February 25, 2021

Irrationality and transcendence ofvalues of special functions.
Michel Waldschmidt
Institut de Mathématiques de Jussieu — Paris VI
http://www.imj-prg.fr/~michel.waldschmidt/

1 / 111

Abstract
There is no general result on the irrationality or transcendenceof values of analytic functions : one needs to restrict to special
functions. A number of results are known concerning analytic
functions satisfying differential equations. We survey this topic.

We consider the values of the E and G functions introducedby Siegel in 1929, the values of the exponential function, of
elliptic or more generally abelian functions, of modular
functions, and also the values of functions satisfying functional
equations.

2 / 111Extended abstract• Irrationality of the values of special functions, transcendence,
algebraic independence.
• The exponential function. Hermite–Lindemann Theorem.
Trigonometric functions, logarithms.
• Weierstrass dream : transcendental functions take transcendental
values. Need to restrict to special functions.
• Siegel E and G–functions. Bessel functions. Hypergeometricfunctions.
• Differential equations : the Schneider–Lang Theorem. Elliptic
functions ; elliptic integrals ; Weierstrass ℘ function, zeta function,
sigma function ; Jacobi functions sn and cn. Modular invariants j
and J, related by j(τ ) = J(e2iπτ ). Theta functions. Special valuesof Euler Gamma and Beta function. Abelian functions andintegrals.
• Riemann zeta function• Functional equations. Mahler’s functions.
3 / 111

Numbers : irrationality, transcendence, algebraic
independence
We are interested with arithmetic properties of values of
special functions.
Irrationality : is f (z 0 ) rational or irrational ?
Transcendence : is f (z 0 ) algebraic (root of a nonzero
polynomial with integer coefficients) ?
Algebraic independence : are f (z 1 ), f (z 2 ), . . . , f (z m )algebraically dependent ?
i.e. does there exist a nonzero polynomial P with integer
coefficients such that P f (z 1 ), f (z 2 ), . . . , f (z m ) = 0 ?
4 / 111

Functions : rational, algebraic, transcendental,
algebraically independent
Rational functions : C(z).

Algebraic functions : P z, f (z) = 0
Example :X 2n
1√=znn
1 − 4zn≥0(|z| < 1).

Transcendental functions. Examples :
X zn
ez =
·
n!
n≥0
X zn
1
log(1 − z) = −
z
n+1
n≥0

(|z| < 1).
5 / 111

The exponential functionez = 1 + z +

zn
z2 z3
+
+ ··· +
+ ···
2
6
n!

d z
e = ez ,dz

e0 = 1.

L. Euler
eiπ = −1.

Leonhard Euler
1707 – 1783

6 / 111

Charles Hermite and Ferdinand Lindemann

Hermite (1873)
Transcendence of ee = 2.718 281 828 4 . . .

Lindemann (1882)
Transcendence of ππ = 3.141 592 653 5 . . .

7 / 111

Hermite – Lindemann Theorem (1882)

Ch. Hermite

von Lindemann1822 – 1901

1852 – 1939

Theorem. If w is a nonzero complex number, one at least of
the two numbers w, ew is transcendental.
Consequences : transcendence of e, π, log α, eβ , foralgebraic α and β assuming α 6= 0, β 6= 0, log β 6= 1.
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hermite.html
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lindemann.html
8 / 111

Transcendental functions
A complex function is called transcendental if it istranscendental over the field C(z), which means that thefunctions z and f (z) are algebraically independent : if
P ∈ 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.
A meromorphic function in C is transcendental if and only if it
is not rational.
Example. The transcendental entire function ez takes analgebraic value at an algebraic argument z only for z = 0.
9 / 111

Transcendence, linear independence, algebraic
independence• An analytic function f which is algebraic over Q takes
algebraic values at all algebraic points where f is defined.
• If f 1 , . . . , f n are analytic functions which are linearly
dependent over Q, then for all α ∈ C (where they are defined)
the values f 1 (α), . . . , f n (α) are linearly dependent over Q.
• If f 1 , . . . , f n are analytic functions which are algebraicallydependent over Q, then for all α ∈ C the values
f 1 (α), . . . , f n (α) are algebraically dependent over Q.

10 / 111

Weierstrass question
Is it true that a transcendental
entire function f takes usuallytranscendental values at algebraicarguments ?

Karl Weierstrass1815 - 1897

Exceptional set of f : bi-algebraic points on the graphE(f ) = α ∈ Q | f (α) ∈ Q
Examples :• For f (z) = ez , E(f ) = {0}.
• For f (z) = eP (z) , where P ∈ Q[z] is a non–constant polynomial,E(f ) is the set of roots of P (Hermite–Lindemann).
• For f (z) = ez + e1+z , E(f ) = ∅ (Lindemann–Weierstrass).
• For f (z) = 2z or eiπz , E(f ) = Q (Gel’fond and Schneider).
11 / 111

Exceptional sets
Answers by Weierstrass (letter to Strauss in 1886), Strauss,Stäckel, Faber, van der Poorten, Gramain, Surroca. . .
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 sometranscendental entire function.
Also multiplicities can be included.
van der Poorten : there are transcendental entire functions fsuch that Dk f (α) ∈ Q(α) for all k ≥ 0 and all algebraic α.

12 / 111

Further results on exceptional sets
For each countable subset A of C and each family of dense
subsets E α,s of C indexed by (α, s) ∈ A × N, there exists atranscendental entire function f : C → C such thatf (s) (α) ∈ E α,s for each (α, s) ∈ A × N.
Jingjing Huang, Diego Marques & Martin Mereb ;Algebraic values of transcendental functions at algebraic points.
Bull. Aust. Math. Soc. 82 (2010), 322–327
http://dx.doi.org/10.1017/S0004972710000316

Example : A = Q, E α,s = Q or C \ Q.
For each α ∈ Q and s ≥ 0, choose whether f (s) (α) is
algebraic or transcendental.

13 / 111

Recent results on exceptional sets
There exists uncountably many transcendental entire functions
f with the property that both f and its inverse function
assume algebraic values at algebraic points.
Diego Marques & Carlos Gustavo Moreira ;A positive answer for a question proposed by K. Mahler.
Math. Ann. 368, No. 3-4, 1059–1062 (2017).
On a stronger version of a question proposed by K. Mahler.
J. Number Theory 194, 372–380 (2019).

