γ = lim

#

n=1

! 1 n

− 1 s

"

γ = $

1

1 2 t ( t + 1) F ! 1 , 2 , 2 3 , t +2 " dt .

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Riemann zeta function

The function

ζ ( s )= #

n≥1

1 n

w as studied by Euler (1707– 1783) fo r integer values of s and by Riemann (1859) fo r complex values of s .

Euler : fo r any even integer value of s ≥ 2 , the numb er ζ ( s ) is a rational multiple of π

.

Examples : ζ (2) = π

/ 6 , ζ (4) = π

/ 90 , ζ (6) = π

/ 945 , ζ (8) = π

/ 9450 ·· ·

Co e ffi cients : Bernoulli numb ers.

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Intro ductio in analysin infinito rum

Leonha rd Euler

(1707 – 1783)

Intro ductio in analysin in finito rum

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Divergent series

Euler : 1 − 1+1 − 1+1 − 1+ ·· · = 1 2

1 + 1 + 1 + 1 + 1 + ·· · = − 1 2

1 + 2 + 3 + 4 + 5 + ·· · = − 1 12

1

+2

+3

+4

+5

+ ·· · =0 .

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Srinivas a Ramanujan (1887 – 1920)

Letter of Ramanujan to M.J.M. Hill in 1912

1 + 2 + 3 + ·· · + ∞ = − 1 12

1

+2

+3

+ ·· · + ∞

=0

1

+2

+3

+ ·· · + ∞

= 1 120

Riemann zeta function

The numb er

ζ (3) = #

n≥1

1 n

=1 , 202 056 903 159 594 285 399 738 161 511 .. .

is irrational (Ap ´ery 1978) .

Recall that ζ ( s ) / π

is rationa l fo r any even value of s ≥ 2 .

Op en question : Is the numb er ζ (3) / π

irrational ?

Riemann zeta function

Is the numb er

ζ (5) = #

n≥1

1 n

=1 . 036 927 755 143 369 926 331 365 486 457 .. .

irrational ?

T. Rivoal (2000) : infinitely many ζ (2 n + 1) are irrational.

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Infinitely many odd zeta are irrational

T an guy Rivoal (2000)

Let $ > 0 . F or any su ffi ciently la rge o dd integer a , the dimension of the Q –vecto r space spanned by the numb ers 1 , ζ (3) , ζ (5) , ·· · , ζ ( a ) is at least

1 − $ 1 + log 2 log a .

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Op en problems (irrationalit y)

• Is the numb er

e + π =5 . 859 874 482 048 838 473 822 930 854 632 .. .

irrational ? • Is the numb er

e π =8 . 539 734 222 673 567 065 463 550 869 546 .. .

irrational ? • Is the numb er

log π =1 . 144 729 885 849 400 174 143 427 351 353 .. .

irrational ?

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Catalan’s constant Is Cata lan ’s constant #

n≥1

( − 1)

(2 n + 1)

=0 . 915 965 594 177 219 015 0 .. . an irrational numb er ?

This is the value at s =2 of the Dirichlet L –function L ( s , χ

) asso ciated with the Kroneck er cha racter χ

( n )= ' n 4 ( =    0 if n is even , 1 if n ≡ 1 (mo d 4) , − 1 if n ≡− 1 (mo d 4) .

which is the quotient of the Dedekind zeta function of Q ( i ) and the Riemann zeta function.

Euler Gamma function

Is the numb er

Γ (1 / 5) = 4 . 590 843 711 998 803 053 204 758 275 929 152 .. .

irrational ?

Γ ( z )= e

z

-

n=1

' 1+ z n (

e

= $

e

t

· dt t

Here is the set of rational values fo r z fo r which the answ er is kno wn (and, fo r these arguments, the Gamma value is a transcendental numb er) :

r ∈ . 1 6 , 1 4 , 1 3 , 1 2 , 2 3 , 3 4 , 5 6 / (mo d 1) .

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Kno wn res ults

Irrationalit y of the numb er π :

¯Ary abhat

. a , b. 476 AD : π ∼ 3 . 1416 .

