If one asks a question: what are the relationships between Heisenberg commutators and the Riemann Hypothesis? What is the spectrum? Are their ground states the same? Start with $s=B+iH>1$ and¹ the sequence $s^{+}=B-i{H}^{+}>1$ ( $B\in R$) [Heisenberg commutators in quantum mechanics (Q.M.)] and with widely-known that $\zeta \left(1/2+iH\right)=0$². Where let

$B\equiv 1/2$(1)

If $s^2=B\cdot B+2iBH-H\cdot H^{+}>1$ can be established so that

$\begin{array}{l}1/4+iH-H\cdot H^{+}>1,\\ iH-H\cdot H^{+}>3/4\end{array}$(2)

Obviously

$2\left(iH-H\cdot H^{+}\right)>3/2$(3)

With operator $H=n+1/2$ making quantum numbers of $n\in R$ in a particular phase to be mapping onto s-components in Riemann zeta functions in a physical system.

Therefore,

$H=s+1/2,s:=n$(4)

And simultaneously the adjoint-operator in terms of $H^{+}=s^{+}-1/2$ is kept for enclosure. Therefore,

$2\left(s+1/2\right)\left(i-H^{+}\right)>3/2$(5)

This paper demonstrates that there exists a spectrum set being both for real and the complex numbers as structures below since the s-region is smooth and continuous. In case of simple harmonic oscillators (SHO), the superposition of wave-functions with $n=0,1,2,3,\cdots$ is overcome with a complete set expressed as sinc-function that the wave function of laser by the spectrum of observable Heisenberg photons appears. Therefore,

$\begin{array}{l}\zeta \left(2\left(s+1/2\right)\left(i+H\right)\right)>\zeta \left(3/2\right),\forall s>1,s\in R\\ \zeta \left(2\left(s+1/2\right)\left(i-H^{+}\right)\right)>\zeta \left(3/2\right),\forall s^{+}\in C\end{array}\}$(6)

And ${\left(\zeta \left(3/2\right)\right)}^n$ to the power $n\in R$ (in later sections, one can see that $n=p=3,7$ definitely in BEC phases). As far as one can see the problem of Riemann hypothesis must be relative to $\zeta \left(3/2\right)$ in expressions of representation by Heisenberg commutators³. Above is clear and powerful as it pertains directly to the Riemann Hypothesis (RH).

To be continued with Sect. 2.1 here, cited here as a widely known formula:

$T_C={\left(\frac{n}{\zeta \left(3/2\right)}\right)}^{2/3}\frac{2\pi {\hslash }^2}{m{k}_B}\propto \frac{1}{{\left(\zeta \left(3/2\right)\right)}^{2/3}}$(7)

If the densities of the particles in BEC phases are fixed, then the relationships between the critical temperature and the Riemann zeta function ${\left(\zeta \left(3/2\right)\right)}^{-2/3}=const$ shown as Equation (7). This approach offers a reputable way of significantly simplifying a problem through pure physics; for example, the prime number 3 can be considered in mathematics. This paper suggests that the critical temperature of BEC phases can be represented as Riemann zeta function of 3/2. Moreover, associated with spectrum in Sect. 2.1, $\zeta \left(3/2\right)$ is considered ( Figure 1 ).

Given Riemann zeta functions as

$\begin{array}{l}\zeta \left(s\right)=\underset{p}{{\displaystyle \prod }}\frac{1}{1-p^{-s}},s>1,\\ \zeta \left(s\right)=\frac{1}{\left(1-\frac{1}{2^s}\right)\left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{5^s}\right)\left(1-\frac{1}{7^s}\right)\left(1-\frac{1}{{11}^s}\right)\cdots },s>1\end{array}$(8)

where $s:=n$ is denoted as the quantum numbers of BEC phases in complex-planes. As far as one can see, since it is associated with

$T_p=\left(1+\frac{1}{{10}^p}\right)T_c\ne 0,-\infty <p<\infty$(9)

Note that Equation (9) is an empirical expression based on data from the Cornell University Laboratory (1995), denoted as p = 7 for ~10⁻⁷ K (i.e., the energy spectrum of photons of the BEC-laser).

