In 1980 F. Wattenberg constructed the Dedekind completion *R d of the Robinson non-archimedean field *R and established basic algebraic properties of *R d. In 1985 H. Gonshor established further fundamental properties of *R d. In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion *R d in transcendental number theory were considered. Given any analytic function of one complex variable , we investigate the arithmetic nature of the values of at transcendental points . Main results are: 1) the both numbers and are irrational; 2) number e e is transcendental. Nontrivial generalization of the Lindemann-Weierstrass theorem is obtained.
In 1873 French mathematician, Charles Hermite, proved that
Conjecture 1. Whether the both numbers
Conjecture 2. Whether the numbers
However, the same question with
Theorem. (Nesterenko, 1996 [
Throughout of 20-th century,a typical question: whether
Theorem. (Siegel C. L.) Suppose that
Then
Let
Conjecture 3. Whether
Conjecture 4. Whether
In this paper we investigate the arithmetic nature of the values of
Definition 1.1. Let
We will call any function given by Equation (1.2)
Definition 1.2. [
Definition 1.3. [
Example. Number
(1) function
(2)
Main results are.
Theorem 1.1. [
From theorem 1.1 immediately follows.
Theorem 1.2. Number ee is transcendental.
Theorem 1.3. [
Theorem 1.4. For any
Theorem 1.5. [
Theorem 1.6. [
Assume that for any
and
Then
Let
For example,
Definition 2.1. Let
A Dedekind hyperreal
1.
4.
Compare the Definition 2.1 with original Wattenberg definition [
Designation 2.1. Let
Designation 2.2. Let
Remark 2.1. The monad of
Supremum of
Let
Example 2.1. 1)
Remark 2.2. Unfortunately the set
Definition 2.2. We define:
1. The additive identity (zero cut)
2. The multiplicative identity
Given two Dedekind hyperreal numbers
3. Addition
It is easy to see that
It is easy to see that
Another fundamental property of cut addition is associativity:
This follows from the corresponding property of
4. The opposite
5. We say that the cut
The absolute value of
6. If
In general,
7. The cut order enjoys on
(i) transitivity:
(ii) trichotomy: eizer
(iii) translation:
Definition 2.3. [
(i) For every
(iii)
Definition 2.4. [
If
Remark 2.3. [
Remark 2.4. [
If
whereas the definition of
but this is a contradiction.
Remark 2.5. Note that in the usual treatment of Dedekind cuts for the ordinary real numbers both of the latter sets are regarded as equivalent so that no serious problem arises [
Remark 2.6. H. Gonshor [
Definition 2.5. (Wattenberg embeding) We embed
and
or in the equivalent way,i.e. if
Thus if
Such embeding
Lemma 2.1. [
(i) Addition
Remark 2.7. Notice, here again something is lost going from
Lemma 2.2. [
(i)
(iv)
(v) Suppose that
(vi) Suppose that
Remark 2.8. Note that in general case
given in [
Example 2.2. [
Let
Lemma 2.3. [
(i) If
Proof. (v) By (iv):
(1) Suppose now
(2)
(4) Note that: (since and imply but this is a contradiction)
(5) Thus
(6) By similar reasoning one obtains:
(7) Note that:
Lemma 2.4. (i)
Proof. (i) For
(1) Suppose
(2)
(4) Note that: (since and imply but this is a contradiction)
(5) Thus
(6) By similar reasoning one obtains:
(7) Note that:
(ii) Immediately follows from (i) by Lemma 2.3.
Definition 2.6. Suppose
Definition 2.7. Suppose
Case (2)
Case (3)
Lemma 2.5. [
(ii) Multiplication
(iii)
Lemma 2.6. Suppose
Proof. We choose now:
(1)
(2) Note that
Then from (2) by Lemma 2.4. (ii) one obtains
(3)
(5) Then from (4) by Lemma 2.5. (v) one obtains
Then from (6) by Lemma 2.4. (ii) one obtains
Definition 2.8. Suppose
Lemma 2.7. [
Lemma 2.8. [
Theorem 2.1. Suppose that
Proof. Let
any
One of standard ways of defining the completion of
Definition 2.9. [
Example 2.5.
