# Ken Ono and Hausdorff dimensions

chis
Hi All, In Ken Onos lecture he mentions Hausdorff dimensions appertaining to prime numbers:
5,7,11 relate to 0 dimension and primes from 13 to 31 as 1 dimension.
Are there more dimensions and if so do they also easily appertain to a set of prime numbers?

Thanks
Chris

chis
Nobody?

JeremyEbert
Hi All, In Ken Onos lecture he mentions Hausdorff dimensions appertaining to prime numbers:
5,7,11 relate to 0 dimension and primes from 13 to 31 as 1 dimension.
Are there more dimensions and if so do they also easily appertain to a set of prime numbers?

Thanks
Chris

doesn't seem to be...
http://en.wikipedia.org/wiki/Ramanujan's_congruences

Raphie
Hi All, In Ken Onos lecture he mentions Hausdorff dimensions appertaining to prime numbers:
5,7,11 relate to 0 dimension and primes from 13 to 31 as 1 dimension.
Are there more dimensions and if so do they also easily appertain to a set of prime numbers?

Thanks
Chris

Hi Chris,

I wish I could answer your question, but not only would I not be qualified to do so, but also I don't believe that just about anyone, including most professional mathematicians would be qualified to do so. This is such a novel and original finding (although, honestly, I am not surprised by it.) and I would love to hear more if you should find out more. For instance, do you have a link to the text of the lecture?

For whatever it's worth, (via Wikipedia) here is another set of numbers with Hausdorff Dimension of 1

---------------------------------------------------
Smith–Volterra–Cantor set
Built by removing a central interval of length 1/2^2n of each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of 1/2

So too the Takagi or Blancmange curve. For more, see...

List of Fractals by Hausdorff Dimension
http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
---------------------------------------------------

As for the specific primes you mention, however, I do have to say I am rather surprised in a good way because I am already well familiar with those particular ones because they are the solutions to a descriptive equation I came up with some time back:

(y - 1)*d(y - 1) = K_(x + 2)

for...
y = {5, 7, 11, 13, 31}
y-1 = {4, 6, 10, 12, 30}
d(y-1) = {3, 4, 4, 6, 8}
x = {1, 2, 3, 4, 6}
(x + 2) = {3,4,5,6,8}

K_(3,4,5,6,8) = {12,24,40,72,240}

...where d(n) denotes the divisor function and K_n denotes a maximal (proven) lattice sphere packing for dimension n, associated with the Lie Groups A3/D3, D4, D5/E5, E6 & E8 and 1, 2, 3, 4, 6 are the solutions to 2*cos ((2*pi)/n) is in N (related to the Crystallographic Restriction Theorem)

The point being, in general, that it would be pretty amazing should it be found that the fractal pattern Ono has found in relation to the partition numbers (the division of number space) can be related in some manner to the optimal packing of n-dimensional physical space and/or the allowable n-fold Rotational Symmetries of a Periodic Crystal.

Best,
RF

A FEW RELATED DETAILS IN REGARDS TO THE ABOVE (if interested)
1, 2, 3, 4 & 6, which are the proper divisors of 12 (=Totient (13)), are also the only integers with a totient of 1 or 2, as well as being the only integers such that d(p_x - 1) = pi (p_x) e.g. pi (p_6) = 6

2,3,5,7 & 13 = p_x are the first 5 Mersenne Prime Exponents that Frampton and Kephart associated with anomaly cancellations in 26-Dimensional String Theory, and have the distiction of being the unique prime factors of the Leech Lattice ( a 24-D Euclidean Lattice constructed from a "light ray" in 26-D Lorentz Space).

Also worth noting in relation to the "Ono Primes" is that 5 is the 3rd prime number and 11 is the the (2*3 - 1)-th = 5th prime number, while 13 is the 6th prime number and 31 is the (2*6 - 1)-th = 11th prime number. If, hypothetically, the progression continued in similar manner for two more iterations, then the next prime ranges in the series would be primes from 37 = p_(12) --> 87 = p_(23), followed by 89 = p_(24) --> 211 = p_(47). That would be kind of "neat" since:

A) 89 is not only the 24th prime, but also the 10th Mersenne Prime Exponent and 11-th Fibonacci Number, and the divisors of 89-1 = 8, which cleanly divides 24, the first prime after 13 (= p_6) for which this is the case. (Other Mersenne Prime Exponents that are also Fibonacci Numbers: 2, 3, 5 & 13, the 1st, 2nd, 3rd and 5th, the index numbers of which map to the number of vertices of a Point, Line, Triangle and Square. A Pentagon has 10 vertices.)

