On Some Connections Between Number Theory and Physics (1.1.5)

Abstract Some connections there are between number theory -especially the Riemann Hypothesis- and physics are explored.


We begin with Euler’s classic argument (1) that the product continued to infinity of this fraction

\frac{2\ 3\ 5\ 7\ 11\ 13\ 17\ 19\text{...}}{2\ 4\ 6\ 10\ 12\ 16\ 18\text{...}}

in which the numerators are prime numbers and the denominators are one less than the numerators, equals the sum of the infinite series


and they are both infinite. To prove his point to Euler invites us to imagine the extraction from the second series a prime denominator and all remaining multiples of that prime denominator until everything except the first term 1 has been eliminated. Let




This leaves


To eliminate the denominators that are divisible by 3, we divide both sides to get

\frac{x}{2\ 3}=\frac{1}{3}+\frac{1}{9}+\frac{1}{15}+\frac{1}{21}\text{...}

Subtracting again eliminates all remaining denominators that are multiples of 3 leaving

\frac{2 x}{2\ 3}=1+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}\text{...}

Applying this eliminatory process to all of the prime numbers leaves

\left(\frac{2\ 4\ 5\ 10\ 12\ 16\ 18}{2\ 3\ 5\ 7\ 11\ 13\ 17\ 19}\text{...}\right) x=1

This is a thought-experiment -mere imagination- but if these eliminations could be performed in the physical world, they would result in the disappearance of any distinction between the form and the content of a coordinate system, and therefore the shrinking of space and the slowing of time to a zero-dimensional point. With all of reality contracted to a zero-dimensional point, the distinction between the world and the mind that surveys it is lost. This is the singularity at root of general relativity. It is not -as is often maintained- all of the mass in the universe compressed down to a point, an infinitely heavy object. This is a gross misconception resulting from the assumption of atomism, and the prioritization of space over light, a misconception responsible for all that is ugly in contemporary physics, including black holes as infinite at their centers (2), the incongruity of relativity and quantum mechanics (3), the infinities of quantum field theory (4), and the flat rotation curves of distant galaxies that seem to call for dark matter (5, 6)… to name a handful of significantly troublesome things. Rather, this singular state is a state such that all of the space and time -all the gaps and holes- in the universe are from the perspective of this singularity excised, leaving only infinitely concentrated light, an infinitely light object in both senses of the term “light”. On this account there are two forms of curvature resulting either from an imbalance of light and space in favor of light (the most extreme form of this imbalance is the origin of the universe) or from an imbalance of light and space in favour of space (the most extreme forms of this imbalance are black holes).

The first idea we can take from Euler’s thought-experiment is that, since both prime-density and energy-density must at this point be infinite, the spatio-temporal development of the universe from a central singular point towards an ever-increasing state of decentralization is a process involving the distribution of the prime numbers. We can add to this that this process of decentralization takes place according to something known as the Riemann Hypothesis (7, 8), which says that the thinning of primes -the spreading of prime-energy over time and space- with arithmetic increase cannot exceed the upper and lower bounds such as those marked in red and blue in the graphs below:

\pi (x)

\sum _{n=2}^x \frac{1}{n \log }-2 \left(\text{Re} \sum _{n=1}^{\infty } \text{Ei}\left(\rho _{-n} (\log x)\right)\right)

\sum _{n=2}^x \frac{1}{H_n}-2 \left(\text{Re} \sum _{n=1}^{\infty } \text{Ei}\left(\rho _{-n} (\log x)\right)\right)

The Generalized Riemann Hypothesis (9, 10) extends the Riemann Hypothesis by reference arithmetic progressions associated with the equation q n + a where q and n have no common factor greater than 1. In a universe whose fundamental condition is an infinite state of prime and energy density which is diffused from the point of view of any and every frame of reference according to the Generalized Riemann Hypothesis, time has a forwards direction associated with the loss of prime and energy density and a backwards direction associated with a gain in prime and energy density. Because the loss of prime-density predominates over any gains in prime-density, the direction of time is given by the GRH, a balance of the two forms of curvature is maintained by the equality

\rho _n=\frac{1}{2}

The waveform (\frac{2 \left(\text{Re} \sum _{n=1}^x \text{Ei}\left(\rho _n (\log x)\right)\right)}{x}) we obtain by taking difference between the number of primes not greater than x and an approximating formula such as

\left(\frac{\sum _{n=2}^x \frac{a_{1\ 1}}{H_n}+\sum _{n=2}^x \frac{a_{2\ 1}}{n \log }+\text{...}}{n}\right)

and dividing by x is a superposition of smooth waves:

-\frac{2 \left(\text{Re} \sum _{n=1}^1 \text{Ei}\left(\rho _n (\log x)\right)\right)}{x},-\frac{2 \left(\text{Re} \sum _{n=2}^2 \text{Ei}\left(\rho _n (\log x)\right)\right)}{x},-\frac{2 \left(\text{Re} \sum _{n=3}^3 \text{Ei}\left(\rho _n (\log x)\right)\right)}{x}

\frac{\pi (x)-\frac{\sum _{n=2}^x \frac{a_{1\ 1}}{H_n}+\sum _{n=2}^x \frac{a_{2\ 1}}{n \log }}{n}}{x},-\frac{2 \left(\text{Re} \sum _{n=1}^{10} \text{Ei}\left(\rho _n (\log x)\right)\right)}{x}

Or from another perspective on the same phenomenon, we can obtain the superposition 2 \left(\text{Re} \sum _{n=1}^{\infty } \frac{\left(p_x\right){}^{\rho _n}}{\rho _n}\right) by taking the difference between the primes themselves and an approximating formula such as