Lemma 2.9. [
(i)
(ii)
Remark 2.9. By Lemma 2.7
definition is that the real interest lies in the non-negative numbers. A technicality occurs if
Remark 2.10. By Lemma 2.7(ii),
Lemma 2.10. [
(i)
(ii)
(iii) If
Lemma 2.11. [
(i)
(v)
Theorem 2.2. [
Theorem 2.3. [
Theorem 2.4. [
Theorem 2.5. [
Theorem 2.6. [
Theorem 2.7. [
(i)
(iv) Let
Among elements of
Definition 2.10. [
(i)
(ii)
(iii)
(iv)
Theorem 2.8. [
(ii)
(iii) The map
(vii)
Theorem 2.9. [
(i)
(ii)
(iii) Suppose
(iv) Suppose
(v) If
(vi) If
Proof (iii) Let
It is clear that
Conversely, suppose
By definition of
Choose
Hence
Examples.
(i)
(ii) Suppose
(iii) Suppose
(iv) Suppose
(v) Suppose
Remark 2.11. Note that in general case, i.e. if
Remark 2.12. Suppose
Lemma 2.12. [
Definition 2.11. [
Lemma 2.13. [
The set of all positive Wattenberg hyperintegers is called the Wattenberg hypernaturals and is denoted by
Definition 2.12. Suppose that (i)
If
Definition 2.13. Suppose that (i)
(1)
If
Theorem 2.10. (i) Let
(ii)
(iii)
(iv) Suppose
(v) Suppose
(vi) Suppose
(vii) Suppose
Proof. (i) Immediately follows from definitions (2.12)-(2.13).
(iv) Let
It is clear that a works to show that
Conversely, suppose
By definition of
Choose
Hence
Definition 2.14. Suppose
Obviously there are two possibilities:
1. A set
Property I:
Since
2. A set
Property II:
Definition 2.15. Suppose
Note that obviously:
This subsection contains key definitions and properties of summ of countable sequence of Wattenberg hyperreals.
Definition 2. 16. [
(i)
(iii)
Then external sum (#-sum)
Theorem 2.11. (i) Let
(ii) Let
(iii) Let
(iv) Let
(v) Let
(1)
Then
Proof. (i) Let
Thus
Therefore from (1) by Robinson transfer one obtains (2)
Using now Wattenberg embedding from (2) we obtain (3)
From (3) one obtains (4)
Note that
From (4) and (5) one obtains (6)
Thus (i) immediately from (6) and from definition of the idempotent
Proof.(ii) Immediately from (i) by Lemma 2.3 (v).
Proof.(iii) Let
Therefore from (1) by Robinson transfer one obtains (2)
Using now Wattenberg embedding from (2) we obtain (3)
From (3) one obtains (4)
From (4) by Definition 2.16 (i) one obtains
Note that
From (5)-(6) follows (7)
Thus Equation (2.23) immediately from (7) and from definition of the idempotent
Proof.(iv) Immediately from (iii) by Lemma 2.3 (v).
Proof.(v) From Definition 2.16.(iii) and Equation (2.23)-Equation (2.24) by Theorem 2.7.(iii) one obtains
Theorem 2.12. Let
Theorem 2.13. (i) Let
(ii) Let
(iii) Let
(1)
(2) infinite series
(3) infinite series
Then the equality is satisfied:
Proof. (i) From Definition 2.16. (i) by Theorem 2.1, Theorem 2.11. (i) and Lemma (2.4) (ii) one obtains
(ii) Straightforward from Definition 2.16. (i) and Theorem 2.1, Theorem 2.11. (ii) and Lemma (2.4) (ii) one obtains
(iii) By Theorem 2.11. (iii) and Lemma (2.4). (ii) one obtains
But other side from (i) and (ii) follows
Definition 2.17. Let
(i) there exists
(ii) there exists
(iii) there exists infinite sequence
(a)
(b) there exists infinite sequence
Then: (i) external upper sum (#-upper sum) of the corresponding countable sequence
(ii) external lower sum (#-lower sum) of the corresponding countable sequence
Theorem 2.14. (1) Let
(i) there exists
(ii) there exists
(iii) there exists infinite sequence
(a)
(b) there exists infinite sequence
Then
and
Proof. (i), (ii), (iii) straightforward from definitions.
