Recent Diophantine results on zeta values: a survey

TheInstituteofMathematicalSciencesMathematicsColloquiumDecember9,2009http://www.imsc.res.in/

Recent Diophantine results on zeta values: a survey

Michel W alds chmidt

Institut de Math ´ematiques de Jussieu & P aris VI

http://www.math.jussieu.fr/∼miw/

1/75

Abs tract

After the pro of by R. Ap ´ery of the irra tionalit y of ζ (3) in 1976, a numb er of pap ers have b een devoted to the study of Diophantine prop erties of values of the Riemann zeta function at p ositive integers.

A survey has b een written by S. Fischler fo r the Bourbaki Semina r in Novemb er 2002.

W e review mo re recent results including contributions by S. Fischler, M. Hata, C. Krattenthaler, R. Ma rcovecchio, R. Murt y, G. Rhin, T. Rivoal, C. Viola, W. Zudilin.

W e plan also to sa y a few w ords on the analog of this theo ry in fini te cha racteristic, with w orks of W.D. Bro wna w ell, M. P appanik olas, D. Thakur, Chieh-Y u Chang, Jing Y u and others.

2/75

Zeta

•

Riemann zeta values

•

Multizeta values

•

W eierstraß zeta function

•

Fib onacci zeta values

•

Hurwitz zeta function

•

Ca rlitz zeta va lues

•

(Other zeta functions : Dedekind, Hasse-W eil, Lerch, Selb erg, Witten, Milno r, dynamical systems. ..)

•

L – functions. ..

3

Riemann zeta function

ζ ( s )=

!

n≥1

1 n

s

=

"

p

1 1

−

p

−s

Euler : s

∈R

. Riemann : s

∈C

.

4

Sp ecial values of Ri emann zeta function

Leona rd Euler (1739) ζ ( s ) fo r s

∈Z

ζ (2) = π

2

6 ,

ζ (4) = π

4

90 ,

ζ (6) = π

6

945 ,

ζ (8) = π

8

9450

·

π

−2k

ζ (2 k )

∈Q

fo r k

≥

1

5/75

Bernoulli numb ers

Jacques Bernoulli (1654–1705), Bernoulli numb ers : B

2

=1 / 6 B

4

=

−

1 / 30 B

6

=1 / 42 B

8

=

−

1 / 30 B

10

=5 / 66

z e

z−

1 =

∞!

n=0

B

n

z

n

n ! ,

ζ (2 k )=(

−

1)

k−1

2

2k−1

B

2k

(2 k )! π

2k

( k

≥

1) .

6/75

T rans cendence of even zeta values

•

F. Lindemann : π is transcendental, hence ζ (2 k ) also fo r k

≥

1 .

Theo rem (H ermite–Lindemann). F or any non-zero complex numb er z , one at least of the tw o numb ers z and e

z

is transcendental.

Corollaries.

T ranscendence of log α and of e

β

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

%

=0 .

Diophantine ques tion Odd p ositive integers : ζ (2 k + 1) , k

≥

1 ?

Question. For n

≥

1 , is the kumb er

ζ (2 k + 1) π

2k+1

rational ? Describ e all algeb raic relations among the numb ers

ζ (2) , ζ (3) , ζ (5) , ζ (7) ,. ..

Conjecture.

There is no relation at all : the numb ers

ζ (2) , ζ (3) , ζ (5) , ζ (7) ,. .. are algeb raically indep endent. In pa rticula r the numb ers ζ (2 k + 1) and ζ (2 k + 1) / π

2k+1

fo r k

≥

1 are conjectured to b e transcendental.

V alues of ζ at odd p os itive integers

•

Ap ´ery (1978) : The numb er

ζ (3) =

!

n≥1

1 n

3

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

is irrational.

•

Rivoal (2000) & Ball, Zudilin. .. Infinitely many ζ (2 k + 1) are irrational & lo w er b ound fo r the dimension of the

Q

-span.

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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.

10/75

W adim Zudilin

•

A t least one of the four numb ers ζ (5) , ζ (7) , ζ (9) , ζ (11) is irrational.

•

There exists an o dd integer j in the range [5 , 69] such that the three numb ers 1 , ζ (3) , ζ ( j ) are

Q

–linea rly indep endent.