N ¯ıl ak an

actual value ? Because it (exact value) cannot b e exp ressed. app ro ximate value b een mentioned here leaving b ehind the . t. ha Soma y¯aj ¯ı , b. 1444 AD : Why then has an

K. Ramas ub ramanian, The Notion of Pro of in Indian Science , 13th W orld Sanskrit Conference, 2006.

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Irrationalit y of π

Johann Heinrich Lamb ert (1728 - 1777) M ´emoire sur quelques prop ri ´et ´es rema rquables des quantit ´es trans cend antes circulaires et loga rithmiques , M ´emoires de l’Acad ´emie des Sciences de Berlin, 17 (1761), p. 265-322 ; read in 1767 ; Math. W erk e, t. II.

tan( v ) is irrational fo r any rational value of v ( =0 and ta n( π / 4) = 1 .

Lamb ert and F rederick II, King of Prus sia

— Que savez vous, Lamb ert ? — T out, Sire. — Et de qui le tenez–vous ? — De moi-m ˆeme !

Continued fraction expans ion of tan( x )

tan( x )= 1 i tanh( ix ) , tanh( x )= e

− e

e

+ e

·

tan( x )= x

1 − x

3 − x

5 − x

7 − x

9 − x

.. . ·

S.A. Shirali – Continued fraction fo r e , Resonance, vol. 5 N

1, Jan. 2000, 14–28. http ://www.ias.ac.in/resonance/

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Leona rd Euler (1707 – 1783)

Leonha rd Euler (1707 – 1783) De fractionibus continuis dissertatio , Commenta rii Acad. Sci. P etrop olitanae, 9 (1737), 1744, p. 98–137 ; Op era Omnia Ser. I vol. 14 , Commentationes Analyticae, p. 187–215.

e = lim

(1 + 1 / n )

=2 . 718 281 828 459 045 235 360 287 471 352 .. .

= 1 + 1 + 1 2 · (1 + 1 3 · (1 + 1 4 · (1 + 1 5 · (1 + ·· · )))) .

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Continued fraction expans ion fo r e (Euler)

e = 2+ 1

1+ 1

2+ 1

1+ 1

1+ 1

4+ 1 .. . = [2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 ,. .. ]

= [2; 1 , 2 m , 1]

.

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Continued fraction expans ion fo r e (Euler)

The continued fraction expansion fo r e is infi nite not p erio dic.

Leonha rd Euler (1707– 1783) Johann Heinrich Lamb ert (1728 - 1777) Joseph-Louis Lagrange (1736 - 1813)

e is neither rational ( J-H. Lamb ert, 1767) no r qua dra tic irrational ( J-L. Lagrange, 1770).

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Continued fraction expans ion fo r e 1 / a Sta rting p oint : y = tanh( x / a ) satisfi es the di ff erential equation ay

+ y

=1 . This leads Euler to

e

= [1 ; a − 1 , 1 , 1 , 3 a − 1 , 1 , 1 , 5 a − 1 ,. .. ]

=[ 1 , (2 m + 1) a − 1 , 1]

.

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Geometric pro of of the irrationalit y of e

Jonathan Sondo w

A geometric pro of that e is irrational and a new measure of its irrationalit y, Amer. Math. Monthly 113 (2006) 637-641.

Sta rt with an interval I

with length 1 . The interval I

will b e obtained by splitting the interval I

into n intervals of the same length, so that the length of I

will b e 1 / n ! .

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Geometric pro of of the irrationalit y of e

The origin of I

will b e

1+ 1 1! + 1 2! + ·· · + 1 n ! ·

Hence w e sta rt from the interval I

= [2 , 3] . F or n ≥ 2 , w e construct I

inductively as follo ws : split I

into n intervals of the same length, and call the second one I

:

I

= 0 1+ 1 1! , 1+ 2 1! 1 = [2 , 3] , I

= 0 1+ 1 1! + 1 2! , 1+ 1 1! + 2 2! 1 = 0 5 2! , 6 2! 1 , I

= 0 1+ 1 1! + 1 2! + 1 3! , 1+ 1 1! + 1 2! + 2 3! 1 = 0 16 3! , 17 3! 1 ·

Irrationalit y of e , follo wing J. Sondo w

The origin of I

is

1+ 1 1! + 1 2! + ·· · + 1 n ! = a

n

! , the length is 1 / n ! , hence I

=[ a

/ n ! , ( a

+ 1) / n !] . The numb er e is the intersection p oint of all thes e intervals, hence it is inside each I

, therefo re it cannot b e written a / n ! with a an integer. Since p q = ( q − 1)! p q ! ,

w e deduce that the numb er e is irrational.

Jos eph F ourier (1768 - 1830)

Course of analysis at the ´Ecole P olytechnique P aris, 1815.