Diego Marques

Gustavo Moreira

14 / 111

Differential equations, functional equations
Siegel : E and G functions : linear differential equations ;hypergeometric functions, Bessel functions.
Schneider – Lang : nonlinear differential equations ;exponential function, elliptic functions, abelian functions.
Mahler : functional equations

C.L. Siegel

Th. Schneider

S. Lang

L. Mahler1896 - 1981

1903 - 1988

1927 – 2005

1903 - 1988

15 / 111

Siegel E–functionsLetf (z) =

X an
n!
n≥0

z n ∈ Q[[z]]be such that• an increases at most exponentially in n (hence f is an entire
function)• f satisfies a linear differential equation with coefficients in
Q(z)• The common denominator of a0 , a1 , . . . , an increases atmost exponentially in n.
Examples. Algebraic constants, polynomials with algebraic
coefficients, the exponential function ez , the trigonometricfunctions cos z and sin z.
16 / 111

Bessel function
The Bessel function
X (−1)n  z 2n
J 0 (z)=(n!)2 2n≥0
 z 2 1  z 4
1  z 6+−
+ · · · , F.W. Bessel
=1−
2
4 2
36 2
1784 - 1846

is an E function, solution of the Bessel differential equation1
y 00 + y 0 + y = 0.
z

17 / 111

References on special functions
Shigeru Kanemitsu, Haruo Tsukada,
Vistas of Special Functions, World Scientific, 2007
Kalyan Chakraborty, Shigeru Kanemitsu & Haruo Tsukada,
Vistas of Special Functions II World Scientific,
Singapore, 2009.
Kalyan Chakraborty, Shigeru Kanemitsu & T. Kuzumaki,
A Quick Introduction to Complex Analysis, World
Scientific, 2016.

K. Chakraborty

S. Kanemitsu
18 / 111

Srinivasa Ramanujan and Friedrich Wilhelm Bessel
Henri Cohen,
Some formulas of Ramanujan involving Bessel functions
Publications mathématiques de Besançon (2010), 59–68.
http://pmb.univ-fcomte.fr/2010/Cohen.pdf

F.W. Bessel

S. Ramanujan

1784 - 1846

1887 - 1920

H. Cohen

Ramanujan also investigated Bessel q series.

19 / 111

Euler Gamma and Beta functionsZ∞dt
e−t tz ·t0Yz −1 z/ne .
= e−γz z −1
1+
n
n≥1

Γ(z) =

B(a, b) =L. Euler
1707 - 1783

Γ(a)Γ(b)Γ(a + b)Z1=

xa−1 (1 − x)b−1 dx.

0

20 / 111

Pochhammer symbol : rising factorial power(x)n = x(x + 1) · · · (x + n − 1)
Γ(x + n)=·Γ(x)Leo August Pochhammer
1841-1920

http://scienceworld.wolfram.com/biography/Pochhammer.html

21 / 111

Siegel hypergeometric E functions
Let a1 , . . . , a` , b1 , . . . , bm be rational numbers with m > ` andb1 , . . . , bm not in {0, −1, −2, . . .} and bm = 1. Definecn =(a1 )n · · · (a` )n(b1 )n · · · (bm )n

Set t = m − `.
Thenf (z) =

X

cn z tnn≥1

is an E–function.

22 / 111

Siegel’s Theorem (1929)
For λ ∈ Q \ {−1, −2, . . .}, consider the E–function
K λ (z) =(−1)n  z 2n
,
(λ
+
1)
n!
2
n
n≥0

X

solution of the second order differential equation
y 00 +

2λ + 1 0y + y = 0.
z

For λ ∈ Q not in {± 12 , −1, ± 23 , −2, . . .}, for any algebraic
number α 6= 0, the two numbers K λ (α) and K 0λ (α) arealgebraically independent.

23 / 111

Bessel functions
Bessel functions of the first kind :J λ (z)=X (−1)n (z/2)2n+λn!Γ(n + 1 + λ)
 z λ1
K λ (z),=Γ(λ + 1) 2n≥1solution of the differential equationz 2 y 00 + zy 0 + (z 2 − λ2 )y = 0.
Also J −λ (z) is a solution of the same differential equation.
Modified Bessel functions of the first kind :(z/2)2n+λ= i−λ J λ (iz).
I λ (z) =n!Γ(n + 1 + λ)n≥1X

24 / 111

Bessel functions and continued fractions
From Siegel’s 1929 Theorem, it follows that the number
I 1 (2)= [0; 1, 2, 3, . . . ] = 0.697774658 . . .
I 0 (2)(Sloane’s A052119, A001053 and A001040)
is transcendental.

Weinstein, Eric W. Continued Fraction Constant. From MathWorld – A
Wolfram Web Resource.
http://mathworld.wolfram.com/ContinuedFractionConstant.html

25 / 111

Bessel functions and continued fractions√
2 e + e−1·,
I −1/2 (1) =π2√
I 1/2 (1) =2 e − e−1··π2

I −1/2 (1)e2 + 1[1; 3, 5, 7 . . . ] = 2=,
e −1
I 1/2 (1)[2; 6, 10, 14 . . . ] =I −1/2 (1/2)
e+1
=
e−1
I 1/2 (1/2)

B. Sury

B. Sury, Bessel contain continued fractions of progressions ;Resonance 10 3 (2005) 80–87.
http://www.ias.ac.in/article/fulltext/reso/010/03/0080-0087
26 / 111

Siegel–Shidlovskiı̆ theory
Generalization of C.L. Siegel 1929 results by C.L. Siegel
himself in 1949, by A.B. Shidlovskiı̆ in 1953 – 1955.
Given a set {f 1 , . . . , f n } of E–functions satisfying a system of
linear differential equations and an algebraic number α, the transcendence degree of the field Q f 1 (α), f 2 (α), . . . , f n (α)
over Q is equal to the transcendence degree of the field
C z, f 1 , f 2 , . . . , f n over C(z).
A.B. Shidlovskiı̆
Transcendental numbers.
Studies in mathematics, 12,
Walter de Gruyter (1989).