B) (Totient (37))^2 + (Totient (37))^1 = 1332, which comes very close to giving the sum of the first (4^2 + 4)/2 Maximal (known) Sphere Packings (to Dimension 10). If the Maximal Sphere Packing for dimension 10 were ever found to be 504 (7!/10) rather than 500 (as I have previously conjectured), as is currently the maximal known, then the match would be 100%, which in and of itself would be quite interesting since (Totient (13))^2 + (Totient (13))^1 = 156, (Totient (5))^2 + (Totient (5))^1 = 20, and (Totient (2))^2 + (Totient (2))^1 = 2 already give exact matches for the sums of maximal (known) sphere packings to dimension (3^2 + 3)/2, (2^2 + 2)/2 and (1^2 + 1)/2, respectively.

Note: 2, 5, 13 and 37 are the 1st, 3rd, 6th and 12th primes, or 1/2 the maximal sphere packings for Dimensions 1, 2, 3 & 4 (= 2, 6, 12, 24. Just add 6 and multiply by dimension number to get 40, 72, 126 and 240, the maximal proven lattice sphere packings for Dimensions 5, 6, 7 & 8)

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Raphie

It is seen to have dimension 0 only in the cases where l = 5,7 or 11 and since the partition function can be written as a linear combination of these functions this can be considered a formalization and proof of Ramanujan's observation.

This says nothing about the existence or lack thereof of dimensions beyond D=0. It only notes that Ono et al. have proven the special case of D=0.

- RF

JeremyEbert
Hi Chris,

I wish I could answer your question, but not only would I not be qualified to do so, but also I don't believe that just about anyone, including most professional mathematicians would be qualified to do so. This is such a novel and original finding (although, honestly, I am not surprised by it.) and I would love to hear more if you should find out more. For instance, do you have a link to the text of the lecture?

For whatever it's worth, (via Wikipedia) here is another set of numbers with Hausdorff Dimension of 1

---------------------------------------------------
Smith–Volterra–Cantor set
Built by removing a central interval of length 1/2^2n of each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of 1/2

So too the Takagi or Blancmange curve. For more, see...

List of Fractals by Hausdorff Dimension
http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
---------------------------------------------------

As for the specific primes you mention, however, I do have to say I am rather surprised in a good way because I am already well familiar with those particular ones because they are the solutions to a descriptive equation I came up with some time back:

(y - 1)*d(y - 1) = K_(x + 2)

for...
y = {5, 7, 11, 13, 31}
y-1 = {4, 6, 10, 12, 30}
d(y-1) = {3, 4, 4, 6, 8}
x = {1, 2, 3, 4, 6}
(x + 2) = {3,4,5,6,8}

K_(3,4,5,6,8) = {12,24,40,72,240}

...where d(n) denotes the divisor function and K_n denotes a maximal (proven) lattice sphere packing for dimension n, associated with the Lie Groups A3/D3, D4, D5/E5, E6 & E8 and 1, 2, 3, 4, 6 are the solutions to 2*cos ((2*pi)/n) is in N (related to the Crystallographic Restriction Theorem)

The point being, in general, that it would be pretty amazing should it be found that the fractal pattern Ono has found in relation to the partition numbers (the division of number space) can be related in some manner to the optimal packing of n-dimensional physical space and/or the allowable n-fold Rotational Symmetries of a Periodic Crystal.

Best,
RF

A FEW RELATED DETAILS IN REGARDS TO THE ABOVE (if interested)
1, 2, 3, 4 & 6, which are the proper divisors of 12 (=Totient (13)), are also the only integers with a totient of 1 or 2, as well as being the only integers such that d(p_x - 1) = pi (p_x) e.g. pi (p_6) = 6

2,3,5,7 & 13 = p_x are the first 5 Mersenne Prime Exponents that Frampton and Kephart associated with anomaly cancellations in 26-Dimensional String Theory, and have the distiction of being the unique prime factors of the Leech Lattice ( a 24-D Euclidean Lattice constructed from a "light ray" in 26-D Lorentz Space).