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Irrationalit y of e , follo wing J. F ourier

e =

#

n=0

1 n ! + #

m≥N+1

1 m ! ·

Multiply by N ! and set

B

= N ! , A

=

#

n=0

N ! n ! , R

= #

m≥N+1

N ! m

! , so that B

e = A

+ R

. Then A

and B

are in Z , R

> 0 and

R

= 1 N +1 + 1 ( N + 1)( N + 2) + ·· · < e N +1 ·

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Irrationalit y of e , follo wing J. F ourier

In the fo rmul a B

e − A

= R

, the numb ers A

and B

= N ! are integers, while the right hand side is > 0 and tends to 0 when N tends to infinit y. Hence N ! e is not an integer, therefo re e is irrational.

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Irrationalit y of e − 1 , follo wi ng C.L. Siegel

C.L. Siegel (1949) : even simpler by considering e

(alternating series).

The sequence (1 / n !)

is decreasing and tends to 0 , hence fo r o dd N ,

1 − 1 1! + 1 2! − ·· · − 1 N ! < e

< 1 − 1 1! + 1 2! − ··· + 1 ( N + 1)! ·

Set

a

= N ! − N ! 1! + N ! 2! − ·· · + ( N − 1)! N ! − 1 ∈ Z

Then 0 < N ! e

− a

< 1 , and therefo re N ! e

is not an integer.

The numb er e is not quadratic

Since e is irrational, the same is true fo r e

when b is a p ositive integer. That e

is irrational is a stronger statement.

Recall (Euler, 1737) : e = [2; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , 1 , 8 ,. .. ] which is not a p erio dic expansion. J.L. Lagrange (1770) : it follo ws that e is not a quadratic numb er.

Assume ae

+ be + c =0 . Replacing e and e

by the series and truncating do es not w ork : the denominato r is to o la rge and the remainder do es not tend to zero.

Liouville (1840) : W rite the quadratic equation as ae + b + ce

=0 .

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Jos eph Liouville

J. Liouville (1809 - 1882) proved also that e

is not a quadratic irrational numb er in 1840.

Sur l’irrationalit ´e du nomb re e =2 , 718 .. . , J. Math. Pures Appl. (1) 5 (1840), p. 192 and p. 193-194.

1844 : J. Liouville proved the existence of transcendental numb ers by giving explicit examples (continued fractions, series).

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Existence of trans cendental numb ers (1844)

J. Liouville (1809 - 1882)

gave the first examples of transcendental numb ers. F or instance #

n≥1

1 10

=0 . 110 001 000 000 0 .. .

is a transcendental numb er.

The numb er e 2 is not quadratic

The irrationalit y of e

, hence of e

fo r b a p ositive integer, follo ws.

It do es not seem that this kind of argument will su ffi ce to prove the irrational it y of e

, even less to prove that the numb er e is not a cubic irrational.

F ourier’s argument rests on truncating the exp onential series, it amounts to app ro ximate e by a / N ! where a ∈ Z . Better rational app ro ximations exist, involving other denominato rs than N ! .

The denominato r N ! arises when one app ro ximates the exp onential series of e

by p olynomials 2

z

/ n ! .

Idea of Ch. Hermite

Ch. Hermite (1822 - 1901). app ro ximate the exp onential function e

by rational fractions A ( z ) / B ( z ) .

F or proving the irrationalit y of e

, ( a an integer ≥ 2 ), app ro ximate e

pa r A ( a ) / B ( a ) .

If the function B ( z ) e

− A ( z ) has a zero of high multiplicit y at the origin, then this function has a small mo dulus nea r 0 , hence at z = a . Therefo re | B ( a ) e

− A ( a ) | is small.

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Cha rles Hermite

A rational function A ( z ) / B ( z ) is close to a complex analytic function f if B ( z ) f ( z ) − A ( z ) has a zero of high multiplicit y at the origin.

Goal : find B ∈ C [ z ] such that the T aylo r expansion at the origin of B ( z ) f ( z ) has a big gap : A ( z ) will b e the pa rt of the expansion b efo re the gap, R ( z )= B ( z ) f ( z ) − A ( z ) the remainder.

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Irrationalit y of e r and π (Lamb ert, 1767)

Cha rles Hermite (1873)

Ca rl Ludwig Siegel (1929, 1949)

Y uri Nesterenk o (2005)

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Irrationalit y of e r and π (Lamb ert, 1767)

W e wis h to prove the irrationalit y of e

fo r a a p ositive integer.