11

Zudilin’s home page

http://wain.mi.ras.ru/zw/index.html

References to w orks on zeta values by 2000 M. Hata, T. Rivoal 2001 K. Ball and T. Rivoal, L.A. Gutnik, G. Rhin and C. Viola, T. V asily ev, W. Zudilin 2002 T. Rivoal, V.N. So rokin, W. Zudilin 2003 Y u.V. Nesterenk o, T. Rivoal, J. Sondo w, C. Viola, W. Zudilin 2004

S.Fischler

, W. Zudilin 2005 F. Calega ri, S. Zlobin 2006 M. Huttner, C. Krattenthaler, T. Rivoal and Zudilin 2007 C. Krattenthaler and T. Rivoal 2008 F. Beuk ers 2009 S. Fis chler and W. Zudilin

LastmodifiedonSeptember19,2009

12

Irrationalit y of zeta values

S. Fischler Irrationalit ´e de valeurs de zˆeta, (d’ap r`es Ap ´ery , Rivoal, ...) , S ´em. Nicolas Bourbaki, 2002-2003, N

◦

910 (Novemb re 2002). Ast ´erisque

294

(2004), 27-62

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

13/75

Christian Krattenthaler and T anguy Rivoal

http://www-fourier.ujf-grenoble.fr/∼rivoal/articles.html

C. Krattenthaler et T. Rivoal, Hyp erg ´eom ´etrie et fonction zˆeta de Riemann, Mem. Amer. Math. So c.

186

(2007), 93 p.

14/75

Irrationalit y measures : the state of the art

ϑ

∈R####

ϑ

−

p q

#### ≥

1 q

µ+"

µ ( ϑ ) < +

∞⇐⇒

ϑ is not a Liouville numb er

ϑ yea r autho r µ ( ϑ ) < π 2008 V.Kh. Salikhov 7 . 6063085 ζ (2) = π

2

/ 6 1996 G. Rhin and C. Vi ol a 5 . 441243 ζ (3) 2001 G. Rhin and C. Vi ol a 5 . 513891 log 2 2008 R. Ma rcovecchio 3 . 57455391

Irrationalit y measure fo r π : his to ry

K. Mahler 1953 : π is not a Liouville numb er and µ ( π )

≤

30 M. Mignotte 1974 : µ ( π )

≤

20 G.V. Chudnovsky 1984 : µ ( π )

≤

14 . 5 M. Ha ta 1992 : µ ( π )

≤

8 . 0161 V.Kh. Salikhov 2008 : µ ( π )

≤

7 . 6063

A b ound µ ( ϑ

2

)

≤

κ fo r some ϑ

∈R

implies µ ( ϑ )

≤

2 κ . Hence the result of Rhin and Viola µ ( ζ (2))

≤

5 . 441 .. . implies only µ ( π )

≤

11 . 882 .. . .

Conversely , a b ound fo r the irrationalit y exp onent of ϑ do es not yield any b ound fo r µ ( ϑ

2

) !

Irrationalit y measure fo r ζ (2) and ζ (3) : histo ry

ζ (2)

R. Ap ´ery 1978, F. Beuk ers 1979 µ ( ζ (2)) < 11 . 85 R. Dvo rnicich and C. Viola 1987 µ ( ζ (2)) < 10 . 02 M. Ha ta 1990 µ ( ζ (2)) < 7 . 52 G. Rhin and C. Viola 1993 µ ( ζ (2)) < 7 . 39 G. Rhin and C. Viola 1996 µ ( ζ (2)) < 5 . 44

ζ (3)

R. Ap ´ery 1978, F. Beuk ers 1979 µ ( ζ (3)) < 13 . 41 R. Dvo rnicich and C. Viola 1987 µ ( ζ (3)) < 12 . 74 M. Hata 1990 µ ( ζ (3)) < 8 . 83 G. Rhin and C. Viola 2001 µ ( ζ (3)) < 5 . 51

17/75

Irrationalit y measure fo r log 2 : histo ry

log 2

Hermite–Lindemann, Mahl er, Bak er, Gel’fond, F eldman ,. .. : transcendence measures G. Rhin 1987 µ (log 2) < 4 . 07 E.A. Rukhadze 1987 µ (log 2) < 3 . 89 R. Ma rcovecchio 2008 µ (log 2) < 3 . 57

Reference : R. Ma rcovecchio , The Rhin-Viola metho d fo r log 2, Acta Arithmetica vol.

139

no.2 (2009), 147–184.

