Hello dear Habrians, the text below outlines the proof of the Riemann hypothesis, which is awaiting verification at CMI (Clay mathematical institute). So let's get straight to the proof
The Todd function
In this section I summarize the properties of the Todd function T(s), constructed in [2].
T is what I will call a weakly analytic function meaning that it is a weak limit of a
Family of analytic functions. So, on any compact set K in C, T is analytic. If K is convex,
T is actually a polynomial of some degree k(K). For example a step function is weakly
Analytic and, for any closed interval K on the line, the degree is 0. This shows that a weakly
Analytic function can have compact support, in contrast to an analytic function. Weakly
2.6 If ƒ and g are power series with no constant term, then
T{1+f(s)] [1+ g(s)]} = T{1+f(s)+g(s)}.LP
Remark. Weakly analytic functions have a formal expansion as a power series near the origin. Formula 2.6 is just the linear approximation of this expansion (more precisely this is
On the branched double cover of the complex s-plane given by √5). This implies
2.6 T(3)T(s) or
2.7 √T(1+8)T(1+8/2)
Which gives us the uniform constant ½ needed in 3.3 of section 3.
3. THE PROOF OF RH
In this section I will use the Todd function T(s) to prove RH. The proof will be by contradiction assume there is a zero & inside the critical strip but off the critical line. To prove RH, it is then sufficient to show that the existence of 6 leads to a contradiction..
Given b, take a = b in 2.1 then, on the rectangle Ka), T is a polynomial of degree k{a}.
(3.1)
F(s)T(1+((s+b)}-1
Consider the composite function of s, given by From its construction, and the hypothesis that ((b) = 0, it follows that 3.2 F is analytic at 80 and F(0) = 0. Now take fg F in 2.6 and we deduce the identity
3.3 F(s) 2F(s).
Since C is not of characteristic 2, it follows that F(s) is identically zero. 2.3 ensures that T is not the zero polynomial and so it is invertible in the field of meromorphic functions of s. The identity F(s)=0 then implies the identity ((s)= 0. This is clearly not the case and gives the required contradiction.
This completes the proof of RH.
The proof of RH that has just been given is sometimes referred to as the search for the first Siegel zero. The idea is to assume there is a counterexample to RH, study the first such zero b, and hope to derive a contradiction.
This is exactly what we did. Using the composite function F(s) of 3.1 with a zero at b, off
The critical line, we found another zero b' which halves the distance is to the critical line..
Continuing this process gives an infinite sequence of distinct zeros, converging to a point (on
The critical line). But an analytic function which vanishes on such an infinite sequence must
Be identically zero. Applying this to F(s) (using 2.8 now instead of 2.6) shows that F(s) is
Identically zero and this then leads to a contradiction as argued in the last few lines after 3.3.
Family of analytic functions. So, on any compact set K in C. T is analytic. If K is convex, T is actually a polynomial of some degree k(K). For example a step function is weakly. Analytic and, for any closed interval K on the line, the degree is 0. This shows that a weakly analytic function can have compact support, in contrast to an analytic function. Weakly
Analytic functions are weakly dense in L² and in their weak duals. They are well adapted for Fourier transforms on all I spaces. They are also composable: a weakly analytic function of a weakly analytic function is weekly analytic.
Define Ka] to be the closed rectangle
(2.1)
Re(s-1/2)≤1/4, Im (s)≤a
Then, on Ka, T is a polynomial of degree k{a} = k(Ka).
This terminology is formally equivalent to that of Hirzebruch [3], with his Todd polynomials. But Hirzebruch worked with formal power series and did not require convergence. That was adequate for his applications which were essentially algebraic and arithmetic, as the appearance of the Bernoulli numbers later showed.
However, to relate to von Neumann's analytical theory it is necessary to take weak limits as has just been done. This provides the crucial link between algebra/arithmetic and analysis which is at the heart of the function.
This makes it reasonable to expect that RH might emerge naturally from the fusion of the different techniques in [2].
I return now to other properties of T(s) explained in [2]:
2.2 T is real i.e. T(5) T(s).
2.3 T(1) = 1
2.4 T maps the critical strip into the critical strip and the critical line into the critical line.
(This is not explicitly stated in [2] but it is included in the mimicry principle 7.6, which asserts that T is compatible with any analytic formula, so in particular Im(T(-1/2)= T(Im(s-1/2)).)
The main result of [2], identifying a with 1/ж, was
2.5 on Re(s) ½, Im(s)>0, T is a monotone increasing function of Im(s) whose limit, as Im(s) tends to infinity, is ж.
As was noted above, on a given compact convex set, the Todd polynomials stabilize as the
Degree increases. In [3] Hirzebruch expressed this stability in the form of an equation:
P
Degree increases. In [3] Hirzebruch expressed this stability in the form of an equation:
2.6 if ƒ and g are power series with no constant term, then
T1+f(s) [1+g(s)}=T(1+f(s)+g(8)}.