Goal : write B

( z ) e

= A

( z )+ R

( z ) with A

and B

in Z [ z ] and R

( a ) ( =0 , lim

R

( a ) = 0 .

Substitute z = a , set q = B

( a ) , p = A

( a ) and get

0 < | qe

− p | < $ .

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Rational app ro ximation to exp

Given n

≥ 0 , n

≥ 0 , find A and B in R [ z ] of degrees ≤ n

and ≤ n

such that R ( z )= B ( z ) e

− A ( z ) has a zero at the origin of multiplicit y ≥ N +1 with N = n

+ n

.

Theo rem There is a non-trivial solution, it is unique with B monic. F urther, B is in Z [ z ] and ( n

! / n

!) A is in Z [ z ] . F urthermo re A has degree n

, B has degree n

and R has multiplicit y exactly N +1 at the origin.

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B ( z ) e z = A ( z )+ R ( z )

Pro of. Unicit y of R , hence of A and B . Let D = d / dz . Since A has degree ≤ n

,

D

R = D

( B ( z ) e

) is the pro duct of e

with a p olynomial of the same degree as the degree of B and sa me leading co e ffi cient. Since D

R ( z ) has a zero of multiplicit y ≥ n

at the ori gi n, D

R = z

e

. Hence R is the unique function satisfying D

R = z

e

with a zero of multiplicit y ≥ n

at 0 and B has degree n

.

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Siegel’s al geb raic p oint of view

C.L. Siegel, 1949. Solve D

R ( z )= z

e

. The op erato r J ϕ = $

0

ϕ ( t ) dt ,

inverse of D , satis fies

J

ϕ = $

0

1 n

! ( z − t ) ϕ ( t ) dt .

Hence R ( z )= 1 n

! $

0

( z − t )

t

e

dt . Also A ( z )= − ( − 1+ D )

z

and B ( z ) = (1 + D )

z

.

Irrationalit y of loga rithms including π

The irrationalit y of e

fo r r ∈ Q

, is equivalent to the irrationalit y of log s fo r s ∈ Q

.

The same argument gives the irrationalit y of log ( − 1) , meaning log ( − 1) = i π (∈ Q ( i ) .

Hence π (∈ Q .

Simultaneous app ro ximation and trans cendence

Irrationalit y pro ofs involve rational app ro ximation to a single real numb er ϑ .

W e wish to prove transcendence results.

A complex numb er ϑ is transcendental if and only if the numb ers 1 , ϑ , ϑ

,. .. , ϑ

,. .. are Q –linea rly indep endent.

Hence our goal is to prove linea r indep endence, over the rational numb er field, of complex numb ers.

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L = a 0 + a 1 ϑ 1 + ·· · + a m ϑ m Let ϑ

,. .. , ϑ

b e rea l numb ers and a

, a

,. .. , a

rational integers, not all of which are zero. W e wish to prove that the numb er L = a

+ a

ϑ

+ ·· · + a

ϑ

is not zero. App ro ximate simultaneously ϑ

,. .. , ϑ

by rational numb ers b

/ b

,. .. , b

/ b

. Let b

, b

,. .. , b

b e ra tional integers. F or 1 ≤ k ≤ m set

$

= b

ϑ

− b

. Then b

L = A + R with A = a

b

+ ·· · + a

b

∈ Z and R = a

$

+ ·· · + a

$

∈ R .

If 0 < | R | < 1 , then L ( =0 .

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Simultaneous app ro ximation to the exp onential function

Irrationalit y results follo w from rational app ro ximations A / B ∈ Q ( x ) to the exp onential function e

.

One of Hermite’s ideas is to consider simultaneous ra tional app ro ximations to the exp onential function , in analogy with Diophantine app ro ximation.

Let B

, B

,. .. , B

b e p olynomials in Z [ x ] . F or 1 ≤ k ≤ m define R

( x )= B

( x ) e

− B

( x ) .

Set b

= B

(1) , 0 ≤ j ≤ m and

R = a

+ a

R

(1) + ·· · + a

R

(1) . If 0 < | R | < 1 , then a

+ a

e + ·· · + a

e

( =0 .

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Cha rles Hermite and F erdinand Lindemann

Hermite (1873) : T ranscendence of e e =2 . 718 281 828 4 .. . Lindemann (1882) : T ranscendence of π π =3 . 141 592 653 5 .. .

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Hermite–Lindemann Theo rem

F or any non-zero complex numb er z , one at least of the tw o numb ers z and e

is transcendental .

