NumberTheoryConference2003,BangaloreDecember13–15,2003 MichelWaldschmidt Newvariationsonthesixexponentialstheorem

Number Theory Conference 2003, Bangalore

December 13–15, 2003

New variations on the six exponentials theorem

Michel Waldschmidt

S. Ramanujan.– Highly composite numbers, Proc. London Math.

Soc. 2 14 (1915), 347–409.

For n ≥ 1 let d(n) denote the number of divisors of n. A superior highly composite number is a positive integer n for which there exists > 0 such that the function d(m)/m reaches its maximum at n. The first superior highly composite numbers are

2, 6, 12, 60, 120, 360, 2 520, 5 040, 55 440, 720 720, . . . and their successive quotients are prime numbers

3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3. . .

L. Alaoglu, P. Erd˝os.– On highly composite and similar numbers, Trans. Amer.

Math. Soc., 56 (1944), 448–469.

Colossally abundant numbers : replace

d(n) = X

d|n

1

by

σ(n) = X

d|n

d

Quotients of consecutive colossally abundant numbers are prime.

¿ If x is a real number such that p^x₁ and p^x₂ are rational integers for two

L. Alaoglu, P. Erd˝os.– On highly composite and similar numbers, Trans. Amer.

Math. Soc., 56 (1944), 448–469.

S. Lang.– Nombres transcendants, S´em. Bourbaki 18`eme ann´ee (1965/66), N^◦ 305.

S. Lang.– Algebraic values of meromorphic functions, II, Topology, 5 (1966), 363–370.

S. Lang.– Introduction to Transcendental Numbers; Addison-Wesley 1966.

L. Alaoglu, P. Erd˝os.– On highly composite and similar numbers, Trans. Amer.

Math. Soc., 56 (1944), 448–469.

S. Lang.– Nombres transcendants, S´em. Bourbaki 18`eme ann´ee (1965/66), N^◦ 305.

S. Lang.– Algebraic values of meromorphic functions, II, Topology, 5 (1966), 363–370.

S. Lang.– Introduction to Transcendental Numbers; Addison-Wesley 1966.

K. Ramachandra.– Contributions to the theory of transcendental numbers (I);

Acta Arith., 14 (1968), 65–72; (II), id., 73–88.

K. Ramachandra.– Lectures on transcendental numbers; The Ramanujan

Six Exponentials Theorem (special case). If x is a real number

such that p

, p

and p

are rational integers for three distinct

primes p

, p

and p

, then x ∈ Z.

Six Exponentials Theorem (special case). If x is a real number such that p

, p

and p

are rational integers for three distinct primes p

, p

and p

, then x ∈ Z.

Six Exponentials Theorem (again a special case). If x is a

complex number such that α

, α

and α

are algebraic for three

multiplicatively independent numbers α

, α

and α

, then x ∈ Q.

Six Exponentials Theorem. If x

, x

are two complex numbers which are Q-linearly independent, if y

, y

, y

are three complex numbers which are Q-linearly independent, then one at least of the six numbers

e

(i = 1, 2, j = 1, 2, 3)

is transcendental.

Six Exponentials Theorem. If x

, x

are two complex numbers which are Q-linearly independent, if y

, y

, y

are three complex numbers which are Q-linearly independent, then one at least of the six numbers

e

(i = 1, 2, j = 1, 2, 3) is transcendental.

Set

x_iy_j = λ_ij (i = 1,2; j = 1,2,3).

A 2 × 3 matrix has rank one iff it is of the form x y x y x y

Denote by Q the set of algebraic numbers and define

L = {log α ; α ∈ Q

}. = {λ ∈ C ; e

∈ Q

}.

Six Exponentials Theorem (logarithmic form). For i = 1, 2 and j = 1, 2, 3, let λ

∈ L. Assume λ

, λ

, λ

are linearly independent over Q and also λ

, λ

are linearly independent over Q. Then the matrix

λ

λ

λ

Four Exponentials Conjecture History.

A. Selberg (50’s).

Th. Schneider(1957) - first problem.