Remark. Weakly analytic functions have a formal expansion as a power series near the origin. Formula 2.6 is just the linear approximation of this expansion (more precisely this is on the branched double cover of the complex s-plane given by √3). This implies
2.6 T(9)T(s) or
2.7 T(1+8)= (1 + 5/2)
Which gives us the uniform constant ½ needed in 3.3 of section 3.
3. THE PROOF OF RH
In this section I will use the Todd function T(s) to prove RH. The proof will be by contradiction: assume there is a zero 6 inside the critical strip but off the critical line. To prove RH, it is then sufficient to show that the existence of b leads to a contradiction.
Given b, take a = b in 2.1 then, on the rectangle K[a], T is a polynomial of degree k{a}. Consider the composite function of s, given by
(3.1)
F(8)T(1+((+6)}-1 From its construction, and the hypothesis that ((b)=0, it follows that 3.2 F is analytic at s = 0 and F(0) = 0. Now take fg=F in 2.6 and we deduce the identity 3.3 F(s) 2F(s).
Since C is not of characteristic 2, it follows that F(s) is identically zero. 2.3 ensures that T is not the zero polynomial and so it is invertible in the field of meromorphic functions of s. The identity F(s)=0 then implies the identity ((s)= 0. This is clearly not the case and gives the required contradiction.
This completes the proof of RH.
The proof of RH that has just been given is sometimes referred to as the search for the first Siegel zero. The idea is to assume there is a counterexample to RH, study the first such zero b, and hope to derive a contradiction.
This is exactly what we did. Using the composite function F(s) of 3.1 with a zero at b, off
The critical line, we found another zero b' which halves the distances to the critical line.
Continuing this process gives an infinite sequence of distinct zeros, converging to a point (on
5. FINAL COMMENTS
In this final section I will comment on possible future developments in Arithmetic Physics. These comments are on two levels.
At the first level there are firm expectations. At the second level there are speculations.
Starting with the first level, some comments on RH. Using our new machinery, RH and the mystery of a, were solved. But RH was a problem over the rational field Q, and there are many generalizations to other fields or algebras. I firmly anticipate much work in this direction.
There are also logical issues that will emerge. To be explicit, the proof of RH in this paper is by contradiction and this is not accepted as valid in ZF, it does require choice. I fully expect that the most general version of the Riemann Hypothesis will be an undecidable problem in the Gödel sense.
RH should be the bench mark for other famous problems in mathematics, such as the Birch-Swinnerton Dyer conjectures. I expect most cases will be undecidable.
I now pass to the second level. Following the example of o, and the more difficult case of the Gravitational constant G (see 2.6 in [2]), I expect that mathematical physics will face issues where logical undecidability will get entangled with the notion of randomness.
In 4-dimensional s
mooth geometry I expect the famous 11/8 conjecture of Donaldson theory will prove to be undecidable, as will the smooth Poincare conjecture.
2.6 If ƒ and g are power series with no constant term, then
T{1+f(s)] [1+ g(s)]} = T{1+f(s)+g(s)}.
Remark. Weakly analytic functions have a formal expansion as a power series near the origin. Formula 2.6 is just the linear approximation of this expansion (more precisely this is
on the branched double cover of the complex s-plane given by √5). This implies
2.6 T(3)T(s) or
2.7 √T(1+8)T(1+8/2)
which gives us the uniform constant 1/2 needed in 3.3 of section 3.
3. THE PROOF OF RH
In this section I will use the Todd function T(s) to prove RH. The proof will be by contradiction assume there is a zero & inside the critical strip but off the critical line. To prove RH, it is then sufficient to show that the existence of 6 leads to a contradiction..
Given b, take a = b in 2.1 then, on the rectangle Ka), T is a polynomial of degree k{a}.
(3.1)
F(s)T(1+((s+b)}-1
Consider the composite function of s, given by From its construction, and the hypothesis that ((b) = 0, it follows that 3.2 F is analytic at 80 and F(0) = 0. Now take fg F in 2.6 and we deduce the identity
3.3 F(s) 2F(s).
Since C is not of characteristic 2, it follows that F(s) is identically zero. 2.3 ensures that T is not the zero polynomial and so it is invertible in the field of meromorphic functions of s. The identity F(s)=0 then implies the identity ((s)= 0. This is clearly not the case and gives the required contradiction.
This completes the proof of RH.
The proof of RH that has just been given is sometimes referred to as the search for the first Siegel zero. The idea is to assume there is a counterexample to RH, study the first such zero b, and hope to derive a contradiction.
This is exactly what we did. Using the composite function F(s) of 3.1 with a zero at b, off
the critical line, we found another zero b' which halves the distance is to the critical line..