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Geo rges Rhin and Ca rlo Viol a

On a p ermutation group related to ζ (2) , Acta Arith.

77

(1996), no.1, 23–56. The group structure fo r ζ (3) , Acta Arith.

97

(2001), no.3, 269–293. The p ermutation group metho d fo r the diloga rithm, Ann. Scuola No rm. Sup. Pisa Cl. Sci. (

5

)

4

(2005), no.3, 389–437.

19

Criterion of Y u. V. Nesterenk o (qualitative)

Let ϑ

1

,. .. , ϑ

m

b e complex numb ers.

Y u.V.Nesterenk o (1985) Let m b e a p ositive integer and α a p ositive real numb er satisfying α >m

−

1 . Assume there is a sequence ( L

n

)

n≥0

of linea r fo rms in

Z

X

0

+

Z

X

1

+ .. . +

Z

X

m

of height

≤

e

n

such that

|

L

n

(1 , ϑ

1

,. .. , ϑ

m

)

|

= e

−αn+o(n)

. Then 1 , ϑ

1

,. .. , ϑ

m

are linea rly indep endent over

Q

. Example : m =1 – irrationalit y criterion.

20

Simplified pro of of Nesterenk o’s Theo rem

F rancesco Amo roso Pierre Colmez

Refinements : Ra

ff

aele Ma rcovecchio, Pierre Bel .

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Recent developments

St ´ephane Fischler and W adim Zudilin , A refinement of Nesterenk o’s linea r indep endence criterion with applications to zeta values. T o app ea r in Math. Annalen.

Prep rint MPIM 2009-35.

22/75

Fischler and Zudilin, 2009

There exist p ositive o dd integers i

≤

139 and j

≤

1961 such that the numb ers 1 , ζ (3) , ζ ( i ) , ζ ( j ) are linea rly indep endent over

Q

.

There exist p ositive o dd integers i

≤

93 and j

≤

1151 such that the numb ers 1 , log 2 , ζ ( i ) , ζ ( j ) are linea rly indep endent over

Q

.

Multizeta val ues

For s

1

,. .. ,s

k

p ositive integers with s

1≥

2 ,

ζ ( s

1

,. .. ,s

k

)=

!

n1>···>nk≥1

1 n

s11···

n

skk

P . Ca rtier. – F onctions p olyloga rithmes, nomb res p olyz ˆetas et group es pro-unip otents. S ´em. Bourbaki no. 885 Ast ´erisque

282

(2002), 137-173.

M. Ho ff man’s w eb site

http://www.usna.edu/Users/math/meh/biblio.html

References on multizeta values and Euler sums

A Double ha rmonic series

48references

B T riple ha rmonic series

8references

C Multiple ha rmonic series/multiple zeta values

137references

D Multiple zeta values over function fields

5references

E Alternating series

16references

F Multiple p olyloga rithms/nested sums

46references

G Finite multiple ha rmonic sums

25references

In 2008 : 62 references In 2009 : 30 references + prep rints : 66 references

LastmodifiedonAugust18,2009

25/75

EZF ace calculato r at CECM

http://oldweb.cecm.sfu.ca/projects/EZFace/

CentreforExperimentalandConstructiveMathematicsatSimonFraserUniversity

The calcula to r gives numerical values of MZVs with up to 100 decimal pla ces accuracy . The calcula to r also has a function to lo ok fo r relations of linea r dep endence ; lindep([ a, b, c ]) lo oks fo r a vanishing linea r combination of a , b , c with integer co e

ffi

cients. This ma kes it easy (EZ ?) to di scover new identities ! J. Bl ¨umlein, D.J. Broadhurst, J.A.M. V ermaseren The Multiple Zeta V alue Data Mine

arXiv:0907.2557v1[math-ph]

26/75

Gamma and Beta values

Γ

( z )=

$∞

0

e

−t

t

z·

dt t

= e

−γz

z

−1 ∞"

n=1 %

1+ z n

&−1

e

z/n

.

Γ

( n + 1) = n ! , ( n

≥

0);

Γ

(1 / 2) =

√

π ,

Γ %

(1) =

−

γ .

B ( a, b )=

Γ

( a )

Γ

( b )

Γ

( a + b )

=

$1

0

x

a−1

(1

−

x )

b−1

dx.