27 / 111

A refined version of the Siegel–Shidlovskiı̆ theorem
Algebraic relations among values of E–functions at analgebraic point arise from algebraic relations among thefunctions.
Linear independence of the values f i (α) when f 1 , . . . , f n arelinearly independent E–functions.
F. Beukers. A refined version of
the Siegel-Shidlovskiı̆ theorem.
Ann. Math. (2) 163, No. 1,
369-379 (2006).
https ://arxiv.org/abs/math/0405549

F. Beukers

28 / 111

Algorithm for determining the exceptional set of an
E–function
B. Adamczewski & T. Rivoal Exceptional values ofE-functions at algebraic points. Bull. Lond. Math. Soc. 50(2018), no. 4, 697–708.
https://doi.org/10.1112/blms.12168

B. Adamczewski

T. Rivoal

29 / 111

Siegel’s Conjecture (1949)
Is-it true that any E–function can be represented as apolynomial with algebraic coefficients in a finite number ofhypergeometric E- functions of the form p F q (z q−p+1 ),q ≥ p ≥ 1, with rational parameters ?
Answer : No.

S. Fischler

T. Rivoal

J. FresánP. Jossen

S. Fischler & T. Rivoal. On Siegel’s problem for E-functions,Rend. Semin. Mat. Univ. Padova, (2021).
https://arxiv.org/pdf/1910.06817.pdf

J. Fresán & P. Jossen. A non-hypergeometric E-function, (2021).
https://arxiv.org/abs/2012.11005
30 / 111

Siegel G–functionsLetg(z) =

X

an z n ∈ Q[[z]]n≥0

be such that• g has a positive radius of convergence• g satisfies a linear differential equation with coefficients in
Q(z)• The common denominator of a0 , a1 , . . . , an increases atmost exponentially in n.
Examples.
• Algebraic functions• Hypergeometric functions with rational parameters• Solutions of Picard–Fuchs equations over Q(z).
31 / 111

Gauss Hypergeometric function

2F 1

a b
z
c


=

X (a)n (b)nn≥0(c)n n!

zn.

It satisfies second order linear differential equationz(z − 1)y 00 + (a + b + 1)z − c y 0 + aby = 0.

Resources on this topic are available on the website (in French) :https://danielduverney.fr/fh/

32 / 111

Examples of hypergeometric functions

2F 1

1 1
z
2


2F 1


2F 1



1
= − log(1 − z)
z

1/2 1
z
1

1/2 1/2z1= (1 − z)−1/2

Z
2 1
dt
p
=
·
2
π 0
(1 − t )(1 − zt2 )F. Beukers. Hypergeometric Functions, How Special Are They ?
Notices Am. Math. Soc. 61, No. 1, 48–56 (2014).
http://dx.doi.org/10.1090/noti1065

33 / 111

Eisenstein hypergeometric function
The G function
X
n≥0

5n
n



(−1)n

4n + 1z 4n+1
F.G.M. Eisenstein1823 - 1852z9z 13
= z − z + 10 − 15 · 14
+ ··· ,
2!
3!
which converges for |z| < 5−5/4 , is a solution of the quintic
equation x5 + x = z.
5

J. Stillwell, Eisenstein’s footnote, Math. Intelligencer 17 (1995), no. 2,
58–62.
F.G.M. Eisenstein, Allgemeine Auflösung der Gleichungen von den ersten
vier Graden. J. Reine Angew. Math. 27 (1844), 81–83. Mathematische
Werke I, 7–9.
34 / 111

Connections between E and G functions
Ifg(z) =

X

an z nn≥0

is a G–function, thenf (z) =

X an
n!
n≥0

zn

is an E–function and conversely.
Remarks:• Siegel’s definitions involve algebraic coefficients, not only
rational coefficients.
• Apéry’s proof of the irrationality of ζ(3) is related with thetheory of G–functions.
S. Fischler & T. Rivoal, Approximants de Padé et séries
hypergéométriques équilibrées, J. Math. Pures Appl. 82 (2003), no. 10,
1369–1394.
35 / 111

The ring G
S. Fischler and T. Rivoal
introduce the set G of allvalues taken by any analytic
continuation of any
G–function at any algebraic
point.

S. Fischler

T. Rivoal

They prove that G is a countable subring of C which contains
the field Q of algebraic numbers and the logarithms ofalgebraic numbers.
Conjecturally, G is not a field.
S. Fischler & T. Rivoal, On the values of G-functions,
Comment. Math. Helv. 89 (2014), 313–341.
http://dx.doi.org/10.4171/CMH/321
36 / 111

Conjecture of Bombieri and Dwork
According to a conjecture of Bombieri and Dwork, G should
coincide with the set of periods of algebraic varieties definedover Q.

E. Bombieri

B. Dwork
1923–1998

Connection with the periods of Kontsevich and Zagier

M. Kontsevich

D. Zagier37 / 111

Euler’s constant γ
1
1 1
γ= lim 1 + + + · · · + − log n
n→∞
2 3
n
= 0.577 215 664 901 532 860 606 512 090 082 . . .
1− log 1 +
γ=
k
k
k≥1

Z ∞
1
1
=
−
dx
bxc x
1
Z 1Z 1(1 − x)dxdy·=−
0 0 (1 − xy) log(xy)X 138 / 111

Euler’s constant γ
J. Sondow double integral was inspired by F. Beukers’s workon Apéry’s proof of the irrationality ofζ(3) =

X 1
n3
n≥1= 1.202 056 903 159 594 285 399 738 161 511 . . .

in 1978.

J. Sondow

F. Beukers
39 / 111

Quoting Tanguy Rivoal
It is believed that γ 6∈ Q. Why ? Because if γ = p/q ∈ Q with
gcd(p, q) = 1, then |q| ≥ 10 242 080.
It is also believed that γ 6∈ G. Why ? Because Euler andRamanujan would have found various formulas proving thisfact.
It is also plausible that γ does not even belong to the field of
fractions of G.
Further, it is expected that e = exp(1) does not belong to thefield of fractions of G.