Co rolla ries : T ranscendence of log α and of e

fo r α and β non-zero algeb raic complex numb ers, provided log α (=0 .

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Hermite : app ro ximation to the functions 1 , e α

x ,. .. , e α

x Let α

,. .. , α

b e pairwise distinct complex numb ers and n

,. .. , n

b e ra tional integers, all ≥ 0 . Set N = n

+ ·· · + n

.

Hermite constructs explicitly p olynomials B

, B

,. .. , B

with B

of degree N − n

such that each of the functions

B

( z ) e

− B

( z ) , (1 ≤ k ≤ m )

has a zero at the origin of multipli ci ty at least N .

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P ad ´e app ro ximants

Henri Eug `ene P ad ´e (1863 - 1953) App ro ximation of complex analytic functi ons by rational functions.

T rans cendental 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 algeb raically indep endent : if P ∈ C [ X , Y ] is a non-zero p olynomial, 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 p olynomial.

Example. The transcendental entire function e

tak es an algeb raic value at an algeb raic argument z only fo r z =0 .

W eiers tras s ques tion

Is–it true that a transcendental entire function f tak es usually transcendental va lues at algeb raic arguments ?

Answ ers by W eierstrass (letter to Strauss in 1886), Strauss , St ¨ack el , F ab er , van der P o orten , 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 w ell as all its derivatives . Also there are transcendental entire functions f such that D

f ( α ) ∈ Q ( α ) fo r al l k ≥ 0 and all algeb raic α .

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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

is an integer valued entire function, not a p olynomial.

Question : Are-there integer va lued entire function gro wing slo w er than 2

without b eing a p olynomial ?

Let f b e a transcendental entire function in C . F or R > 0 set

| f |

= sup

| f ( z ) | .

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Integer valued entire functions

G. P´ oly a (1914) : if f is not a p olynomial and f ( n ) ∈ Z fo r n ∈ Z

, then lim sup

2

| f |

≥ 1 .

F urther w orks on this topic by G.H. Ha rdy , G. P´ oly a , D. Sato , E.G. Straus , A. Selb erg , Ch. Pisot , F. Ca rls on , F. Gross ,. ..

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Arithmetic functions

P´ oly a ’s pro of sta rts by expanding the function f into a Newton interp olation series at the p oints 0 , 1 , 2 ,. .. :

f ( z )= a

+ a

z + a

z ( z − 1) + a

z ( z − 1)( z − 2) + ·· · Since f ( n ) is an integer fo r all n ≥ 0 , the co e ffi cients a

are rational and one can b ound the denomi nato rs. If f do es not gro w fast, one deduces that these co e ffi cients vanish fo r su ffi ciently la rge n .

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Newton interp olation series

Sir Isaac Newton (1643 - 1727)

F rom f ( z )= f ( α

)+( z − α

) f

( z ) ,

f

( z )= f

( α

)+( z − α

) f

( z )+ .. .

w e deduce

f ( z )= a

+ a

( z − α

)+ a

( z − α

)( z − α

)+ ·· ·

with

a

= f ( α

) , a

= f

( α

) ,. .. , a

= f

( α

) .

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An identit y due to Ch. Hermite

1 x − z = 1 x − α + z − α x − α · 1 x − z ·

Rep eat :

1 x − z = 1 x − α

+ z − α

x − α

· ! 1 x − α

+ z − α

x − α

· 1 x − z " ·

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An identit y due to Ch. Hermite

Inductively w e deduce the next fo rmula due to Hermite :

1 x − z =

#

j=0

( z − α

)( z − α

) ·· · ( z − α

) ( x − α

)( x − α

) ·· · ( x − α

) + ( z − α

)( z − α

) ·· · ( z − α

) ( x − α

)( x − α

) ·· · ( x − α

) · 1 x − z ·

Newton interp olation expans ion

Application. Multiply by (1 / 2 i π ) f ( z ) and integrate :

f ( z )=

#

j=0

a

( z − α

) ·· · ( z − α

)+ R

( z )

with

a

= 1 2 i π $

C

F ( x ) dx ( x − α

)( x − α

) ·· · ( x − α

) (0 ≤ j ≤ n − 1)

and

R

( z )=( z − α

)( z − α

) ·· · ( z − α

) · 1 2 i π $

C

F ( x ) dx ( x − α

)( x − α

) ·· · ( x − α

)( x − z ) ·

Integer valued entire function on Z [ i ]

A.O. Gel’fond (1929) : gro wth of entire functions mapping the Gaussian integers into themselves. Newton interp olation series at the p oints in Z [ i ] .