27

W eiers traß functions

Let

Ω

=

Z

ω

1

+

Z

ω

2

b e a lattice in

C

. The

canonicalproduct

attached to

Ω

is the W eierstraß sigma function σ ( z )= σ

Ω

( z )= z

"

ω∈Ω\{0} %

1

−

z ω

&

e

(z/ω)+(z2/2ω2)

The loga rithmic derivative of the sigma function is the

Weierstraßzetafunction

σ

%

σ = ζ

and the derivative of ζ is

−

℘ , where ℘ is the W eierstra ß elliptic function :

℘

%2

=4 ℘

3−

g

2

℘

−

g

3

,

℘ ( z + ω )= ℘ ( z ) , ζ ( z + ω )= ζ ( z )+ η .

28

Complex multiplication : Q ( i )

℘

%2

=4 ℘

3−

4 ℘ ,g

2

=4 ,g

3

=0 ,

ω

1

=

$1

0

dx

√

x

−

x

3

= 1 2 B(1 / 4 , 1 / 2) =

Γ

(1 / 4)

2√

8 π =2 . 622 057 554 2 .. . ω

2

= i ω

1

. η

1

= π ω

1

= (2 π )

3/2

Γ

(1 / 4)

2

, η

2

=

−

i η

1

.

29/75

Complex multiplication : Q ( - )

- = e

2iπ/3

℘

%2

=4 ℘

3−

4 ,g

2

=0 ,g

3

=4 ,

ω

1

=

$1

0

dx

√

1

−

x

3

= 1 3 B(1 / 3 , 1 / 2) =

Γ

(1 / 3)

3

2

4/3

π =2 . 428 650 648 .. . ω

2

= -ω

1

, η

1

= 2 π

√

3 ω

1

= 2

7/3

π

2

3

1/2Γ

(1 / 3)

3

, η

2

= -

2

η

1

.

30/75

T rans cendence of sp ecial values of W eiers traß functions

Th. Schneider (

1934

). The numb ers

Γ

(1 / 4)

4

/ π

3

and

Γ

(1 / 3)

3

/ π

2

are transcendental.

Diophantine app ro ximation

Γ

(1 / 4)

4

/ π

3

and

Γ

(1 / 3)

3

/ π

2

are not Liouville numb ers.

Lo w er b ounds fo r linea r combinations of elliptic loga rithms : Bak er, Coates, Anders on .. .in the CM case, Philipp on-W aldschmidt in the general case, refinements by N. Hirata Kohno, S. David, ´E. Gaudron - use Arakh el ov’s Theo ry ( J-B. Bost : slop es inequalities ).

Motivation

: metho d of S. Lang fo r solving Diophantine equations (integer p oints on elliptic curves).

Sinnou David and No rik o Hi rata

David, Sinnou ; Hirata-Kohno, No rik o Linea r fo rms in elliptic loga rithms. J. Reine Angew. Math.

628

(2009), 37–89.

33/75

Ab elian va rieties

Th. Schneider (

1948

). For a and b in

Q

with a , b and a + b not in

Z

, the numb er

B ( a, b )=

Γ

( a )

Γ

( b )

Γ

( a + b )

is trans cendental .

The pro of involves Ab elian integrals of higher genus, related with the Jacobian of a F ermat curve.

34/75

Chudnovsky’s algeb raic indep endence Theo rem

G.V. Chudnovsky (1978)

Theorem

Tw o at least of the numb ers

g

2

,g

3

, ω

1

, ω

2

, η

1

, η

2

are algeb raically indep endent.

Corollary:

π and

Γ

(1 / 4) = 3 . 625 609 908 2 .. . are algeb raically indep endent. Also π and

Γ

(1 / 3) = 2 . 678 938 534 7 .. . are algeb raically indep endent.

35

Diophantine app ro ximation

TranscendencemeasuresforΓ

(1 / 4) ( P . Philipp on, S. Bruiltet ) For P

∈Z

[ X ,Y ] with degree d and height H ,

log

|

P

'

π ,

Γ

(1 / 4)

(|

>

−

10

326 '

(log H + d log ( d + 1)

(

·d2 '

log ( d + 1)

(2

Corollary:Γ

(1 / 4) is not a Liouville numb er :

#### Γ

(1 / 4)

−

p q

####

> 1 q

10330·

36

Chudnovsky’s metho d

( K.G. V asil

%

ev 1996, P . Grinspan 2002). Tw o at least of the three numb ers π ,

Γ

(1 / 5) and

Γ

(2 / 5) are algeb raically indep endent.