40 / 111

Open problems
Nothing is known on the arithmetic nature of Catalan’sconstantG=

X (−1)n= 0.915 965 594 177 219 015 054 603 5 . . .
(2n + 1)2n≥1

nor of the valueΓ(1/5) = 4.590 843 711 998 803 053 204 758 275 929 152 0 . . .
of Euler’s Gamma function, nor of the value
ζ(5) = 1.036 927 755 143 369 926 331 365 486 457 034 1 . . .
of Riemann’s zeta function.
41 / 111

The units of the ring G
×

The group of units of G contains Q and the values B(a, b),(a, b in Q) of Euler Beta function.
The numbers Γ(a/b)b , a/b ∈ Q \ {0, −1, −2, . . .}, are units
in the ring G.
For instance, π = Γ(1/2)2 is a unit.
Proof:π=X 4(−1)n2n + 1
n≥1

,


2n 3
1 X (42n + 5) n
=
·
12n+4
π
2
n≥1

42 / 111

Walt Disney Productions and 1/π
The formula(1/2)n 3
16 X=(42n + 5) 3 6n
π
n! 2
n≥0

appeared in the Walt Disney film High School Musical,
starring Vanessa Anne Hudgens, who plays an exceptionallybright high school student named Gabriella Montez. Gabriellapoints out to her teacher that she had incorrectly written theleft-hand side as π8 instead of 16
on the blackboard. After firstπclaiming that Gabriella is wrong, her teacher checks (possibly
Ramanujan’s Collected Papers ?) and admits that Gabriella iscorrect.
N.D. Baruah, B. Berndt & H.H. Chan.
Ramanujan’s Series for 1/π : A Survey.
American Mathematical Monthly 116 (2009) 567–587.
http://www.jstor.org/stable/40391165
43 / 111

Ramanujan series for 1/π√
2 2 X (4k)! 26390k + 11031·=π
992 k≥0 k!4
3964k

N.D. Baruah
B. Berndt
Chan Heng Huat
Nayandeep Deka Baruah, Bruce C. Berndt & Heng Huat Chan.
Ramanujan’s Series for 1/π : A Survey.
American Mathematical Monthly 116 (2009) 567–587.
https://en.wikipedia.org/wiki/Ramanujan-Sato_series
44 / 111

Algebraic values of Siegel G functions
Let f be a G–function which is not algebraic. Is–it true thatf (α) is algebraic for at most finitely many algebraic α ?

F. Beukers
1
2F 1

5
12

12
1
2

1323
1331


=

3√
4
11.
4

F. Beukers. Algebraic values of G-functions. J. Reine Angew.
Math. 434 (1993), 45–65.
45 / 111

Wolfart’s work(1988)


a bz with a, b, c in Q. Let ∆ be thecmonodromy group and
E = α ∈ Q | f (α) ∈ Q
Let f (z) = 2 F 1

the exceptional set of f .
(1) If f is algebraic (∆ finite),then E = Q.
(2) If f is arithmetic, then E
is dense in Q.
(3) Otherwise, E is finite.

Jürgen Wolfart

A. Yafaev & B. Edixhoven : complete the proof of (3).
Subvarieties of Shimura varieties. Ann. of Math. (2) 157(2003), no. 2, 621–645.
46 / 111

Monodromy
Singular points 0, 1, ∞Monodromy matrices : M 0 , M 1 , M ∞ with M 0 M 1 M ∞ = Id.
http://math.uchicago.edu/~drinfeld/Hypergeometric_motives/Beukers-Heckman.pdf
47 / 111

Arithmetic groups
We assume that 0 < a, b, c ≤ 1 and either a, b < c or c < a, b.
Then the monodromy modulo scalars embeds in PSL(2, R).
Let g i ∈ SL(2, R) be a lift of the monodromy aroundi = 0, 1, ∞. Then Id2 , g 0 2 , g 1 2 , g 2∞ generate a quaternion
algebra H defined overk = Q cos2 πc, cos2 π(a − b), cos2 π(c − a − b),cos πc cos π(a − b) cos π(c − a − b) .

We say that our monodromy is arithmetic if H is split atexactly one infinite places of k.

48 / 111

Reference

Paula Tretkoff,
Periods and Special Functionsin Transcendence ;Advanced Textbooks in
Mathematics (2017)
World Scientifichttps://doi.org/10.1142/q0085
Paula Tretkoff

https://aussiemathematician.io/
49 / 111

Modular group
Let a = 1/12, b = 5/12, c = 1/2. Then the quaternion algebrais M (2, Q) and the monodromy group is SL(2, Z). It can beshown that
4
E 4 (τ )1/12 5/12
1
=
1−
2F 1
J(τ )
E 4 (i)1/2
where
E 4 (τ ) = 1 + 240

X n3 q n
1−q
n≥1

,
n

∆(τ ) = qY

(1 − q n )24

n≥1

and q = e2iπτ , J(τ ) = E 4 (τ )3 /1728∆(τ ). In particularJ(i) = 1. From CM-theory it follows that if τ0 ∈ Q(i),Im(τ0 ) > 0, then both J(τ0 ) and E 4 (τ0 )/E 4 (i) are algebraic.
50 / 111

Further examples

1/4 1/2 80
81
3/4





1/3 2/3 27
5/6
32



2F 1

9
= .
5

8
= .
5


1/12 1/4 135
2√
6
=
270.
2F 1
5/6
256
5
2F 1


2F 1

1/24 7/24
210 33 5
−
114
5/6
r
√ 6 11= 6.
55

Akihito Ebisu (2014)



Yifan Yang (2015)
51 / 111

Non arithmetic examples

1 − 3a 3a 1
2
a





2a 1 − 4a 1
1−a
2



2F 1

2F 1


2F 1

= 23−2a cos πa.
= 4a cos πa.