Theorem 2.15. (1) Let
(i) there exists
(ii) there exists
(iii) there exists infinite sequence
(a)
(b) there exists infinite sequence
Then for any
and
Proof. Copy the proof of the Theorem 2.13.
Theorem 2.16. (1) Let
(i) there exists
(ii) there exists
(iii) there exists infinite sequence
(a)
(b) there exists infinite sequence
Then for any
and
Proof. (1) From Equation (2.31) we obtain
From Equation (2.37) by Theorem 2.1 we obtain directly
(2) From Equation (2.32) we obtain
From Equation (2.39) by Theorem 2.1 we obtain directly
Remark 2.13. Note that we have proved Equation (2.35) and Equation (2.36) without any reference to the Lemma 2.4.
Definition 2.18. (i) Let
Then external countable upper sum (#-sum) of the countable sequence
In particular if
(ii) Let
Then external countable lower sum (#-sum) of the countable sequence
In particular if
Theorem 2.17. (i) Let
(ii) Let
Then for any
Proof. Immediately from Definition 2.18 by Theorem 2.1.
Definition 2.19. Let
correspondingly.
Note that any properties of this sum immediately follow from the properties of the real external sum.
Definition 2.20. (i) We define now Wattenberg complex plane
Theorem 2.18. Let
Definition 2.21. [
Note that
(ii) S has no maximum, then
Let f be a continuous strictly increasing function in each variable from a subset of
Definition 2.22. [
Theorem 2.19. [
Theorem 2.20. [
Theorem 2.21. [
Remark 2.14. For any function
Theorem 2.22. [
For any
(2) For any
(3) For any
(4) For any
Note that we must always beware of the restriction in the domain when it comes to multiplication.
Theorem 2.23. [
In this section we will prove the #-transcendence of the numbers
In this section we remind the basic definitions of the Shidlovsky quantities [
where
and consequently
Lemma 3.1. [
Proof. ([
where p is a prime. By using equality
Thus
Lemma 3.2. [
Proof. ([
By using equality
and by subsitution Equation (3.10) into RHS of the Equation (3.9) one obtains
Lemma 3.3. [
where sequences
Proof. ([
and
Substitution inequalities (3.13)-(3.14) into RHS of the Equation (3.3) by simple calculation gives
Statement (i) follows from (3.15). Statement (ii) immediately follows from a statement (ii).
Lemma 3.4. [
Proof. From Equation (3.5) one obtains
From Equation (3.17) by using Lemma 3.3. (ii) one obtains (3.17).
Remark 3.1. We remind now the proof of the transcendence of
Theorem 3.1. The number
Proof. ([
where
From Equation (3.19) one obtains
We rewrite the Equation (3.20) for short in the form
We choose now the integers
and
and therefore
By using Lemma 3.4 for any
From (3.25) and Equation (3.21) we obtain
From (3.26) and Equation (3.24) one obtains the contradiction.This contradiction finalized the proof.
In this subsection we will replace using Robinson transfer the Shidlovsky quantities
perties of the standard quantities
1. Using Robinson transfer principle [
From Equation (3.11) using Robinson transfer principle one obtains
Using Robinson transfer principle from inequality (3.15) one obtains
Using Robinson transfer principle, from Equation (3.5) one obtains
Lemma 3.5. Let
Proof. From Equation (3.30) we obtain
From Equation (3.32) and (3.29) we obtain (3.31).
In this subsection we will replace by using Wattenberg imbedding [
1. By using Wattenberg imbedding
2. By using Wattenberg imbedding
3. By using Wattenberg imbedding
Lemma 3.6. Let
Proof. Inequality (3.36) immediately follows from inequality (3.31) by using Wattenberg imbedding
To prove that
Suppose that e is w-transcendental, i.e., there exists an
such that the equality is satisfied:
In this subsection we obtain an reduction of the equality given by Equation (3.39) to equivalent equality given by Equation (3). The main tool of such reduction that external countable sum defined in Subsection 2.8.