Also worth noting in relation to the "Ono Primes" is that 5 is the 3rd prime number and 11 is the the (2*3 - 1)-th = 5th prime number, while 13 is the 6th prime number and 31 is the (2*6 - 1)-th = 11th prime number. If, hypothetically, the progression continued in similar manner for two more iterations, then the next prime ranges in the series would be primes from 37 = p_(12) --> 87 = p_(23), followed by 89 = p_(24) --> 211 = p_(47). That would be kind of "neat" since:

A) 89 is not only the 24th prime, but also the 10th Mersenne Prime Exponent and 11-th Fibonacci Number, and the divisors of 89-1 = 8, which cleanly divides 24, the first prime after 13 (= p_6) for which this is the case. (Other Mersenne Prime Exponents that are also Fibonacci Numbers: 2, 3, 5 & 13, the 1st, 2nd, 3rd and 5th, the index numbers of which map to the number of vertices of a Point, Line, Triangle and Square. A Pentagon has 10 vertices.)

B) (Totient (37))^2 + (Totient (37))^1 = 1332, which comes very close to giving the sum of the first (4^2 + 4)/2 Maximal (known) Sphere Packings (to Dimension 10). If the Maximal Sphere Packing for dimension 10 were ever found to be 504 (7!/10) rather than 500 (as I have previously conjectured), as is currently the maximal known, then the match would be 100%, which in and of itself would be quite interesting since (Totient (13))^2 + (Totient (13))^1 = 156, (Totient (5))^2 + (Totient (5))^1 = 20, and (Totient (2))^2 + (Totient (2))^1 = 2 already give exact matches for the sums of maximal (known) sphere packings to dimension (3^2 + 3)/2, (2^2 + 2)/2 and (1^2 + 1)/2, respectively.

Note: 2, 5, 13 and 37 are the 1st, 3rd, 6th and 12th primes, or 1/2 the maximal sphere packings for Dimensions 1, 2, 3 & 4 (= 2, 6, 12, 24. Just add 6 and multiply by dimension number to get 40, 72, 126 and 240, the maximal proven lattice sphere packings for Dimensions 5, 6, 7 & 8)

I'm going to need a week to take this all in...THANK YOU.

chis
Thanks folks.

Chis

Raphie
Thanks folks.

Chis

I'm going to need a week to take this all in...THANK YOU.

No problem, Chris and Jeremy. Without reading too much into it, here is a very simple way to numerically relate the five primes mentioned by Ono in terms of the prime counting function and two well-known multiplicative functions.

for pi (31) = 11 and pi (pi (31)) = 5
...where pi (n) denotes the prime counting function, sigma(n) the sum of divisors function, and totient(n) the Euler Phi function, then...

sigma (pi (31)) = totient (13) = 12
sigma (pi ( pi (31))) = totient (7) = 6

totient (pi (31)) = totient (11) = 10
totient (pi (pi (31))) = totient (5) = 4

And, worth noting, 31 == pi (127) == pi (2^7 - 1) == pi (8*T_5 + 7) == pi (11^2 + 7)== pi (1/10*T_5 (Tetra_7) + 1/10*T_5)), expressions which can all be related to the Ramanujan-Nagell Equation.

- RF

A RELATED NUMBER SEQUENCE
Primeth recurrence: a(n+1) = a(n)-th prime.
http://oeis.org/A007097

AND A PAPER RELATED TO THE PRIMETH RECURRENCE...
Lubomir Alexandrov,"The Eratosthenes Progression p(k+1)=p(p(k)), k=0,1,2,..., p(0)=1,4,6,... Determines an Inner Prime Number Distribution Law", Second Int. Conf. "Modern Trends in Computational Physics", Jul 24-29, 2000, Dubna, Russia, Book of Abstracts, p. 19
http://arxiv.org/abs/math/0105154

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chis
2) The Hausdorff dimension for each prime ell equals (in almost every case) is [(ell+1)/12], where [*] is the usual greatest integer function. So notices that this number is 0 for the primes 5, 7, 11, and equals 1 for ell=13.