Continuing this process gives an infinite sequence of distinct zeros, converging to a point (on
the critical line). But an analytic function which vanishes on such an infinite sequence must
be identically zero. Applying this to F(s) (using 2.8 now instead of 2.6) shows that F(s) is
identically zero and this then leads to a contradiction as argued in the last few lines after 3.3.
family of analytic functions. So, on any compact set K in C. T is analytic. If K is convex, T is actually a polynomial of some degree k(K). For example a step function is weakly. analytic and, for any closed interval K on the line, the degree is 0. This shows that a weakly analytic function can have compact support, in contrast to an analytic function. Weakly
analytic functions are weakly dense in L² and in their weak duals. They are well adapted for Fourier transforms on all I spaces. They are also composable: a weakly analytic function of a weakly analytic function is weekly analytic.
Define Ka] to be the closed rectangle
(2.1)
Re(s-1/2)≤1/4, Im (s)≤a
Then, on Ka, T is a polynomial of degree k{a} = k(Ka).
This terminology is formally equivalent to that of Hirzebruch [3], with his Todd polynomials. But Hirzebruch worked with formal power series and did not require convergence. That was adequate for his applications which were essentially algebraic and arithmetic, as the appearance of the Bernoulli numbers later showed.
However, to relate to von Neumann's analytical theory it is necessary to take weak limits as has just been done. This provides the crucial link between algebra/arithmetic and analysis which is at the heart of the function.
This makes it reasonable to expect that RH might emerge naturally from the fusion of the different techniques in [2].
I return now to other properties of T(s) explained in [2]:
2.2 T is real i.e. T(5) T(s).
2.3 T(1) = 1
2.4 T maps the critical strip into the critical strip and the critical line into the critical line.
(This is not explicitly stated in [2] but it is included in the mimicry principle 7.6, which asserts that T is compatible with any analytic formula, so in particular Im(T(-1/2)= T(Im(s-1/2)).)
The main result of [2], identifying a with 1/ж, was
2.5 on Re(s) 1/2, Im(s)>0, T is a monotone increasing function of Im(s) whose limit, as Im(s) tends to infinity, is ж.
As was noted above, on a given compact convex set, the Todd polynomials stabilize as the
degree increases. In [3] Hirzebruch expressed this stability in the form of an equation:
p
degree increases. In [3] Hirzebruch expressed this stability in the form of an equation:
2.6 if ƒ and g are power series with no constant term, then
T1+f(s) [1+g(s)}=T(1+f(s)+g(8)}.
Remark. Weakly analytic functions have a formal expansion as a power series near the origin. Formula 2.6 is just the linear approximation of this expansion (more precisely this is on the branched double cover of the complex s-plane given by √3). This implies
2.6 T(9)T(s) or
2.7 T(1+8)= (1 + 5/2)
which gives us the uniform constant 1/2 needed in 3.3 of section 3.
3. THE PROOF OF RH
In this section I will use the Todd function T(s) to prove RH. The proof will be by contradiction: assume there is a zero 6 inside the critical strip but off the critical line. To prove RH, it is then sufficient to show that the existence of b leads to a contradiction.
Given b, take a = b in 2.1 then, on the rectangle K[a], T is a polynomial of degree k{a}. Consider the composite function of s, given by
(3.1)
F(8)T(1+((+6)}-1 From its construction, and the hypothesis that ((b)=0, it follows that 3.2 F is analytic at s = 0 and F(0) = 0. Now take fg=F in 2.6 and we deduce the identity 3.3 F(s) 2F(s).
Since C is not of characteristic 2, it follows that F(s) is identically zero. 2.3 ensures that T is not the zero polynomial and so it is invertible in the field of meromorphic functions of s. The identity F(s)=0 then implies the identity ((s)= 0. This is clearly not the case and gives the required contradiction.
This completes the proof of RH.
The proof of RH that has just been given is sometimes referred to as the search for the first Siegel zero. The idea is to assume there is a counterexample to RH, study the first such zero b, and hope to derive a contradiction.
This is exactly what we did. Using the composite function F(s) of 3.1 with a zero at b, off
the critical line, we found another zero b' which halves the distances to the critical line.
Continuing this process gives an infinite sequence of distinct zeros, converging to a point (on
5. FINAL COMMENTS
In this final section I will comment on possible future developments in Arithmetic Physics. These comments are on two levels.
At the first level there are firm expectations. At the second level there are speculations.
Starting with the first level, some comments on RH. Using our new machinery, RH and the mystery of a, were solved. But RH was a problem over the rational field Q, and there are many generalizations to other fields or algebras. I firmly anticipate much work in this direction.
There are also logical issues that will emerge. To be explicit, the proof of RH in this paper is by contradiction and this is not accepted as valid in ZF, it does require choice. I fully expect that the most general version of the Riemann Hypothesis will be an undecidable problem in the Gödel sense.
RH should be the bench mark for other famous problems in mathematics, such as the Birch-Swinnerton Dyer conjectures. I expect most cases will be undecidable.
I now pass to the second level. Following the example of o, and the more difficult case of the Gravitational constant G (see 2.6 in [2]), I expect that mathematical physics will face issues where logical undecidability will get entangled with the notion of randomness.
In 4-dimensional smo
oth geometry I expect the famous 11/8 conjecture of Donaldson theory will prove to be undecidable, as will the smooth Poincare conjecture.