The pro of involves a simple facto r of dimens ion 2 of the Jacobian of the F ermat curve

X

5

+ Y

5

= Z

5

which is an Ab eli an va riet y of dimension 6 .

37/75

Ramanujan F unctions

P ( q ) = 1

−

24

∞!

n=1

nq

n

1

−

q

n

,

Q ( q ) = 1 + 240

∞!

n=1

n

3

q

n

1

−

q

n

,

R ( q ) = 1

−

504

∞!

n=1

n

5

q

n

1

−

q

n ·

38/75

Eis ens tein Series _E

2k

( z ) = 1 + (

−

1)

k

4 k B

k ∞!

n=1

n

2k−1

z

n

1

−

z

n ·

F. G. M. Eisenstein (1823 - 1852) P ( z )= E

2

( z ) ,

Q ( z )= E

4

( z ) , R ( z )= E

6

( z ) .

Sp ecial values

τ=i,q=e −2π,ω1= Γ(1/4)2√8π =2.6220575542... P(q)= 3π ,Q(q)=3 %ω1π &4,R(q)=0. τ=$,q=−e −π √3,ω1= Γ(1/3) 3

24/3π =2.428650648... P(q)= 2 √3π ,Q(q)=0,R(q)= 272 %ω1π &6.

Y u. V. Nes terenk o

Theorem

( Nesterenk o ,

1996

). F or any q

∈C

with 0 <

|

q

|

< 1 , three at least of the four numb ers q, P ( q ) ,Q ( q ) ,R ( q ) are algeb raically indep endent.

Tools:

The functions P , Q , R are algeb raicaly indep endent over

C

( q ) ( K. Mahler ) and satisfy a system of di

ff

erential equations fo r D = q d/dq :

12 DPP =P− QP ,3 DQQ =P− RQ ,2 DRR =P− Q 2

R ·

41/75

Consequences of Nes terenk o’s Theo rem

The three numb ers

π ,e

π

,

Γ

(1 / 4)

are algeb raically indep endent.

The three numb ers

π ,e

π √3

,

Γ

(1 / 3)

are algeb raically indep endent.

42/75

Sp ecial values of W eierstraß sigma functions

The numb er

σ

Z[i]

(1 / 2) = 2

5/4

π

1/2

e

π/8Γ

(1 / 4)

−

is transcendental.

43

Fib onacci zeta values

F

0

=0 ,F

1

=1 ,F

n+1

= F

n

+ F

n−1

ζ

F

( s )=

!

n≥1

1 F

sn

Iek ata Shiok aw a (joint w orks with Ca rsten Elsner and Shun Shimomura , 2006)

ζ

F

(2) , ζ

F

(4) , ζ

F

(6) are algeb raically indep endent. Consequence of Nesterenk o ’s Theo rem.

44

Fib onacci zeta values ζ F ( s )= )

n

≥

1 1 F

sn

u = ζ

F

(2) ,v = ζ

F

(4) ζ

F

(4 s + 2)

∈Q

( u, v ) fo r s

≥

0 , s

∈Z

.

ζ

F

(4 s )

−

r

s

ζ

F

(4)

∈Q

( u, v ) fo r s

≥

2 , s

∈Z

, with some r

s∈Q\{

0

}

.

For s

1

, s

2

, s

3

, distinct p ositive integers, the numb ers ζ

F

(2 s

1

) , ζ

F

(2 s

2

) , ζ

F

(2 s

3

) are algeb raically dep endent if and only if the three integers s

i

are o dd.

45/75

Standa rd relations among Gamma values

Translation:

Γ

( a + 1) = a

Γ

( a )

Reflexion:

Γ

( a )

Γ

(1

−

a )= π sin( π a )

Multiplication:foranypositiveinteger

n

,

n−1"

k=0 Γ *

a + k n

+

= (2 π )

(n−1)/2

n

−na+(1/2)Γ

( na ) .

46/75

Rohrlich’s Conjecture

Conjecture(D.Rohrlich)

Any multiplicative relation

π

b/2 "

a∈Q Γ

( a )

ma∈Q

with b and m

a

in

Z

lies in the ideal generated by the standa rd relations. Examples :

Γ *

1 14

+

Γ *

9 14

+

Γ *

11 14

+

=4 π

3/2

"

1≤k≤n(k,n)=1 Γ

( k /n )=

,

(2 π )

ϕ(n)/2

/

√

p if n = p

r

is a prime p ow er, (2 π )

ϕ(n)/2

otherwise.