\frac{a_1 x H_x+a_2 x \log (x)+\text{...}}{n x}

\frac{2 \left(\text{Re} \sum _{n=1}^1 \frac{\left(p_x\right){}^{\rho _n}}{\rho _n}\right)}{x},\frac{2 \left(\text{Re} \sum _{n=2}^2 \frac{\left(p_x\right){}^{\rho _n}}{\rho _n}\right)}{x},\frac{2 \left(\text{Re} \sum _{n=3}^3 \frac{\left(p_x\right){}^{\rho _n}}{\rho _n}\right)}{x}

\frac{e^{2 \gamma } \sqrt{\frac{p_x}{e^{2 \gamma }}}{}^2-\frac{\int_1^x a_1 x (x \log +\gamma ) \, dn+\int_1^x a_1 x (x \log ) \, dn+\left(\int_1^x a_2 x (x \log +\gamma ) \, dn+\int_1^x a_2 x (x \log ) \, dn\right)}{2 x}}{x},\frac{2 \left(\text{Re} \sum _{n=1}^{10} \frac{\left(p_x\right){}^{\rho _n}}{\rho _n}\right)}{x}

The intimate connection there is between number theory and quantum field theory can be simply illustrated (11) by associating the creation operators \left(b_n\right){}^{\dagger } and \left(f_n\right){}^{\dagger } to the prime numbers p_n… Now we have identified the unique ‘factorization’ of a state into creation operators acting on the ‘vacuum’ with the unique factorization of an integer into prime numbers (and we have a hierarchy of states: |1> is the ‘vacuum’; |2> and |3> and |5> are one-particle states; |6> is a two-particle state… and so on). By reference to the Witten index (12) -the number of bosonic minus the number of fermionic zero-energy states- we see that the Mobius inversion function

\mu n={1 = n has an even number of distinct factors,
-1 = n has an odd number of distinct factors, 0 = n has a repeated factor}

is equivalent to the operator (-1)^F that distinguishes bosonic from fermionic states, with \mu n = 0 when n has a repeated factor being equivalent to the Pauli exclusion principle. If we re-express the Mertens function (which sums the 1s and -1s of the Mobius function) as \sum _{n=1}^{p_x} \mu (n), we see that sums of these states give us essentially the same composite spiral-wave as before.

\frac{2 \Re\left(\sum _{n=1}^1 \frac{\left(p_x\right){}^{\rho _n}}{\rho _n \zeta '\left(\rho _n\right)}\right)}{x},\frac{2 \Re\left(\sum _{n=2}^2 \frac{\left(p_x\right){}^{\rho _n}}{\rho _n \zeta '\left(\rho _n\right)}\right)}{x},\frac{2 \Re\left(\sum _{n=3}^3 \frac{\left(p_x\right){}^{\rho _n}}{\rho _n \zeta '\left(\rho _n\right)}\right)}{x}

\frac{2 \Re\left(\sum _{n=1}^{10} \frac{\left(p_x\right){}^{\rho _n}}{\rho _n \zeta '\left(\rho _n\right)}\right)}{x},\frac{\sum _{n=1}^{p_x} \mu (n)+2}{x}

Assuming that there are an equal number of non-zero-energy bosonic and fermionic states, this wave depicts the zero-energy fluctuations of these particles, the energy fluctuations of the vacuum. This is to say that the vacuum is the basis of everything -everything emanates from the vacuum- and that the vacuum is far from vacuous. There is here a distinction between symmetric superpositions such as \frac{2 \Re\left(\sum _{n=1}^{10} \frac{\left(p_x\right){}^{\rho _n}}{\rho _n \zeta '\left(\rho _n\right)}\right)}{x} (where the associated progression is arithmetically continuous) and asymmetric superpositions familiar from quantum mechanics (where the associated progression is arithmetically discontinuous). The intersections of the x-axis by these fluctuations can be identified with the end of one asymmetric superposition and the beginning of another… These examples concern the case where q =1 and a = 0, but we can easily construct approximating formulas and wave-forms for all the other possible values of q and a. The value \rho _n=\frac{1}{2}, associated as it is with a very particular balance of prime-density and sparsity, signifies the dividing line between classical objects that are constrained to travel in time in an arithmetic manner, and quantum objects that transcend arithmetic and are not so constrained. A key aspect of this mathematical scheme is that we can take any arithmetic progression associated to q n + a and to an L-Function and associate it to a formula such as \frac{\sum _{n=2}^x \frac{a_{1\ 1}}{H_n}+\sum _{n=2}^x \frac{a_{2\ 1}}{n \log }+\text{...}}{n} and a wave-form such as -2 \left(\text{Re} \sum _{n=1}^x \text{Ei}\left(\rho _{-n} (\log x)\right)\right)). The first perspective is the arithmetic perspective from which things are distributed, unit by unit, in mathematical and physical space and in time (the time perspective), the second perspective (the frequency perspective) is a “trans-arithmetic” perspective from which everything is in some sense always present. From the fact that the initial, foundational, state of the universe is state such that there is no space and time and no separation between abstract and/or physical units (a state such that prime and energy-density are infinite), it follows that the second perspective has priority over the first, that the first perspective is a well-founded illusion -a projection- arising from the second and governed by the Riemann Hypothesis.

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(1) Euler, L (1737), Various observations concerning infinite series

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(4) Feynman, Richard (1948), A Relativistic Cut-Off for Quantum Electrodynamics

(5) Rubin, V et al (1980), Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R = 4kpc) to UGC 2885 (R = 122kpc)

(6) de Swart, J. et al (2017), How dark matter came to matter

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(12) Witten, E (1982), Constraints on supersymmetry breaking