Associated with the Bogoliubov transformation in Bardeen–Cooper–Schrieffer (BCS) theory,

$E=\frac{hc}{\lambda }=3.52k_B{T}_c\sqrt{1-\frac{T}{T_c}}$(10)

Therefore, in the fine-tuning process, where the parameter p must be self-regulated as $p=\left|\left[3.52\right]\right|=3$ in progress⁴. As for the other terms of prime numbers, they need to be moved onto the left-hand side (LHS). Hence:

$K\cdot \zeta \left(s\right)=\frac{1}{\left(1-\frac{1}{3^s}\right)},s>1$(11)

Where defined as

$K\equiv \left(1-\frac{1}{2^s}\right)\left(1-\frac{1}{5^s}\right)\left(1-\frac{1}{7^s}\right)\left(1-\frac{1}{{11}^s}\right)\cdots ,s>1$ for all prime numbers(12)

e.g., let $s=3/2$ in a BEC phase, and next let $K\zeta \left(3/2\right)$ in terms of ${\zeta }^{\prime }\left(3/2\right)$. Namely,

$\begin{array}{l}{\zeta }^{\prime }\left(3/2\right)=\frac{1}{1-3^{-3/2}}=-0.2383135547\equiv C\ne 0,\\ ⋮\\ {\zeta }^{\prime }\left(s\right)=\frac{1}{1-3^{-s}}=C,\forall s>1,\\ {\zeta }^{\prime }\left(s\right)-C=0,\forall s>1\end{array}$(13)

On the critical line where $\mathrm{Re}\left(s\right)=1/2$, it obviously turns to $\zeta \left(s\right)=0,\forall s>1$ for all trivial zeros (definitely with the critical line where $\mathrm{Re}\left(s\right)=1/2$). This implies that a BEC phase on arbitrary atoms could possibly be denoted by Equation (13).

The famous fine structure constant, whose reciprocal approximates the prime p =137, could have been included by this mathematical induction if p = 137 were picked. Hence, the method stated by the “Todd function” (2019) was in vain in the results⁵ $\alpha =1/137$.

As a running point, α is here assumed to be represented by a matrix and calculated via a statement of interactions. Note that the above p is not denoted as a quantum number, so one should be careful with it.

Equation (14) implies that the valence electrons of Si atoms or Si-topological superconductors (e.g., Yb₂Si₂O₇) could form a fermionic condensate (Cooper pairs), which is permitted. The total bulk of this material transitions into BEC phases. Additionally, due to the similar chemical periodic properties of Na/Si and Cs/Yb, the connection of the Riemann Hypothesis (RH) to BEC phase materials (e.g., NaCs or Yb₂Si₂O₇) and the molecule numbers of a BEC phase as an observable distribution of prime numbers establishes a clear relation between the RH and material science, which is universal.

Referring to the paper cited by Figure 5 of Ref. [25] , we have solved this problem in perspective: Figure 5 shows that the molecule quantities as prime numbers are clearly computed: 41, 53, 79, 97, 127, 163, 179, 181, 211, 241, 251, 257 on the list from RHS to LHS of the x-axis (Hold time as shown) in Figure 5 of Ref. [25] . From a quantum mechanical point of view, the coherent state of the unique BEC phase broadly reveals that the state of one molecule is inseparable from another (molecule-to-molecule, i.e., molecular clouds) into two arbitrarily different states and must retain a unique state (i.e., a coherent state). In quantum mechanics (e.g., Heisenberg commutators), the distribution of prime numbers naturally satisfies the characteristics of BEC phases. As a result, such matter strongly supports our viewpoints in this paper, sufficiently and convincingly. Likewise, our relevant derivations are undoubtedly confirmed.