7/48 31/48
1
−
29/24
3



F. Beukerss= 25/24 3−11/12 5·

sin π/24·
sin 5π/24

52 / 111

Schneider – Lang Theorem (1949, 1966)

Theodor Schneider

Serge Lang

1911 – 1988

1927 – 2005

Let f 1 , . . . , f m be meromorphic functions in C. Assume f 1
and f 2 are algebraically independent and of finite order. Let K
be a number field. Assume f 0j belongs to K[f 1 , . . . , f m ] forj = 1, . . . , m. Then the set
S = {w ∈ C | w not pole of f j , f j (w) ∈ K for j = 1, . . . , m}is finite.
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schneider.html
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lang.html

53 / 111

Hermite – Lindemann Theorem (again)

Charles Hermite
1822 – 1901

von Lindemann1852 – 1939

Carl Louis Ferdinand
Corollary. If w is a nonzero complex number, one at least of
the two numbers w, ew is transcendental.
Proof. Let K = Q(w, ew ). The two functions f 1 (z) = z,
f 2 (z) = ez are algebraically independent, of finite order, andsatisfy the differential equations f 01 = 1, f 02 = f 2 . The set S
contains {`w | ` ∈ Z}. Since w 6= 0, this set is infinite ; itfollows that K is not a number field. 
54 / 111

The exponential function (again)d z
e = ez ,dz

ez1 +z2 = ez1 ez2

exp : C → C×z 7→ ezker exp = 2iπZ.
The function z 7→ ez is the exponential map of themultiplicative group Gm .
The exponential map of the additive group Ga is
C → C
z 7→ zThe only period is 0.

55 / 111

Elliptic curves and elliptic functions
Elliptic curves : ∆ = g 2 3 − 27g 3 2 6= 0.
E = (t : x : y) ; y 2 t = 4x3 − g 2 xt2 − g 3 t3 ⊂ P2 (C).
Elliptic functions℘02 = 4℘3 − g 2 ℘ − g 3 ,℘(z 1 + z 2 ) = R ℘(z 1 ), ℘(z 2 )expE : C →
z 7→

E(C)

1, ℘(z), ℘0 (z)56 / 111

Weierstraß elliptic functionΩ = Zω1 + Zω2 ⊂ C


X 
1
1
1
−
.
℘(z) = 2 +z(z − ω)2 ω 2ω∈Ω\{0}Karl Weierstrass℘0 (z) =−2·(z − ω)3ω∈Ω
X

1815–1897

57 / 111

Jacobi 12 elliptic functions
Elliptic curve as an intersection of quadrics : the functions sn
and cn.

Karl Jacobi
1804–1851

sn sc sd ns nc nd cs cn cd ds dn dc
https://en.wikipedia.org/wiki/Jacobi_elliptic_functions

58 / 111

Periods of a Weierstrass elliptic function
The set of periods of an elliptic function is a lattice :Ω = {ω ∈ C ; ℘(z + ω) = ℘(z)} = Zω1 + Zω2 .
A pair of fundamental periods (ω1 , ω2 ) is given by
Z ∞
dt
p
,
(i = 1, 2)
ωi = 23
4t − g 2 t − g 3ei
where
4t3 − g 2 t − g 3 = 4(t − e1 )(t − e2 )(t − e3 ).

59 / 111

Examples
Example 1 : g 2 = 4, g 3 = 0, j = 1728
A pair of fundamental periods of the elliptic curvey 2 t = 4x3 − 4xt2 .
is given byZ∞
1
Γ(1/4)2dt√= B(1/4, 1/2) = 3/2 1/2 = 2.6220575542 . . .
ω1 =
2
2 π
t3 − t1andω2 = iω1 .

60 / 111

Examples (continued)
Example 2 : g 2 = 0, g 3 = 4, j = 0
A pair of fundamental periods of the elliptic curvey 2 t = 4x3 − 4t3 .
is
Z∞
ω1 =
1

√

1
dt
Γ(1/3)3= B(1/6, 1/2) = 4/3= 2.428650648 . . .
32 πt3 − 1andω2 = %ω1 ,where % = e2iπ/3 .

61 / 111

Chowla–Selberg Formula

S. Chowla

A. Selberg

1907 - 1995

1917 - 2007X

(m + ni)−4 =(m,n)∈Z2 \{(0,0)}Γ(1/4)826 · 3 · 5 · π 2andX(m + n%)−6 =(m,n)∈Z2 \{(0,0)}Γ(1/3)18
28 π 6

Formula of Chowla and Selberg (1966) : the periods ofelliptic curves with complex multiplication are products ofvalues of the Gamma function at rational points.
62 / 111

Chowla–Selberg Formula : an example

F. Adiceam
Faustin Adiceam (2011) :63 / 111

Elliptic integrals and ellipses
An ellipse with radii a and b has equation
x2 y 2+ 2 =1a2band the length of its perimeter is
Z br
a2 x 2dx.
1+ 4
2
b − b2 x 2−b
In the same way, the perimeter of a lemniscate(x2 + y 2 )2 = 2a2 (x2 − y 2 )is given by an elliptic integral
Z1
4a (1 − t4 )−1/2 dx.
064 / 111

Hypergeometry and elliptic integrals
Recall Gauss Hypergeometricseries Xa b(a)n (b)n z n
·
z =
2F 1
(c)n n!
c
n≥0C.F. Gauss
1777 - 1855

Z1

dx
p
2
(1 
− x )(1 − z 2 x2 )0π
1/2 1/2 2=· 2F 1z .
2
1

K(z) =

65 / 111

Weierstrass sigma function

Karl Weierstrass
Let Ω = Zω1 + Zω2 be a lattice in C.
1815–1897
The canonical product of Weierstraß associated with Ω is thesigma function σΩ defined by


Y 
z
z
z2
1−
exp
+
σΩ (z) = zωω 2ω 2ω∈Ω\{0}This function has a simple zero at each point of Ω.

