1.7.5 Discussion and Summary

The prime-count­ing func­tion $\pi(n)$ de­fines the number of primes oc­cur­ring for $\leq n\in \mathbb{N}$ and the prime number the­o­rem states that $\pi(n)$ is as­ymp­tot­ic to $n$ over the log­a­rithm of $n$

which, as an es­ti­mate, is at­trib­ut­able to Gauss and Le­gendre. The prime number the­o­rem states that the ratio $\pi(n)$ to $\frac{n}{\ln(n)}$ ap­proach­es $1$ as $n$ ap­proach­es in­fin­i­ty

and that the prime num­bers are as­ymp­tot­ic in their dis­tri­b­u­tion, be­com­ing less fre­quent in ap­pear­ance for larger num­bers. The prob­a­bil­i­ty that a chosen nat­ur­al number $n$ is prime is ap­prox­i­mate­ly

In Figure 1.7.40, the prime-count­ing func­tion $\pi(n)$ is rep­re­sent­ed as the floor func­tion $\left\lfloor \left. \ \pi(n) \right\rfloor \right.$, which is com­pared with the rec­i­p­ro­cal log­a­rithm ap­prox­i­ma­tion $\pi(n)\sim\frac{n}{\ln(n)}$. A better es­ti­mate re­lates the prime-count­ing func­tion to the offset log­a­rith­mic in­te­gral

which yields small­er and small­er rel­a­tive errors for in­creas­ing $n$.

In 1859, Bernard Rie­mann¹ used an­a­lyt­ic con­tin­u­a­tion to extend the study of har­mon­ic series from real to com­plex vari­ables and fur­ther to es­tab­lish the re­la­tion­ship be­tween the zeta zeroes and the dis­tri­b­u­tion of the primes. In the form of a series, the zeta func­tion is de­fined as

for the com­plex vari­able $s\in \mathbb{C}$. Like the Euler prod­uct over the reals, $\zeta(s)$ has the prime number equiv­a­lence

For $Re(s) > 1$, we can rep­re­sent the Rie­mann zeta func­tion as the con­ver­gent in­te­gral

in which the gamma func­tion $\Gamma(s)$ is also con­ver­gent

and when $s \in \mathbb{Z}^{+}$, the gamma func­tion is re­lat­ed to the fac­to­r­i­al func­tion by

The an­a­lyt­ic con­tin­u­a­tion begins with the re­la­tion­ship be­tween the $\zeta(s)$ func­tion and the Dirich­let eta func­tion $\eta(s)$

for which the con­ver­gence of the zeta func­tion on the right-hand side ex­tends to all $Re(s) > 0$, except for $\left( 1 - \frac{2}{2^{s}} \right) = 0$, oc­cur­ring when $n\in \mathbb{Z}$ in the fol­low­ing ex­pres­sion for $s$

The con­ver­gence issues for these zeros can be solved using the limit ap­proach. By the re­la­tion­ship with the $\eta(s)$ func­tion, the zeta func­tion can be de­fined as

The in­fa­mous Rie­mann Hy­poth­e­sis con­cerns the zeros out­side the region of con­ver­gence for the in­fi­nite series stated in Eq. (1.7.400). Rie­mann showed the an­a­lyt­ic con­tin­u­a­tion of the $\zeta(s)$ func­tion sat­is­fies the func­tion­al equa­tion

in which the triv­ial zeros man­i­fest when $s$ equals a neg­a­tive even in­te­ger value. The non-triv­ial zeros all lie in what is de­fined as the crit­i­cal strip be­tween real values of $0$ and $1$. Rie­mann fur­ther hy­poth­e­sizes that all the non-triv­ial zeros have a real por­tion $s$ equal to one-half, $Re(s) = \frac{1}{2}$, laying on the crit­i­cal line.

In the chap­ter, we extend the nat­ur­al num­bers to the com­plex domain, taking the form of cyclic groups $G_{n}$, with prime number groups $G_{p}$ being fur­ther clas­si­fied as simple groups. The reader has likely gath­ered that the min­i­mal el­e­ment, ini­tial­ly pre­sent­ed in Sec­tion 1.4.5, is syn­ony­mous with a group's pri­ma­ry gen­er­a­tor(s), ca­pa­ble of self-prod­ucts that yield all el­e­ments of the de­fined group, $\left\langle g^{k} \right\rangle = G$. For $n > 1 \in \mathbb{N}$, a pos­i­tive and neg­a­tive ori­ent­ed pri­ma­ry gen­er­a­tor are guar­an­teed to exist, noting in the case for $n = 2$, these two pri­ma­ry gen­er­a­tors are su­per­im­posed on the com­plex unit circle

When con­sid­ered in the com­plex domain and as groups, the law of net-zero can easily be ap­plied to the prime num­bers. The most straight­for­ward is the con­ju­gate re­la­tion­ship that exists be­tween any pri­ma­ry gen­er­a­tors $\left\langle g_{n}^{+} \right\rangle$, $\left\langle g_{n}^{-} \right\rangle$, which form the prod­uct

For the pos­i­tive and ex­tend­ing wave­form func­tions $g^{+}\left( \mathbb{Z}^{+}(n) \right)$ and $g^{-}\left( \mathbb{Z}^{+}(n) \right)$, de­pict­ed in Fig­ures 1.7.37-39, their net-zero coun­ter­parts are in­verse­ly re­lat­ed in both their spher­i­cal and spa­tial ar­gu­ments. This is seen using the ex­am­ple of the prime number $n = 7$, with

which have the cor­re­spond­ing in­vers­es

The para­met­ric func­tions defin­ing the con­tract­ing rec­i­p­ro­cal wave­forms are

and are plot­ted in Figure 1.7.41. The con­tract­ing rec­i­p­ro­cal nat­ur­al number para­met­ric equa­tions are for the values of $n \in t$, for $n\in \mathbb{N}$

onto the unit disc for the rec­i­p­ro­cal time func­tion $f(t) = \frac{1}{t},\ t\in \mathbb{R}$.