Note that the physical significance of the Simple Harmonic Oscillator (SHO) average energy, where the chemical covalent bond is directed to the electron cloud, acts as an electron-gas shell around absolute zero. This forms simple harmonically-mixed condensed states while Yb and Si are cooling. Therefore, based on the Virial theorem, the Hamiltonian of the system is indicated as $\bar{E}=2\bar{K}=2\bar{V}$. Via approximation, one can only consider Si atoms which contribute the average kinetic energy to Yb₂Si₂O₇ in crystals, leading to Equation (14). Therefore, Yb atoms are regarded as a bound state having an average potential energy $\bar{V}$. The prime number $p\equiv 3$ is naturally options associated with Equation (9) for the Cooper pairs’ energy $E_g\le {10}^{-3}\text{eV}$ required.

The Riemann series of absent s > 1 i.e., $\zeta \left(s\right)=\infty$, s = 0 i.e., $\zeta \left(0\right)=\infty$ [laser unique spot appears in BEC phases by using the famous EIT effects (quantum storages)⁶ this implies singularity point $\zeta \left(0\right)=\infty$]. The BEC phase gives the atomic physical systems of the coherent states in complex spaces to be indicated as photons of laser [UVC ranges with metastable state $i\cdot n$ of $s^{+}$ (see the text on p-10) and $s^{+}=s\to 2$ e.g., $n=s=2^{-}>1$ see Figure 2 ] to incident into a BEC phase to provide this phase as presentation stage, hence $\zeta \left(s\right)=0$, $\forall s=-4,-2$ stands for all trivial zeros. However, if the Electromagnetically Induced Transparency (EIT) photons act on a BEC phase (s = 1 being in a BEC vortex core, see later sections), they could be retained for a longer lifetime due to this vortex core, thereby maintaining the BEC phase⁷. This causes:

$\zeta \left(s\right)=0,\forall s=-4,-2$(15)

The BEC phase (where s = 1 exists in a BEC vortex core, see later sections) can be retained for a longer life-time, and then the BEC phase is remained.

As widely-known as, a beam of laser photons in UVC ranges can sculpt a hole (i.e., with $t=\left[i/2,-i/2\right]$ to product one of results: $s=1/2+it$ where obtained s = 0 (see the famous Riemann spheres). Obviously, s = 0 indicated as absence of bosons (e.g., no photon gas) such naturally directs s = 0 itself to be the lowest energy-level within the material of Yb₂Si₂O₇ stationed in BEC phases. Based on this, the Riemann hypothesis is established, i.e., the statement of the RH would be given by

${\displaystyle \underset{i=prime}{\sum }{\zeta }_i\left(s\right)}=\infty ,s>1$ with $\mathrm{Re}\left(s\right)=\frac{1}{2},0\le s<2,\forall s=0$(16)

Namely, for all prime numbers, there exists a unique pole $\zeta \left(s\right)=1$ as one of the non-trivial zero points of $\zeta \left(s\right)=\infty$, $s=0$ when considering both BEC phases and EIT effects applied in a wave range, e.g., with $\lambda =226.43\text{nm}$ UVC photons (see Appendix C for the calculations). We claim that this is a singular point in BEC phases, i.e., the existence of the Riemann Hypothesis (RH) is supported by Equation (16). This is precisely the widely-expected definite solution to the Clay Mathematics Institute Millennium Prize Problem, which has been well-solved by this paper for the first time. We are justified in making the above state-ments and relevant conclusions. Note that above ${\displaystyle \underset{i=prime}{\sum }{\zeta }_i\left(s\right)}=\infty ,s>1$, actually locates on 2D plane (see RHS of Figure 1 ).