Lemma 3.7. Let
Then
Proof. Suppose there exists k such that
Remark 3.2. Note that from Equation (3.39) follows that in generel case there is a sequence
or there is a sequence
or both sequences
Remark 3.3. We assume now for short but without loss of generelity that (3.41) is satisfied. Then from (3.41) by using Definition 2.17 and Theorem 2.14 (see Subsection 2.8) one obtains the equality [
Remark 3.4. Let
Note that from Equation (3.43)-Equation (3.44) follows that
Remark 3.5. Assume that
Lemma 3.8.
Proof. Suppose there exists
From Equation (3.46) by Theorem 2.11 follows that
Theorem 3.2. [
Proof. Let us consider hypernatural number
From Equation (3.43) and Equation (3.47) one obtains
Remark 3.6. Note that from inequality (3.27) by Wattenberg transfer one obtains
Substitution Equation (3.30) into Equation (3.48) gives
Multiplying Equation (3.50) by Wattenberg hyperinteger
By using inequality (3.49) for a given
Now using the inequality (3.49) we are free to choose a prime hyperinteger
Hence from Equation (3.52) and Equation (3.53) we obtain
Therefore from Equations (3.51) and (3.54) by using definition (2.15) of the function
From Equation (3.55) using basic property I of the function
We will choose now infinite prime integer
Hence from Equation (3.34) follows
Note that
Using Equation(3.35) one obtains
In this subsection we wil prove that main equality (3.56) is inconsistent. This prooff based on the Theorem 2.10 (v), see Subsection 2.6.
Lemma 3.9. The equality (3.56) under conditions (3.59)-(3.60) is inconsistent.
Proof. (I) Let us rewrite Equation (3.56) in the short form
where
From (3.59)-(3.60) follows that
Remark 3.7. Note that
(II) Let
Note that from Equation (3.61) and Equation (3.64) follows that
Lemma 3.10. Under conditions (3.59)-(3.60)
and
Proof. First note that under conditions (3.59)-(3.60) one obtains
Suppose that there exists an
From Equation (3.69) by Theorem 2.17 one obtains
Thus
From Equation (3.71) by Theorem 2.11 follows that
Part (III)
Remark 3.8. (i) Note that from Equation (3.62) by Theorem 2.10 (v) follws that
where
(ii) Substitution by Equation (3.72) into Equation (3.61) gives
Remark 3.9. Note that from (3.74) by definitions follows that
Remark 3.10. Note that from (3.73) by construction of the Wattenberg integer
Therefore
Note that under conditions (3.59)-(3.60) and (3.73) obviously one obtains
From Equation(3.74) follows that
Therefore
From (3.78) follows that
Note that by Theorem 2.8 (see Subsection 2.5) and Formula (3.44) one otains
From Equation (3.81)-Equation (3.82) follows that
Thus
and therefore
But this is a contradiction. This contradiction completed the proof of the Lemma 3.9.
In this section we remind the basic definitions of the Shidlovsky quantities, see [
Theorem 4.1. [
and
Let
Let
where in (4.4) we integrate in complex plane
where
where
From Equation (4.3) one obtains
where
Picture 1. Contour
one obtains
where
From Equation (4.3) and Equation (4.5) one obtains
where
where
Let
Let us rewrite now Equation (4.12) in the following form
where
The polinomial
It well known that
From Equation (4.15) and Equation (4.17) one obtains
Therefore
Let
From Equation (4.6) and Equation (4.20) one obtains
where
From (4.22) follows that for any
where
where
Having substituted RHS of the Equation (4.24) into Equation (4.25) one obtains
From Equation (4.26) by using Equation (4.19) one obtains
We choose now a prime
Theorem 5.1. [
and
Then
We will divide the proof into three parts.
Part I. The Robinson transfer
Let
Let
where in (5.5) we integrate in nonstandard complex plane
where
where
1. Using Robinson transfer principle [
where
2. Using Robinson transfer principle from Equation (5.6) and Equation (4.19) one obtains directly
and therefore
3. Using Robinson transfer principle from Equation (5.7) and Equation (4.21) one obtains directly
where
4. From (5.13) follows that for any
where
5. From Equation (5.5)-Equation (5.7) we obtain
where
Part II. The Wattenberg imbedding
1. By using Wattenberg imbedding
where
2. By using Wattenberg imbedding
and therefore
3. By using Wattenberg imbedding
4. By using Wattenberg imbedding
where
Part III. Main equality
Remark 5.1. Note that in this subsection we often write for a short
instead Equation (5.21).