66 / 111

Hadamard canonical products

For N = {0, 1, 2, . . .} :Yeγzz  −z/n
−
1−
e
.
=z
Γ(−z)
n
n≥1

For Z :

J. Hadamard
1865 - 1963

Y
sin πzz2=z
1− 2 .
πnn≥167 / 111

Wallis formula for πJohn Wallis (Arithmetica
Infinitorum 1655)


4n2
π Y
=
2 n≥1 4n2 − 1=

2 · 2 · 4 · 4 · 6 · 6 · 8 · 8····1 · 3 · 3 · 5 · 5 · 7 · 7 · 9···J. Wallis
1616 - 1703

68 / 111

Weierstraß sigma function : an example
For Ω = Z + Zi :σZ[i] (z) = zYω∈Z[i]\{0}



z
exp
1−
ω



z
z2
+ 2
ω 2ω


·

σZ[i] (1/2) = 25/4 π 1/2 eπ/8 Γ(1/4)−2 = 0.4749493799 . . .
For α ∈ Q(i), the number σZ[i] (α) is algebraic overQ π, eπ , Γ(1/4) .
D.W. Masser, Auxiliary Polynomials in Number Theory, Cambridge Tracts in
Mathematics 207 (2016), Cambridge University Press, Chap. 20.
69 / 111

Weierstraß zeta function
The logarithmic derivative of the Weierstraß sigma function is
the Weierstraß zeta functionσ0=ζ
σand the derivative of ζ is −℘. The minus sign is selected sothat1℘(z) = 2 + a function analytic at 0.
z
The function ζ is therefore quasi–periodic : for any ω ∈ Ωthere exists η = η(ω) such thatζ(z + ω) = ζ(z) + η.

70 / 111

Legendre relation (1811, 1825)

The numbers η(ω) are thequasi–periods of the elliptic
curve.
When (ω1 , ω2 ) is a pair offundamental periods, we set
η1 = η(ω1 ) and η2 = η(ω2 ).
Legendre relation :ω2 η1 − ω1 η2 = 2iπ.

this is Louis Legendre and not
Adrien Marie Legendre(1752 - 1833)

71 / 111

Legendre and FourierPeter Duren, Changing Faces : The Mistaken Portrait ofLegendre.
Notices of American Mathematical Society, 56 (2009)
1440–1443.
The Mathematics Consortium Bulletin, October 2020, vol.2, Issue 2, p. 34–35.
https://www.themathconsortium.in/publication/timcbulletin/vol2/issue2

72 / 111

Examples
For the curve y 2 t = 4x3 − 4xt2 the quasi–periods associatedto the previous fundamental periods areη1 =π(2π)3/2,=
ω1
Γ(1/4)2

η2 = −iη1 ,while for the curve y 2 t = 4x3 − 4t3 they are
2π
27/3 π 2,η1 = √= 1/2
3 Γ(1/3)33ω1

η2 = % 2 η1 .

73 / 111

Transcendence and elliptic functions
Siegel (1932) : elliptic analog of Lindemann’s Theorem on thetranscendence of π.
Schneider (1937) : elliptic analog of Hermite–Lindemann
Theorem. General transcendence results on values of elliptic
functions, on periods, on elliptic integrals of the first andsecond kind.

C.L. Siegel

Th. Schneider

1896 - 1981

1911 - 1988
74 / 111

Elliptic analog of Hermite–Lindemann Theorem
Let w ∈ C, not pole of ℘. Then one at least of the numbers
g 2 , g 3 , w, ℘(w) is transcendental.
Proof as a consequence of the Schneider–Lang Theorem.
Let K = Q(g 2 , w, ℘(w), ℘0 (w)). The two functions f 1 (z) = z,
f 2 (z) = ℘(z) are algebraically independent, of finite order. Set
f 3 (z) = ℘0 (z). From ℘0 2 = 4℘3 − g 2 ℘ − g 3 one deduces
f 01 = 1,

f 02 = f 3 ,f 03 = 6f 2 2 − (g 2 /2).

The set S contains{`w | ` ∈ Z, `w not pole of ℘}which is infinite. Hence K is not a number field. 
75 / 111

Elliptic integrals of the third kind
Quasi–periodicity of the Weierstraß sigma function :σ(z + ωi ) = −σ(z)eηi (z+ωi /2)

The function
σ(z + u) −zζ(u)eF u (z) =σ(z)σ(u)satisfies
F u (z+ωi ) = F u (z)eηi u−ωi ζ(u) .

(i = 1, 2).

J-P. Serre (1979)

76 / 111

Periods of elliptic integrals of the third kind
Theorem (1979). Assume g 2 , g 3 , ℘(u1 ), ℘(u2 ), β are
algebraic and Zu1 ∩ Ω = {0}. Then the numberσ(u1 + u2 ) β−ζ(u1 ) u2eσ(u1 )σ(u2 )is transcendental.
Corollary. Transcendence of periods of elliptic integrals of thethird kind :eωζ(u)−ηu+βω .

77 / 111

Schneider’s Theorem on Euler’s Beta function
Let a, b be rational numbers,
not integers. Then thenumber B(a, b) istranscendental.
Th. Schneider
1911 - 1988

Further results by Th. Schneider (1941) and S. Lang (1960’s)
on abelian functions, abelian varieties and commutative
algebraic groups.

78 / 111

Linear independence of transcendental numbers

A. Baker,J. Coates,

D.W. Masser,

G. Wüstholz,. . .

79 / 111

Values of Euler Beta and Gamma functions

J. Wolfart
G. Wüstholz
G. Wüstholz : any Q–linear relation among periods of anabelian variety arises from its endomorphisms.
(J. Wolfart & G. Wüstholz) : linear independence over thefield of algebraic numbers of the values of the Euler Beta
function at rational points (a, b).
Transcendence of values at algebraic points of hypergeometricfunctions with rational parameters.
J. Wolfart & G. Wüstholz. Der Überlagerungsradius
gewisser algebraischer Kurven und die Werte der Betafunktion anrationalen Stellen. Math. Ann. 273 (1985), no. 1, 1–15.
80 / 111

Transcendence of Γ(1/4) and Γ(1/3) (1976)

Algebraic independence of two numbers among ω1 , ω2 , η1 , η2
when g 2 and g 3 are algebraic.

Γ(1/4) and π are algebraicallyindependent.
Γ(1/3) and π are algebraicallyindependent.