S Lang (60’s).

K. Ramachandra (1968).

Leopoldt’s Conjecture on the p-adic rank of the units of an

algebraic number field (non vanishing of the p-adic regulator).

Four Exponentials Conjecture (exponential form). Let x

, x

be two Q-linearly independent complex numbers and y

, y

also two Q-linearly independent complex numbers. Then one at least of the four numbers

e

, e

, e

, e

is transcendental.

Four Exponentials Conjecture (logarithmic form). For i = 1, 2 and j = 1, 2, let α

be a non zero algebraic number and λ

a complex number satisfying e

= α

. Assume λ

, λ

are linearly independent over Q and also λ

, λ

are linearly independent over Q. Then

λ

λ

6= λ

λ

.

Four Exponentials Conjecture (logarithmic form). For i = 1, 2 and j = 1, 2, let α

be a non zero algebraic number and λ

a complex number satisfying e

= α

. Assume λ

, λ

are linearly independent over Q and also λ

, λ

are linearly independent over Q. Then

λ

λ

6= λ

λ

.

Notice:

Algebraic independence of logarithms of algebraic numbers

Conjecture Let α

, . . . , α

be non zero algebraic numbers. For

1 ≤ j ≤ n let λ

∈ C satisfy e

= α

. Assume λ

, . . . , λ

are

linearly independent over Q. Then λ

, . . . , λ

are algebraically

independent.

Algebraic independence of logarithms of algebraic numbers Conjecture Let α

, . . . , α

be non zero algebraic numbers. For 1 ≤ j ≤ n let λ

∈ C satisfy e

= α

. Assume λ

, . . . , λ

are linearly independent over Q. Then λ

, . . . , λ

are algebraically independent.

Write λ

= log α

.

If log α

, . . . , log α

are Q-linearly independent then they are

algebraically independent.

Algebraic independence of logarithms of algebraic numbers Conjecture Let α

, . . . , α

be non zero algebraic numbers. For 1 ≤ j ≤ n let λ

∈ C satisfy e

= α

. Assume λ

, . . . , λ

are linearly independent over Q. Then λ

, . . . , λ

are algebraically independent.

Open problem:

transc.deg

Q(L) ≥ 2?

Quadratic relations between logarithms of algebraic numbers

Homogeneous quadratic relations (Four Exponentials Conjecture):

λ

λ

= λ

λ

? Transcendence of α

:

(log α)(log β ) = (log γ )(log δ)?

Quadratic relations between logarithms of algebraic numbers

Non homogeneous quadratic relations:

(log α)(log β ) = log γ .

Open problem: Transcendence of 2

:

(log 2)

= log γ ?

Quadratic relations between logarithms of algebraic numbers

Non homogeneous relations

Three Exponentials Conjecture (logarithmic form). Let

λ

, λ

, λ

be three elements in L and γ a non zero algebraic

number. Assume λ

λ

= γλ

. Then λ

λ

= γλ

= 0.

Quadratic relations between logarithms of algebraic numbers

Non homogeneous relations

Three Exponentials Conjecture (logarithmic form). Let λ

, λ

, λ

be three elements in L and γ a non zero algebraic number. Assume λ

λ

= γλ

. Then λ

λ

= γλ

= 0.

Special case: λ

= λ

= log α, γ = 1: transcendence of α

?

Example: transcendence of e

?

Quadratic relations between logarithms of algebraic numbers

Non homogeneous relations

Three Exponentials Conjecture (logarithmic form). Let λ

, λ

, λ

be three elements in L and γ a non zero algebraic number. Assume λ

λ

= γλ

. Then λ

λ

= γλ

= 0.

Three Exponentials Conjecture (exponential form). Let

x

, x

, y be non zero complex numbers and γ a non zero

algebraic number. Then one at least of the three numbers

Five Exponentials Theorem (exponential form). If x

, x

are Q-linearly independent, y

, y

are Q-linearly independent and γ is a non zero algebraic number, then one at least of the five numbers

e

, e

, e

, e

, e

is transcendental.