Assumption 5.1. Let
Note that from Assumption 5.1 follows that algebraic numbers over the field
Assumption 5.2. We assume now that there exists a sequence
and rational number
such that
and
Assumption 5.3. We assume now for a short but without loss of generality that the all numbers
In this subsection we obtain an reduction of the equality given by Equation (5.27) in
Lemma 5.1. Let
Then
Proof. Suppose there exists r such that
Remark 5.2. Note that from Equation (5.27) follows that in generel case there is a sequence
or there is a sequence
or both sequences
Remark 5.3. We assume now for short but without loss of generelity that (5.29) is satisfied. Then from (5.29) by using Definition 2.17 and Theorem 2.14 (see Subsection 2.8) one obtains the equality [
Remark 5.4. Let
Note that from Equation (5.31)-Equation (5.32) follows that
Remark 5.5. Assume that
Lemma 5.2.
Proof. Suppose there exists
From Equation (5.34) by Theorem 2.11 follows that
Theorem 5.2. [
Proof. Let us considered hypernatural number
From Equation (5.31) and Equation (5.35) one obtains
where
Remark 5.6. Note that from inequality (5.12) by Gonshor transfer one obtains
Substitution Equation (5.21) into Equation (5.36) gives
Multiplying Equation (5.39) by Wattenberg hyperinteger
By using inequality (5.38) for a given
Therefore from Equations (5.40) and (5.41) by using definition (2.15) of the function
From Equation (5.42) finally we obtain the main equality
We will choose now infinite prime integer
Hence from Equation (5.16) follows
Note that
Using Equation (5.11) one obtains
Part IV. The proof of the inconsistency of the main equality (5.43)
In this subsection we wil prove that main equality (5.43) is inconsistent. This proof is based on the Theorem 2.10 (v), see Subsection 2.6.
Lemma 5.3. The equality (5.43) under conditions (5.46)-(5.47) is inconsistent.
Proof. (I) Let us rewrite Equation (5.43) in the short form
where
From (5.46)-(5.47) follows that
Remark 5.7. Note that
But the other hand from Equation (5.48) follows that
But this is a contradiction. This contradiction completed the proof of the statement (I).
(II) Let
Note that from Equation (5.43) and Equation (5.53) follows that
Lemma 5.4. Under conditions (5.46)-(5.47)
and
Proof. First note that under conditions (5.46)-(5.47) one obtains
Suppose that there exists
From Equation (5.58) by Theorem 2.17 one obtains
Thus
From Equation (5.60) by Theorem 2.11 follows that
(III)
Remark 5.8. (i) Note that from Equation (5.49) by Theorem 2.10 (v) follws that
where
(ii) Substitution by Equation (5.61) into Equation (5.48) gives
Remark 5.9. Note that from (5.63) by definitions follows that
Remark 5.10. Note that from (5.62) by construction of the Wattenberg integer
Therefore
Note that under conditions (5.46)-(5.47) and (5.66) obviously one obtains
From Equation (5.63) follows that
Therefore
From (5.69) follows that
Note that from (5.70) by Theorem 2.8 (see Subsection 2.5) and Formula (5.32) one otains
From Equation (5.70)-Equation (5.71) follows that
Thus
and therefore
But this is a contradiction. This contradiction completed the proof of the Lemma 5.3.
Remark 5.11. Note that by Definition 2.18 and Theorem 2.18 from Assumption 5.1 and Assumption 5.2 follows
Theorem 5.3.The equality (5.75) is inconsistent.
Proof. The proof of the Theorem 5.3 obviously copies in main details the proof of the Theorem 5.3.
Theorem 5.3 completed the proof of the main Theorem 1.6.
JaykovFoukzon, (2015) Non-Archimedean Analysis on the Extended Hyperreal Line *Rd and the Solution of Some Very Old Transcendence Conjectures over the Field Q. Advances in Pure Mathematics,05,587-628. doi: 10.4236/apm.2015.510056