The $g^{+}(t),g^{-}(t)$ curves in­ter­sect the real-axis at the iden­ti­ty $\epsilon$ and $g_{2}^{+},g_{2}^{-}$, and in­ter­sect the imag­i­nary-axis at $g_{4}^{+},g_{4}^{-}$, and the non-in­te­ger values $g_{\mathcal{L}}^{-},g_{\mathcal{L}}^{+}$, for $t = \frac{4}{3}$. The sub­script $\mathcal{L}$ in­di­cates that these are light num­bers, ev­i­denced by the values of these points on each curve

Two car­dioids in Figure 1.7.41 are formed by the in­ter­sec­tion of the para­met­ric func­tion $g^{+}(t),g^{-}(t)$. The larger one is formed over the in­ter­val $t = \left\lbrack 1,\frac{1}{2} \right\rbrack$ for $g^{+}(t),g^{-}(t)$, be­com­ing a closed curve when $t \geq \frac{1}{2}$. The small­er car­dioid only ap­proach­es a close curve as $t \rightarrow \infty$. This is ap­pre­ci­at­ed by zoom­ing into the origin for the plot on the in­ter­val $t = \lbrack 1,1000\rbrack$, where blue points on the $g^{+}(t)$ para­met­ric curve are plot­ted for $t = 1000$ and $t = 10,000$.

The rec­i­p­ro­cal func­tions are re­lat­ed to log­a­rith­mic spi­rals and the log­a­rithm func­tion itself, seen by con­sid­er­ing that the in­te­grals of the fol­low­ing form are equal to $\ln(n)$

As dis­cussed, many number theory func­tions and series are re­lat­ed to the study of prime num­bers, in­clud­ing the fac­to­r­i­al and the ex­tend­ed gamma func­tions, the digam­ma func­tion, the har­mon­ic series, the log­a­rithm func­tion, and the Euler-Mascheroni con­stant. The Rie­mann zeta func­tion ex­tends the har­mon­ic series by rais­ing the el­e­ments of the series to com­plex powers of $s$. The chap­ter aimed to see nat­ur­al num­bers $\in \mathbb{Z}$, the in­te­gers $\in \mathbb{R}$, and the real num­bers $\in \mathbb{C}$. All on­to­log­i­cal num­bers are in­ter­re­lat­ed, and groups are num­bers in motion. When seen in this frame­work, all the prime num­bers are present in the sin­gu­lar­i­ty. Num­bers and their amal­gams as group rep­re­sen­ta­tions can extend beyond the ini­tial sin­gu­lar state's uni­tary bound­ary, made pos­si­ble by their co­in­ci­dent in­verse form that con­tracts.

Are prime num­bers valid on­to­log­i­cal­ly, mean­ing they have rel­e­vance to the nature of being? Or are they just math­e­mat­ics fan­fare? Prime num­bers play an es­sen­tial role in cryp­tog­ra­phy, but this falls short of the de­f­i­n­i­tion of being on­to­log­i­cal­ly sig­nif­i­cant.

In 1972, Hugh Mont­gomery² shared recent find­ings re­lat­ed to his work on the zeros of the Rie­mann zeta func­tion with Atle Sel­berg, a pro­fes­sor at Prince­ton's In­sti­tute for Ad­vanced Study. Mont­gomery had found the dis­tri­b­u­tion of the zeros on the crit­i­cal line of $\zeta(s)$ tended to repel be­tween the neigh­bor­ing levels. Mont­gomery would also share what is now called the pair cor­re­la­tion con­jec­ture with Free­man Dyson, whose work on random matrix theory was fun­da­men­tal to un­der­stand­ing the quan­tum nature of the heavy nu­cle­us. Dyson's work made it pos­si­ble to un­der­stand the dis­tri­b­u­tion and be­hav­ior of the eigen­val­ues of the random ma­tri­ces and how they ap­plied to the heavy nu­cle­us. When Dyson saw Mont­gomery's pair cor­re­la­tion, he was as­ton­ished, rec­og­niz­ing it was the pair cor­re­la­tion he found for the eigen­val­ues of a random Her­mit­ian matrix a decade ear­li­er.

"His result was the same as mine. They were coming from com­plete­ly dif­fer­ent di­rec­tions, and you get the same answer. It shows that is a lot there that we don't un­der­stand, and when we do un­der­stand it, it will prob­a­bly be ob­vi­ous. But at the moment, it is just a mir­a­cle." – Free­man Dyson

Know­ing this link be­tween the primes and par­ti­cle physics should change the reader's per­spec­tive. The primes are man­i­fes­ta­tions of the simple groups in the form of cyclic groups. These simple groups are the build­ing blocks of finite groups; there­fore, the geo­met­ric primes are fun­da­men­tal to the group rep­re­sen­ta­tions pre­sent­ed in the Arith­moí series.

Are prime num­bers valid on­to­log­i­cal­ly? The answer is ab­solute­ly YES!

1. Rie­mann, Bern­hard, "On the Number of Primes Less than a Given Mag­ni­tude", Nov. 1859, Pruss­ian Acad­e­my of Sci­ences of the Royal Month­ly in Berlin.
2. Thomas, K, "From Prime Num­bers to Nu­clear Physics and Beyond", 2013. https://www.ias.edu/ideas/2013/primes-random-ma­tri­ces