Five Exponentials Theorem (exponential form). If x

, x

are Q-linearly independent, y

, y

are Q-linearly independent and γ is a non zero algebraic number, then one at least of the five numbers

e

, e

, e

, e

, e

is transcendental.

Five Exponentials Theorem (logarithmic form). For

i = 1, 2 and j = 1, 2, let λ

∈ L. Assume λ

, λ

are linearly

independent over Q. Further let γ ∈ Q

and λ ∈ L. Then the

matrix

Sharp Six Exponentials Theorem (logarithmic form). For i = 1, 2 and j = 1, 2, 3, let λ

∈ L and β

∈ Q. Assume λ

, λ

, λ

are linearly independent over Q and also λ

, λ

are linearly independent over Q. Then the matrix

λ

+ β

λ

+ β

λ

+ β

λ

+ β

λ

+ β

λ

+ β

has rank 2.

Sharp Six Exponentials Theorem (exponential form). If x

, x

are two complex numbers which are Q-linearly independent, if y

, y

, y

are three complex numbers which are Q-linearly independent and if β

are six algebraic numbers such that

e

∈ Q for i = 1, 2, j = 1, 2, 3,

then x

y

= β

for i = 1, 2 and j = 1, 2, 3.

Sharp Six Exponentials Theorem (exponential form). If x

, x

are two complex numbers which are Q-linearly independent, if y

, y

, y

are three complex numbers which are Q-linearly independent and if β

are six algebraic numbers such that

e

∈ Q for i = 1, 2, j = 1, 2, 3, then x

y

= β

for i = 1, 2 and j = 1, 2, 3.

The sharp six exponentials Theorem implies the five exponentials

Theorem: set y

= γ/x

and use Baker’s Theorem for checking

A consequence of the sharp six exponentials Theorem:

One at least of the two numbers

e

= α

, e

= α

is transcendental.

rank

1 λ λ

λ λ

λ

= 1.

First proof in 1970 (also by W.D. Brownawell) as a consequence of a result of

Sharp Four Exponentials Conjecture (exponential form).

If x

, x

are two complex numbers which are Q-linearly independent, if y

, y

, are two complex numbers which are Q- linearly independent and if β

, β

, β

, β

are four algebraic numbers such that the four numbers

e

, e

, e

, e

are algebraic, then x

y

= β

for i = 1, 2 and j = 1, 2.

Sharp Four Exponentials Conjecture (logarithmic form). For i = 1, 2 and j = 1, 2, let λ

∈ L and β

∈ Q. Assume λ

, λ

are linearly independent over Q and also λ

, λ

are linearly independent over Q. Then

det

λ

+ β

λ

+ β

λ

+ β

λ

+ β

6= 0.

Sharp Three Exponentials Conjecture (exponential form).

If x

, x

, y are non zero complex numbers and α, β

, β

, γ are algebraic numbers such that the three numbers

e

, e

, e

,

are algebraic, then either x

y = β

or γx

= αx

.

Sharp Three Exponentials Conjecture (exponential form).

If x

, x

, y are non zero complex numbers and α, β

, β

, γ are algebraic numbers such that the three numbers

e

, e

, e

,

are algebraic, then either x

y = β

or γx

= αx

.

Sharp Three Exponentials Conjecture (logarithmic form).

Let λ

, λ

, λ

be three elements of L with λ

λ

6= 0 and

β

, β

, β

, γ four algebraic numbers. Then

Sharp Five Exponentials Conjecture. If x

, x

are Q- linearly independent, if y

, y

are Q-linearly independent and if α, β

, β

, β

, β

, γ are six algebraic numbers with γ 6= 0 such that

e

, e

, e

, e

, e

are algebraic, then x

y

= β

for i = 1, 2, j = 1, 2 and also

γx

= αx

.