The laser cooling (statistical mechanics): Consider that an atomic system of reversal population which has the reversion of Boltzmann distribution

$N\propto {\text{e}}^{-E/kT}$(17)

Let $E=nhv,n=1,2,3,\cdots$ while the laser photons incident into the energy-level of exited state in metastable states, such that

$N\propto {\text{e}}^{inhv/\left(-ikT\right)}$(18)

Moreover,

${\text{e}}^{-nhv/kT}={\text{e}}^{inhv/\left(-ikT\right)}={\text{e}}^{in\hslash \omega /\left(-ikT\right)}$(19)

where

$\kappa \equiv -ik$(20)

Hence

${\text{e}}^{in\hslash \omega /\kappa T}=\cos \left(n\hslash \omega /\kappa T\right)+i\sin \left(i\cdot n\hslash \omega /kT\right)$(21)

Such gives $i\cdot n$ of $s^{+}$ where atoms/or molecules occupied metastable states in case of $n>s^{+}$ since $\begin{array}{l}\hat{s}:=s=n-\frac{1}{2}\equiv \hat{H}\\ {\hat{s}}^{+}:=s^{+}=n+\frac{1}{2}\equiv i\hat{H}\end{array}\}$ where ${\hat{s}}^{+}$ could be denoted as

raising operator and $\hat{s}$ as lowering operator definitely $n:=\hat{n}$ in quantum mechanics. In an atomic system in reversal population (abbreviated as RP) with consideration of sufficient larger numbers of atoms $n\gg 1/2$ while the laser-RP is applied⁸.

Therefore $s^{+}=s=n_1$ if $n\in R$. Defined $s:=s^{+}=i{n}_2$ if $n\in \mathrm{Im}$⁹ and $i\hat{H}:=i\hat{n}$ strongly leaves alone that the real number 1/2 as the critical line such that causes all real parts of non-trivial zeros to be located on $\mathrm{Re}\left(s\right)=1/2=\left[\hat{x},\hat{p}\right]$. See Figure 2 .

Important that by A. Einstein “The Quantum Theory of Radiation” (1917) the detailed derivations are as discussing as below:

Given that $\stackrel{\dot{}}{\rho }\left(v\right)+\nabla \cdot \stackrel{\to }{j}=0$ where ${\left({\stackrel{\dot{}}{n}}_1\right)}_{B_{21}}=0=B_{21}{n}_2\cdot \rho \left(v\right)$, $n_1=Const$, such that ${\stackrel{\dot{}}{\rho }\left(v\right)|}_{T=0\text{K}}=0$. Note that $\rho \left(v\right)=\underset{T\to 0\text{K}}{\lim }\frac{2h{v}^3}{c^3\left({\text{e}}^{hv/kT}-1\right)}=0$ while laser is forcing a BEC phase arrival.

Note that the ideas are at the first time ( Table 1 ).

Remark. “Einstein General Elevator”: We generalize the Einstein’s elevator of gravity (1916-1917) to the version of atoms in a laser system, as shown in Table 2 .

Remark. The above mechanism is named as Einstein’s general elevator. Returning to the previous articles, let us arrange the equation stated by ${\left({\stackrel{\dot{}}{n}}_1\right)}_{B_{21}}=B_{21}{n}_2\rho \left(v\right)$ precisely. Yields

$\rho \left(v\right)=\frac{1}{B_{21}}\frac{{\left({\stackrel{\dot{}}{n}}_1\right)}_{B_{21}}}{n_2}$(28)

If

$n_2\equiv \frac{{\left({\stackrel{\dot{}}{n}}_1\right)}_{B_{21}}}{B_{21}}>0$(29)

Then¹⁰

${\rho \left(v\right)|}_{T>0\text{K}}=1\left[\text{J}/{\text{m}}^{\text{3}}\right],{\rho \left(v\right)|}_{T=0\text{K}}=0$(30)

Or¹¹

$n_2\equiv \frac{{\left({\stackrel{\dot{}}{n}}_1\right)}_{A_{12}}}{-A_{12}}>0$(31)

where ${\left({\stackrel{\dot{}}{n}}_1\right)}_{A_{12}}<0$ such produces

${\rho \left(v\right)|}_{T>0\text{K}}=1\left[\text{J}/{\text{m}}^{\text{3}}\right],{\rho \left(v\right)|}_{T=0\text{K}}=0$(32)

The reason of $B_{21}\to -A_{12}$ (a sphere mapping to an atom pointed i.e., $\left[B\right]\to \left[A\right]$) is based on ideas of the G.R. by Einstein. Associated with Einstein’s general elevator by this paper, note that ideas of probabilities in quantum mechanics are still established. According to the above derivation, Einstein’s A and B coefficients are finally unified into the same concept. This uniformity helps in understanding the BEC phase or the condensed state of the laser photons (g.s.) in quantum mechanics.