G.V. Chudnovsky

81 / 111

Modular functions

P (q)= E 2 (q) = 1 − 24

X nq n,1 − qnn≥1Q(q)= E 4 (q) = 1 + 240
R(q)= E 6 (q) = 1 − 504

X n3 q n
1 − qnn≥1
X n5 q nn≥1∆=

1 − qn,S. Ramanujan·

1887 - 1920

Q3
1
Q3 − R2 and J =.
1728∆
82 / 111

Nesterenko’s Theorem
1996, Yu. V. Nesterenko :Let q ∈ C satisfy 0 < |q| < 1. Then three at least of thenumbers
q, P (q), Q(q), R(q)are algebraically independent.

Y. Nesterenko

83 / 111

Corollary of Nesterenko’s Theorem

The three numbers
π, eπ and Γ(1/4)
are algebraically independent.
Open problem :Show that e and π are algebraically independent.

84 / 111

Transcendence of Γ(i) following D.W. Masser
Γ(i)Γ(i) = Γ(i)Γ(−i) =Γ(i)Γ(1 − i)
π
2π
=
= π
·
−i
−i sin(iπ)
e − e−πD.W. Masser
D.W. Masser, Auxiliary Polynomials in Number Theory,
Cambridge Tracts in Mathematics 207 (2016),
Cambridge University Press.
85 / 111

Standard relations among Gamma values(Translation) :Γ(a + 1) = aΓ(a)(Reflection) :Γ(a)Γ(1 − a) =πsin(πa)(Multiplication) : for any positive number n,
n−1
Y



k
Γ a+
n
k=0



= (2π)(n−1)/2 n−na+(1/2) Γ(na).

86 / 111

Conjecture of Rohrlich
Conjecture (D. Rohrlich)
Any multiplicative relationY
π b/2
Γ(a)ma ∈ Q
a∈Q

with b and ma in Z is in theideal generated by thestandard relations.

David Rohrlich

S. Lang. Relations de distributions et exemples classiquesSéminaire Delange-Pisot-Poitou, 19e année : 1977/78, Théorie desnombres, Fasc. 2, Exp. N◦ 40, 6 p.
Collected Papers, vol. III, Springer (2000), 59–65.
87 / 111

Conjecture of Rohrlich–Lang

D. Rohrlich

S. Lang
1927 - 2005

Conjecture (D. Rohrlich–S. Lang, 1978) Any algebraic
dependence relation among (2π)−1/2 Γ(a) with a ∈ Q is in theideal generated by the standard relations(universal odd distribution).
Consequence of the conjecture of Rohrlich–Lang √(F. Adiceam) : the three numbers Γ(1/5), Γ(2/5) and eπ 5
are algebraically independent. (Not yet know).
88 / 111

Gamma valuesq√Γ(1/24)Γ(11/24) √= 3 2 + 3.
Γ(5/24)Γ(7/24)

Yves André — Groupes deGalois motiviques et périodes.
Séminaire N. Bourbaki, Samedi
7 novembre 2015, 68ème année,
2015-2016, n◦ 1104.
http://www.bourbaki.ens.fr/TEXTES/1104.pdf

Y. André

J. Ayoub : analogs for function fields of the periods
conjectures of Grothendieck and Kontsevich–Zagier.
Une version relative de la conjecture des périodes deKontsevich-Zagier.
Ann. of Math. (2) 181 (2015), no. 3, 905–992.
89 / 111

Riemann zeta function
ζ(s)=

X 1
n≥1

L. Euler
1707 - 1783

=

Y
p

ns
1
1 − p−sB. Riemann1826 - 1866

Euler : s ∈ R.

Riemann : s ∈ C.

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Euler.html
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Riemann.html

90 / 111

Special values of the Riemann zeta function

Jacques Bernoulli

Leonhard Euler

1655–1705

1707 – 1783s ∈ Z : π −2k ζ(2k) ∈ Q for k ≥ 1 (Bernoulli numbers).

91 / 111

Values of the Riemann zeta function at thepositive integers
Even positive integers
ζ(2n) = (−1)n−1 22n−1

B 2n 2nπ(2n)!

(n ≥ 1).

Odd positive integers : ζ(2n + 1), n ≥ 1 ?
Question : for n ≥ 1, is the number
ζ(2n + 1)
π 2n+1
rational ?
92 / 111

Diophantine question
Determine all algebraic relations among the numbers
ζ(2),

ζ(3),

ζ(5),

ζ(7), . . .

Conjecture. There is no algebraic relation : the numbers
ζ(2),

ζ(3),

ζ(5),

ζ(7), . . .

are algebraically independent.
As a consequence, one expects the numbers ζ(2n + 1) andζ(2n + 1)/π 2n+1 for n ≥ 1 to be transcendental.
93 / 111

Values of ζ at the even positive integers• F. Lindemann : π is atranscendental number, hence ζ(2k) also
for k ≥ 1.

F. Lindemann1852 - 1939

94 / 111

Values of ζ at the odd positive integers• Apéry (1978) : The number
ζ(3) =

X 1
n≥1

n3

= 1.202 056 903 159 594 285 399 738 161 511 . . .

is irrational.
• Rivoal (2000) + Ball, Zudilin, Fischler,. . . : Infinitely many
numbers among ζ(2k + 1) are irrational + lower bound for thedimension of the Q-space they span.
95 / 111

Tanguy Rivoal

Let  > 0. For any sufficiently
large odd integer a,the dimension of the Q–spacespanned by the numbers
1, ζ(3), ζ(5), · · · , ζ(a) is at
least
1−
log a.
1 + log 2

T. Rivoal

96 / 111

Wadim Zudilin (2002)• At least one of the four numbers
ζ(5), ζ(7), ζ(9), ζ(11)is irrational.
• There exists an odd number jin the interval [5, 69] such thatthe three numbers
1, ζ(3), ζ(j)are Q–linearly independent.

W. Zudilin

97 / 111

References
S. Fischler
Irrationalité de valeurs de zêta,(d’après Apéry, Rivoal, . . .),Sém. Nicolas Bourbaki, 2002-2003,
N◦ 910 (Novembre 2002).