Small Gamma Pro ducts with Simple V alues

The tw o previous examples are due res p ectively to

Alb ert Nijenhuis , Small Gamma pro ducts with Simple V alues http ://a rxiv.o rg/abs/0907.1689, July 9, 2009.

and to

Greg Ma rtin , A pro duct of Gamma function values at fractions with the same denominato r http ://a rxiv.o rg/abs/0907.4384, July 24, 2009.

Lang’s Conjecture

Conjecture(S.Lang)

Any algeb raic dep endence relation among the numb ers (2 π )

−1/2Γ

( a ) with a

∈Q

lies in the ideal generated by the standa rd relations.

(Universalodddistribution).

49/75

Consequence of the Rohrlich–Lang Conjecture

As an example, the Rohrlich–Lang Conjecture implies that fo r any q> 1 , the transcendence degree of the field generated by numb ers π ,

Γ

( a/q )1

≤

a

≤

q, ( a, q ) = 1

is 1+ ϕ ( q ) / 2 .

50/75

V ariant of the Rohrlich–Lang Conjecture

Conjecture of S. Gun, R. Murt y, P . Rath (2009) : fo r any q> 1 , the numb ers

log

Γ

( a/q )1

≤

a

≤

q, ( a, q )=1

are linea rly indep endent over the field

Q

of algeb raic numb ers.

A consequence is that fo r any q> 1 , there is at mos t one primitive o dd cha racter χ mo dulo q fo r which

L

%

(1 , χ ) = 0 .

51

Ram and Kuma r Murt y (2009)

Ram Murt y Kuma r Murt y

T ranscendental values of class group L –functions.

52

P eter Bundschuh (1979) .

For p/q

∈Q

with 0 <

|

p/q

|

< 1 ,

∞!

n=2

ζ ( n )( p/q )

n

is transcendental. For p/q

∈Q\Z

,

Γ %

Γ *

p q

+

+ γ

is transcendental

53/75

P eter Bundschuh (1979)

(P.Bundschuh):

As a consequence of

Nesterenko

’s Theo rem, the numb er

∞!

n=0

1 n

2

+1 = 1 2 + π 2

·

e

π

+ e

−π

e

π−

e

−π

=2 . 076 674 047 4 .. .

is trans cendental , while

∞!

n=0

1 n

2−

1 = 3 4

(telescoping series). Hence the numb er

∞!

n=2

1 n

s−

1

is trans cendental over

Q

fo r s =4 . The transcendence of this numb er fo r even integers s

≥

4 w ould follo w as a consequence of Schanuel ’s Conjecture.

54/75

)

n

≥

1 A ( n ) /B ( n )

Arithmetic nature of

!

n≥1

A ( n ) B ( n )

where A/B

∈Q

( X ) .

In case B has dis tinct zero es, by decomp osing A/B in simple fractions one gets linea r combinations of loga rithms of algeb raic numb ers ( Bak er ’s metho d). The example A ( X ) /B ( X )=1 /X

3

sho ws that the general case is ha rd. W ork by S.D. Adhik ari, N. Sa radha, T.N. Sho rey and R. Tijdeman (2001), Sanoli Gun, Ram Murt y and Purusottam Rath (2009).

Adolf Hurwitz (1859 - 1919)

Hurwitz zeta function : fo r z

∈C

z

%

=0 and

+

e ( s ) > 1 ,

ζ ( s, z )=

∞!

n=0

1 ( n + z )

s

ζ ( s, 1) = ζ ( s )

( Riemann zeta function )

Conjecture of Cho wl a and Milno r

Sa rvadaman Cho wla (1907 - 1995) John Willa rd Milno r (1931 - )

For k and q integers > 1 , the ϕ ( q ) numb ers

ζ ( k ,a/q ) , 1

≤

a

≤

q, ( a, q )=1

are linea rly indep endent over

Q

.

57/75

Sanoli Gun, Ram Murt y and Purus ottam Rath

The Cho wla-Milno r Conjecture fo r q =4 implies the irrationalit y of the numb ers ζ (2 n + 1) / π

2n+1

fo r n

≥

1 .