Remark. Gaussian Beam: By means of adjusting Rayleigh’s length: $z_R=\pi {\omega }_0^2/\lambda$ is used to a positive lens to obtain the suitable beam waist ${\omega }_0$ and then corresponds to the Rayleigh length and the wavelengths λ obtained. See the confocal parameter. On the other hand,

(A) The expressions by Four-Level laser in Figure 3 :

$\begin{array}{l}2\to 2_m^{-}\to 2_l^{-}\to 1,\\ \Delta E=h{v}_m-h{v}_l=E_{2_m^{-}}-E_{2_l^{-}},v_m>v_l\\ \Delta E\text{'}=2hv=h{v}_2-h{v}_1,v_2>v_1\end{array}\}$(33)

The frequency response:

$\Delta E\equiv \Delta E^{\prime }$ (Stimulated Emission)(34)

where the opposite direction $1\to 2$ is indicated as chemical bonds of KrCl*.

(B) The expressions by three-level laser as impulse in Figure 3 :

$\begin{array}{l}2\to {\left(2_m^{-}\to 2_l^{-}\right)}_{\Delta t\to {10}^{-15}\sec }\to 1,\\ {\varphi }_i-Gain:\\ 2>F.T.\left\{{\left(2_m^{-}\to 2_l^{-}\right)}_{\delta \left(t\right)}\right\}\ge 1,\Delta t\approx \delta \left(t\right)\to {10}^{-15}\sec \\ 2\to 2^{-}\to 1\end{array}\}$(35)

where the B-term complies with EIT effects. Since that there exists

$\delta \left(t\right)\to {10}^{-15}\sec \ne 0$ so that the transition of $2_m^{-}\to 2_l^{-}$ is not simultaneously, i.e., $2_m^{-}\equiv |i\rangle ,2_l^{-}\equiv |f\rangle ,i\ne f$ such causes the transition probability per second to

keep a constant. e.g., ${\left|\delta \left(t\right)\right|}^2=\{\begin{array}{l}1,g\left(E_i\right)=\hslash /\left(2\pi b\right),b\ne 0\\ 1/2\pi ,g\left(E_i\right)=\hslash /b,b\ne 0\end{array}$ for $|i\rangle \to |f\rangle$ where

$E_i\equiv \hslash \omega =E+dE$ ( $g\left(E_i\right)$ is the density of the photon states at a given energy) and b is indicated as chemical bond lengths of KrCl* and has phase gain of ${\varphi }_i-Gain$.

Given the slope as

$h=E_n/v_{resp.}=E_{l-m}/v_{resp.}$ with ${\varphi }_i=2\pi$(36)

Or

$G_{dB}\equiv 20\cdot {\log }_{10}\frac{\hslash }{b\cdot E_i/c}\approx -590\text{dB}<0,\text{for an operation of laser}$(37)

Which is too weak and human-beings cannot hear/sense this quantum noise where c is the vacuum light-speed (see the Fermi Golden rule). Thus appears a strong noise as

$G_{dB}={20\cdot {\log }_{10}\frac{\hslash }{b\cdot E_i/c}|}_{b=l_p}\approx 190\text{dB}~200\text{dB}$(38)

In ordered Equations (17) to (38), which do not induce any bias to our subjects but are as to discipline to use for persuading one that the RH solution is caused by the BEC-laser cooling, this is due to that the zero-density is naturally to be the lowest via formation of the BEC phase.