Sharp Five Exponentials Conjecture. If x

, x

are Q- linearly independent, if y

, y

are Q-linearly independent and if α, β

, β

, β

, β

, γ are six algebraic numbers with γ 6= 0 such that

e

, e

, e

, e

, e

are algebraic, then x

y

= β

for i = 1, 2, j = 1, 2 and also γx

= αx

.

Difficult case: when y

, y

, γ/x

are Q-linearly dependent.

Sharp Five Exponentials Conjecture. If x

, x

are Q- linearly independent, if y

, y

are Q-linearly independent and if α, β

, β

, β

, β

, γ are six algebraic numbers with γ 6= 0 such that

e

, e

, e

, e

, e

are algebraic, then x

y

= β

for i = 1, 2, j = 1, 2 and also γx

= αx

.

Consequence: Transcendence of the number e

.

Denote by L e the Q-vector space spanned by 1 and L

:

L e = (

β

+

`

X

h=1

β

log α

; ` ≥ 0, α’s in Q

, β ’s in Q )

Strong Six Exponentials Theorem (D. Roy). If x

, x

are Q-linearly independent and if y

, y

, y

are Q-linearly

independent, then one at least of the six numbers

Strong Four Exponentials Conjecture. If x

, x

are Q- linearly independent and if y

, y

, are Q-linearly independent, then one at least of the four numbers

x

y

, x

y

, x

y

, x

y

does not belong to L. e

Strong Three Exponentials Conjecture. If x

, x

, y are non zero complex numbers with x

/x

6∈ Q and x

/x

6∈ Q, then one at least of the three numbers

x

y, x

y, x

/x

is not in L. e

Strong Five Exponentials Conjecture. Let x

, x

be Q-linearly independent and y

, y

be Q-linearly independent.

Then one at least of the five numbers

x

y

, x

y

, x

y

, x

y

, x

/x

does not belong to L. e

12 statements

Three exponentials

sharp Four exponentials

strong Five exponentials

Six exponentials

12 statements

Three exponentials

sharp Four exponentials Conjecture

strong Five exponentials Theorem

Six exponentials

12 statements Three exponentials

sharp Four exponentials Conjecture

strong Five exponentials Theorem

Six exponentials

Three exponentials: three conjectures

Four exponentials: three conjectures

Alg. indep. C

⇓

Strong 3 exp C⇐ Strong 4 exp C ⇒ Strong 5 exp C ⇒ Strong 6 exp T

⇓ ⇓ ⇓ ⇓

Sharp 3 exp C ⇐ Sharp 4 exp C ⇒ Sharp 5 exp C ⇒ Sharp 6 exp T

⇓ ⇓ ⇓ ⇓

3 exp C 4 exp C ⇒ 5 exp T ⇒ 6 exp T

Remark:

The sharp 6 exponentials Theorem implies the 5 exponentials Theorem.

Consequences of the 4 exponentials Conjecture

λ

− λ

λ

λ

6= 0, λ

λ

λ

λ

6= 0, λ

λ

− λ

λ

6= 0, λ

λ

− λ

λ

6= 0.

Consequence of the sharp 4 exponentials Conjecture

Let λ

(i = 1, 2, j = 1, 2) be four non zero logarithms of algebraic

numbers.

Consequence of the sharp 4 exponentials Conjecture

Let λ

(i = 1, 2, j = 1, 2) be four non zero logarithms of algebraic numbers.

Assume

λ

− λ

λ

λ

∈ Q.

Then

λ

λ

= λ

λ

.

Proof. Assume

λ

− λ

λ

λ

= β ∈ Q.

Use the sharp four exponentials conjecture with

(λ

− β )λ

= λ

λ

.

Consequence of the strong 4 exponentials Conjecture Let λ

(i = 1, 2, j = 1, 2) be four non zero logarithms of algebraic numbers.

Assume

λ

λ

λ

λ

∈ Q.

Then

λ

λ

λ

λ

∈ Q.

Proof: Assume

λ

λ

λ

λ

= β ∈ Q.

Use the strong four exponentials conjecture with

λ

λ

= βλ

λ

.