StrongChowla-MilnorConjecture

(2009) : For k and q integers > 1 , the 1+ ϕ ( q ) numb ers

1

and

ζ ( k ,a/q ) , 1

≤

a

≤

q, ( a, q ) = 1

are linea rly indep endent over

Q

. For k> 1 o dd, the numb er ζ ( k ) is irrational if and only if the strong Cho wla-Milno r Conjecture holds fo r this value of k and fo r at least one of the tw o values q =3 and q =4 . Hence the strong Cho wla-Milno r Conjecture holds fo r k =3 ( Ap ´ery ) and also fo r infinitely many k ( Rivoal ).

58/75

Linea r indep endence of p olyloga rithms

For k

≥

1 and

|

z

|

< 1 ,

Li

k

( z )=

∞!

n=1

z

n

n

k ·

Thus Li

1

( z ) = log (1

−

z ) and Li

k

(1) = ζ ( k ) fo r k

≥

2 . P olylog Conjecture of S. Gun, R. Murt y, P . Rath : Let k> 1 b e an integer and α

1

,. .. , α

n

algeb raic numb ers such that Li

k

( α

1

) ,. .. , Li

k

( α

n

) are linea rly indep endent over

Q

. Then these numb ers Li

k

( α

1

) ,. .. , Li

k

( α

n

) are linea rly indep endent over the field

Q

of algeb raic numb ers.

S. Gun, R. Murt y, P . Rath : if the p olylog Conjecture is true, then the

Chowla-Milnor

Conjecture is true fo r all k and all q .

59

The digamma function

For z

∈C\{

0 ,

−

1 ,

−

2 ,. ..

}

,

ψ ( x )= d dz log

Γ

( z )=

Γ %

( z )

Γ

( z )

·

ψ ( z )=

−

γ

−

1 z

− ∞!

n=1 *

1 n + z

−

1 n

+

ψ (1 + z )=

−

γ +

∞!

n=2

(

−

1)

n

ζ ( n ) z

n−1

.

60

Sp ecial values of the digamma function

ψ (1) =

−

γ , ψ

*

1 2

+

=

−

2 log (2)

−

γ ,

ψ

*

2 k

−

1 2

+

=

−

2 log (2)

−

γ +

2k−1!

n=1

1 n +1 / 2 ,

ψ

*

1 4

+

=

−

π 2

−

3 log (2)

−

γ , ψ

*

3 4

+

= π 2

−

3 log (2)

−

γ .

Hence

ψ (1) + ψ (1 / 4)

−

3 ψ (1 / 2) + ψ (3 / 4) = 0 .

61/75

Ram Murt y and N. Sa radha

Conjecture (2007) : Let K b e a numb er fi eld over which the q -th cyclotomic p olynomial is irreducible. Then the ϕ ( q ) numb ers ψ ( a/q ) with 1

≤

a

≤

q and ( a, q )=1 are linea rly indep endent over K .

62/75

Ram Murt y and N. Sa radha

Bak er p erio ds : elements of the

Q

–vecto r space spanned by the loga rithms of algeb raic numb ers. A Bak er p erio d is a p erio d in the sense of Kontsevich and Zagier , and is either zero or else transcendental, by Bak er ’s Theo rem. Murt y and Sa radha : one at least of the tw o follo wing statements is true :

•

Euler ’s Constant γ is not a Bak er p erio d

•

the ϕ ( q ) numb ers ψ ( a/q ) with 1

≤

a

≤

q and ( a, q ) = 1 are linea rly indep endent over K , whenever K b e a numb er field over which the q -th cyclotomic p olynomial is irreducible.

Euler cons tant

Euler – Mascheroni constant

γ = lim

n→∞ *

1+ 1 2 + 1 3 +

···

+ 1 n

−

log n

+

=0 . 577 215 664 9 .. .

Neil J. A. Sloane ’s encyclopaedia http ://www.resea rch.att.com/

∼

njas/sequences/A001620

Jonathan Sondo w

http://home.earthlink.net/∼jsondow/

γ =

$∞

0 ∞!

k=2

1

k

2 't+kk (

dt

γ = lim

s→1+ ∞!

n=1 *

1 n

s −

1 s

n +

γ =

$∞

1

1 2 t ( t + 1) F

*

1 , 2 , 2 3 ,t +2

+

dt.

65/75

Euler’s constant γ

A.I. Aptek arev (2007) : App ro xima tion to Euler ’s constant.