An entire function f which is not a p olynomial and satisfies f ( a + ib ) ∈ Z [ i ] fo r all a + ib ∈ Z [ i ] satisfies

lim sup

1 R

log | f |

≥ γ .

F. Gramain (1981) : γ = π / (2 e ) . This is b est p ossible : D.W. Masser (1980).

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T rans cendence of e π

A.O. Gel’fond (1929) .

If e

= 23 , 140 692 632 779 269 005 729 086 367 .. . is rationa l, then the function e

tak es values in Q ( i ) when the argument z is in Z [ i ] .

Expand e

into an interp olation series at the Gaussian integers.

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Hilb ert’s seventh problem

A.O. Gel’fond and Th. Schn ei der (1934). Solution of Hilb ert’s seventh problem : transcendence of α

and of (log α

) / (log α

) fo r algeb raic α , β , α

and α

.

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Dirichlet’s b ox principle

Gel’fond and Schneider use an auxilia ry function , the existence of which follo ws from Dirichlet’s b ox principle (pigeonhole principle, Thue-Siegel Lemma).

Johann P eter Gusta v Lejeune Dirichlet (1805 – 1859)

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Auxilia ry functions

C.L. Siegel (1929) : Hermite’s explicit fo rmulae can b e replaced by Dirichlet’s b ox principle (Thue–Siegel Lemma) which sho ws the existence of suitabl e auxilia ry functions .

M. La urent (1991) : instead of using the pigeonhole principle fo r proving the existence of solutions to homogeneous linea r systems of equations, consider the matri ces of such systems and tak e determinants.

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Slop e inequali ti es in Arak elov theo ry

J–B. Bost (1994) : matrices and determinants require choices of bases. Arak elov’s Theo ry pro duces slop e inequalities which avoid the need of bases.

P ´erio des et isog ´enies des va ri ´et ´es ab ´eliennes sur les co rps de nomb res, (d’ap r`es D. Masser et G. W ¨ustholz). S ´eminaire Nicolas Bourbaki, V ol. 1994/95.

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Rational interp olation

Ren ´e Lagrange (1935).

1 x − z = α − β ( x − α )( x − β ) + x − β x − α · z − α z − β · 1 x − z ·

Iterating and integrating yield

f ( z )=

#

n=0

B

( z − α

) ·· · ( z − α

) ( z − β

) ·· · ( z − β

) + ˜ R

( z ) .

Hurwitz zeta function T. Rivoal (2006) : consider Hurwitz zeta function

ζ ( s , z )=

#

k=1

1 ( k + z )

·

Expand ζ (2 , z ) as a series in

z

( z − 1)

·· · ( z − n + 1)

( z + 1)

·· · ( z + n )

·

The co e ffi cients of the expansion b elong to Q + Q ζ (3) . This pro duces a new pro of of Ap ´ery’s Theo rem on the irrationalit y of ζ (3) . In the same w ay : new pro of of the irrationalit y of log 2 by expanding

#

k=1

( − 1)

k + z ·

Mixing C. Hermite and R. Lagrange

T. Rivoal (2006) : new pro of of the irrationalit y of ζ (2) by expanding

#

k=1

! 1 k − 1 k + z "

as a Hermite–Lagrange series in % z ( z − 1) ·· · ( z − n + 1) &

( z + 1) ·· · ( z + n ) ·

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T aylo r series and interp olation series

T aylo r series are the sp ecial case of Hermite ’s fo rmula with a single p oint and multi plicities — they give rise to P ad ´e app ro ximants.

Multiplicities can also b e intro duced in Ren ´e Lagrange interp olation.

There is another dualit y b et w een the metho ds of Gel’fond and Schneider : F ou rier-Bo rel transfo rm.

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F urther develoments

T ranscendence and algeb raic indep endence of values of mo dula r functions ( m ´etho de st ´ephanoise and w ork of Y u.V. Nesterenk o).

Measures : transcendence, linea r indep endence, algeb raic indep endence. ..

Finite cha racteristic :

F ederico P ella rin - Asp ects de l’ind ´ep endance alg ´eb rique en ca ract ´eristique non nulle [d’ap r`es Anderson, Bro wna w ell, Denis, P apanik olas, Thakur, Y u,. ..] S ´eminaire Nicolas Bourbaki, Dimanche 18 ma rs 2007.

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