Consequence of the strong 4 exponentials Conjecture Let λ

(i = 1, 2, j = 1, 2) be four non zero logarithms of algebraic numbers.

Assume

λ

λ

− λ

λ

∈ Q.

Then

• either λ

/λ

∈ Q and λ

/λ

∈ Q

• or λ

/λ

∈ Q and

Remark:

λ

λ

− bλ

− aλ

bλ

= a b ·

Proof: Assume

λ

λ

− λ

λ

= β ∈ Q.

Use the strong four exponentials conjecture with

λ

(βλ

+ λ

) = λ

λ

.

Question: Let λ

(i = 1, 2, j = 1, 2) be four non zero logarithms of algebraic numbers. Assume

λ

λ

− λ

λ

∈ Q.

Deduce

λ

λ

= λ

λ

.

Question: Let λ

(i = 1, 2, j = 1, 2) be four non zero logarithms of algebraic numbers. Assume

λ

λ

− λ

λ

∈ Q.

Deduce

λ

λ

= λ

λ

.

Answer: This is a consequence of the Conjecture on algebraic

independence of logarithms of algebraic numbers.

Consequences of the strong 6 exponentials Theorem

Let λ

(i = 1, 2, j = 1, 2, 3) be six non zero logarithms of algebraic numbers. Assume

• λ

, λ

are linearly independent over Q and

• λ

, λ

, λ

are linearly independent over Q.

• One at least of the two numbers λ

− λ

λ

λ

, λ

− λ

λ

λ

is transcendental.

• One at least of the two numbers λ

λ

λ

λ

, λ

λ

λ

λ

is transcendental.

• One at least of the two numbers λ

λ

− λ

λ

, λ

λ

− λ

λ

is transcendental.

• Also one at least of the two numbers λ

− λ

, λ

− λ

Replacing λ

by 1.

• One at least of the two numbers

λ

− λ

λ

, λ

− λ

λ

is transcendental.

• The same holds for λ

λ

− λ

, λ

λ

− λ

.

• Finally one at least of the two numbers λ

λ

λ

, λ

λ

λ

is transcendental, and also one at least of the two numbers 1

λ

− λ

λ

, 1

λ

− λ

λ

·

is transcendental.

Missing:

One at least of the two numbers

λ

λ

− λ

λ

, λ

λ

− λ

λ

is transcendental ?

Theorem. Let λ

(i = 1, 2, j = 1, 2, 3, 4, 5) be ten non zero logarithms of algebraic numbers. Assume

• λ

, λ

are linearly independent over Q and

• λ

, . . . , λ

are linearly independent over Q.

Then one at least of the four numbers

λ

λ

− λ

λ

, (j = 2, 3, 4, 5)

is transcendental.

Further related transcendence results

Theorem (W.D. Brownawell, M. W., 1970) For i = 1, 2 and j = 1, 2, let α

be a non zero algebraic number and λ

a complex number satisfying e

= α

. Assume λ

, λ

are linearly independent over Q and also λ

, λ

are linearly independent over Q. Then one at least of the following two statements holds

• λ

λ

6= λ

λ

• the field Q(λ

, λ

, λ

λ

) has transcendence degree ≥ 2.

Consequences

• One at least of the two numbers e

, e

is transcendental.

• For λ ∈ L \ {0}, one at least of the two numbers e

, e

is transcendental.

• One at least of the two following statements is true:

◦ the two numbers e and π are algebraically independent

◦ the number e

is transcendental.

• For λ ∈ L \ {0}, one at least of the two following statements

Generalization

Theorem (D. Roy–M. W., 1995) Let Q ∈ Q[X

, . . . , X

] be a homogeneous quadratic polynomial and λ

, . . . , λ

be elements in L such that

Q(λ

, . . . , λ

) = 0.

Assume the field Q(λ

, . . . , λ

) has transcendence degree 1 over Q. Then the point (λ

, . . . , λ

) belongs to a linear subspace of C

contained in the hypersurface Q = 0.

Next step: investigate the transcendence of numbers

Happy Birthday Professor Ramachandra!