T an guy Rivoal (2009) : App ro xi mation to the function γ + log x . Consequence : app ro ximation to γ and to ζ (2)

−

γ

2

.

OpenProblems

.

•

Is the

Euler

constant γ irrational ?

•

Is γ transcendental ?

•

Kontsevich – Zagier : γ is not a p erio d

66/75

Ca rlitz zeta values

Leona rd Ca rlitz (1907 - 1999)

A =

Fq

[ t ] , A

+⊂

A monic p olynomials, P = prime p olynomials in A

+

, K =

Fq

( t ) , K

∞

=

Fq

((1 /t )) ,

Ca rlitz zeta val ues : fo r s

∈Z

,

ζ

A

( s )=

!

a∈A+

1 a

s

=

"^p_∈

P

(1

−

p

−s

)

−1∈

K

∞

.

67

Thakur Gamma function

Dinesh Tha kur

Γ

( z )= 1 z

"

a∈A+ %

1+ z a

&

68

Thakur Gamma values

Indep endence of Gamma values in p ositive cha racteristic : Linea r relations ( W.D. Bro wna w ell and M. P apanik ola s , 2002) and algeb raic relations (with G. Anderson , 2004).

Dale Bro wna w ell Matt P apanik olas

69/75

Ca rlitz zeta values at even A –integers

Define

-

π =( t

−

t

q

)

1/(q−1) ∞"

n=1 *

1

−

t

qn−

t t

qn+1−

t

+

For m a multiple of q

−

1 ,

-

π

−m

ζ

A

( m )

∈

A

Ca rlitz – Bernoulli numb ers.

70/75

Greg Anders on, Dinesh Thakur, Jing Y u

For m a p ositive integer, ζ

A

( m ) is transcendental over K . For m a p ositive integer not a multiple of q

−

1 , ζ

A

( m ) /

-

π

m

is transcendental over K .

Dinesh Tha kur Jing Y u

Bourbaki Semina r

F ederico PELLARIN Asp ects de l’ind ´ep endance alg ´eb rique en ca ract ´eristique non nulle Asp ects of algeb raic indep endence in non–zero cha racteristic S ´eminaire Bourbaki - V olume 2006/2007 - Exp os ´es 967-981 Ast ´erisque

317

(2008), 205–242

Chieh-Y u Chang, Matthew A. P apanik olas , Jing Y u

Title : Geometric Gamma values and zeta values in p ositive cha racteristic arXiv :0905.2876 Abstract : In analogy with values of the classical Euler Gamma-function at rational numb ers and the Riemann zeta-function at p ositive integers, w e consider Thakur’s geometric Gamma-function evaluated at rati ona l arguments and Ca rlitz zeta-values at p ositive integers. W e prove that, when considered together, all of the algeb raic relations among these sp ecial values arise from the sta nda rd functional equations of the Gamma-function and from the Euler-Ca rlitz relations and F rob enius p -th p ow er relations of the zeta-function.

73/75

Chieh-Y u Chang, Matthew A. P apanik olas , Dines h S. Thak ur, Jing Y u

Title : Algeb raic indep endence of arithmetic gamma values and Ca rlitz zeta values arXiv :0909.0096 Abstract : W e consider the values at prop er fractions of the arithmetic gamma function and the values at p ositive integers of the zeta function fo r

Fq

[ θ ] and provide complete algeb raic indep endence results fo r them.

74/75

Chieh-Y u Chang

Title : P erio ds of third kind fo r rank 2 Drinfeld mo dules and algeb raic indep endence of loga rithms arXiv :0909.0101

Abstract :

Inanalogywiththeperiodsofabelianintegralsofdifferentialsofthirdkindforanellipticcurvedefinedoveranumberfield,weintroduceanotionofperiodsofthirdkindforarank2DrinfeldFq[t]–modulerhodefinedoveranalgebraicfunctionfieldandderiveexplicitformulaeforthem.Whenrhohascomplexmultiplicationbyaseparableextension,weprovethealgebraicindependenceofrho-logarithmsofalgebraicpointsthatarelinearlyindependentovertheCMfieldofrho.Togetherwiththemainresultin[CP08],wecompletelydetermineallthealgebraicrelationsamongtheperiodsoffirst,secondandthirdkindsforrank2DrinfeldFq[t]–modulesinoddcharacteristic.

75