S. Fischler

http//www.math.u-psud.fr/~fischler/publi.html

C. Krattenthaler & T. Rivoal, Hypergéométrie et fonction zêta deRiemann, Mem. Amer. Math. Soc. 186 (2007), 93 p.
http://www-fourier.ujf-grenoble.fr/~rivoal/articles.html

S. Fischler, J. Sprang, W. Zudilin. Many odd zeta values are irrational,(2018).
https://arxiv.org/abs/1803.08905

98 / 111

Hurwitz zeta function
T. Rivoal (2006) : consider the Hurwitz zeta functionX
1·
ζ(s, z) =(k + z)sk≥1
Expand ζ(2, z) as a series inz 2 (z − 1)2 · · · (z − n + 1)2·(z + 1)2 · · · (z + n)2
The coefficients of the expansion belong to Q + Qζ(3). This
produces a new proof of Apéry’s Theorem on the irrationalityof ζ(3).
In the same way : new proof of the irrationality of log 2 by
expandingX (−1)k·k+z
k≥1
99 / 111

P 2n
P n2
On
z and
z
The name Fredholm series is often wrongly attributed to thepower series
X n
z2
n≥0

(see Allouche & Shallit, Notes on chapter 13). However
Fredholm studied rather the theta series
X 2
zn .
n≥0

J-P. Allouche

J. Shallit

Automatic sequences, Theory, applications, generalizations, Cambridge University
Press, Cambridge, 2003,

100 / 111

P 2n
On
z
Let χ(z) =

P

n≥0 z

2n

.

E. Catalan (1875), J. Sondow and W. Zudilin (2006) :
Z1
dx
·
γ=
(χ(x) − x)x(1 + x)0

E. Catalan

J. Sondow

W. Zudilin

1814 - 1894
J. Sondow & W. Zudilin. Euler’s constant, q-logarithms, and formulas ofRamanujan and Gosper. Ramanujan J. 12 (2006) 225–244.
101 / 111

P 2n
P n2
On
z and
z
P 2n P n2
Connection between the seriesz ,z , theta functions,modular functions and paperfolding.
Michel Mendes-France & Ahmed Sebbar, Pliages de papier, fonctions
thêta et méthode du cercle, Acta Math. 183 (1999), 101–139.
Ahmed Sebbar, Paperfolding and modular functions. in Exponential
Analysis of Differential Equations and Related Topics (ed. Y. Takei).
RIMS Kôkyûroku Bessatsu, B 52 (2014), 97–126.

M. Mendes–France
1936–2018

A. Sebbar
102 / 111

Mahler functions
A. J. Kempner (1916) proved the transcendence of the numberXnχ(1/2) =2−2 .
n≥0

Kurt Mahler (1930, 1969) :For d ≥ 2, transcendence ofthe values atalgebraic pointsXnzd .
of χd (z) =n≥0

K. Mahler1903 - 1988

Tool: The function χd satisfies the functional equationχd (z) = z + χd (z d ) for |z| < 1.
103 / 111

Mahler functions : another example
The function
h(z) =

Yn(1 − z 2 )n≥0satisfies the functional equationh(z) = (1 − z 2 )h(z 2 ).
The coefficients of its Taylor series at the originXh(z) =(−1)tn z n|z| < 1n≥0

are given by the Thue–Morse sequence(tn )n≥0 = (0110100110010110 . . . ) :t0 = 0,t2n = tn ,t2n+1 = 1 − tn .
104 / 111

Mahler theory : general framework
ωij (1 ≤ i, j ≤ n) nonnegative integers,Ω : Cn→z = (z 1 , . . . , z n ) 7→

n
Y
j=1

Cn!
z i ωij1≤i≤nf 1 , . . . , f m power series with algebraic coefficients in nvariables,
A a m × m matrix with algebraic entries,
B(z) a 1 × m matrix with entries rational fractions withalgebraic coefficients.
Functional equations :f 1 (Ωz)
f 1 (z) .. 
 . 
 .  = A  ..  + B(z).
f m (z)f m (Ωz)
105 / 111

Mahler theory
The proof of algebraic
independence results forvalues of Mahler functions
reduces to the proof ofalgebraic independence of thefunctions.
Kumiko Nishioka,
Mahler Functions andTranscendence,
Lecture Notes in Mathematics
1631 (1996), Springer Verlag.

K. Mahler1903 - 1988

Kumiko Nishioka

106 / 111

Rational and algebraic points on graphs• E. Bombieri and J. Pila (1989) : on the number of integral
points on arcs and ovals• J. Pila (2004) : on integer points on the dilation of asubanalytic surface• Upper bound for the number of points and the density ofalgebraic points of bounded degree and height on graphs oftranscendental analytic functions.

E. Bombieri

J. Pila107 / 111

Rational and algebraic points on graphs• A. Surroca (2006) : transcendence method related toSchneider’s work.
• D.W. Masser (2011) : There exists a constant C such thatfor any D ≥ 3 there are at most C(log D)(log log D)2 rational
numbers s with 2 < s < 3 and denominator at most D suchthat the Riemann zeta function ζ(s) is also rational withdenominator at most D.
• E. Besson (2014) : Euler Gamma function.
• G. Boxall, G. Jones, P. Villemot, T.P. Chalebgwa. . .

A. Surroca

D.W. Masser
108 / 111

o-minimal structures from model theory• 2006, J. Pila and A.
Wilkie : rational points of adefinable set• 2011, J. Pila : contribution
to the André–Oort conjecture
for Cn using o-minimality.

J. Pila and A. Wilkie

A. Beshenov & M. Bilu & Yu. Bilu & P. Rath.
Rational points on analytic varieties.
EMS Surv. Math. Sci. 2 (2015), 109–130.
http://dx.doi.org/10.4171/EMSS/10

109 / 111

Conclusion

The transcendence theory of values of Siegel E–functions andof functions satisfying Mahler equations is strong, but a lot
remains to be done in the other situations. One open problem
is to prove the Hermite–Lindemann Theorem on thetranscendence of log α for nonzero algebraic number α byusing the logarithmic function (i.e. the theory of G–functions)instead of the exponential function (i.e. the theory of
E–functions).

110 / 111

February 25, 2021

Irrationality and transcendence ofvalues of special functions.
Michel Waldschmidt
Institut de Mathématiques de Jussieu — Paris VI
http://www.imj-prg.fr/~michel.waldschmidt/

111 / 111