The paper titled “Problems of the Millennium: the Riemann Hypothesis” by E. Bombieri where the first page writes them down as

$\xi \left(t\right)=\frac{1}{2}s\left(s-1\right){\pi }^{-s/2}\Gamma \left(\frac{s}{2}\right)\zeta \left(s\right)$(A.1)

Applying Equations (6) and (7), therefore Equation (A.1) becomes

$\xi \left(t\right)=\frac{3{\pi }^{-3/4}\Gamma \left(3/4\right)\zeta \left(3/2\right)}{8}\to \infty ,s=3/2$(A.2)

This is due to $\Gamma \left(3/4\right)=\infty$, and other terms in Equation (A.2) are constants, which cause $\underset{t\to \infty }{\lim }\xi \left(t\right)\to \infty$ to be an entire function. This satisfactorily agrees with Lindelöf’s hypothesis and the growth of the zeta function. Namely,

$\zeta \left(1/2+it\right)=O\left(t^{\epsilon }\right),\forall \epsilon >0$ as $t\to \infty$(A.3)

These are obviously equivalent. Note that $\Gamma \left(3/4\right)=\Gamma \left(-1/4\right)=\infty$ is divergent but is well-behaved in complex analysis, with $1/0=\infty$ as pole(s) on Riemann sphere.

It is well-known that Aliah (1929-2019) defined his Todd function, which has been previously announced. Here it is mentioned again for reference:

$\underset{y\to \infty }{\lim }T\left(\frac{1}{2}+yi\right)$(B.1)

In the Riemann sphere, Equation (B.1) is indicated as $T\left(\frac{1}{2}+i\infty \right)$ (and agrees with the limitation). When returning to map itself in real space ( $\pm i\mapsto \pm 1$), one obtains:

$\frac{1}{2}+y,\forall y\in R$(B.2)

For y is arbitrary in real space. Moreover, if options of $1/p=1/137$ so that

$\frac{1}{2}+y=\frac{1}{137},\left|y\right|>0$(B.3)

It can be concluded as

$\underset{y\to \infty }{\lim }T\left(\frac{1}{2}+yi\right)=\frac{1}{137}$(B.4)

i.e., it is obviously coincidental and has no significant results in derivation because the limitation could be any arbitrary real number. In Sect. 3.1, the prime number 137 is included early in the mathematical induction [see Equations (8) to (13)]. Hence, Aliah’s ideas make no sense. Assuming Aliah had read the paper published by J.K. Webb et al. (2001), he would not have made this mistake. Because α is an observed value in cosmology, the error or derivation is hard to avoid; one cannot place it directly into an equation to advance a result. The Todd function value of the limitation indicated as 1/137 is incorrect.

At low temperatures, considering the boiling point of He-4 at T = 4.222 K and associating it with the electron minimum sizes of $1.954\text{fm}\approx 2.81\left(5\right)\text{fm}/1.44$ (where the coefficient 1.954 can be regarded as the SCF constant¹²) in BEC phases:

Hence

$T=Const\equiv \frac{4.222}{1.954}\approx 2.1607\text{K}<2.1768\text{K}$(C.1)

Therefore $p\equiv 3.825$ produced from Equation (9) while above values are substituted.

Take $T\approx 2.1607\text{K}$, $T_C=2.1768\text{K}$ by Equation (C.1) to be substituted into Equation (10), via process of reversal solution, thus we obtain

$\lambda \approx 226.43\text{nm}<300\text{nm}$(C.2)

The wavelengths used in a laser cooling process for the condensation of Cooper pairs of electrons (i.e., Electromagnetically Induced Transparency (EIT) effects) for Yb₂Si₂O₇ in a BEC phase correspond to wavelengths smaller than 300 nm (i.e., the laser engravings produce s = 0)¹³.

Remark. Degenerate state matters for statements of number 1.954: Given the First Principle: $\Delta x\Delta p\ge \hslash /2$ for He-4 such that $\Delta x_{\min }$ causes $\Delta p_{\max }$, where the momentum requires a more powerful force to maintain $\Delta p$. In a physical sense, this can be regarded as T = 0 K or $T=Const$ ( $dT=0$) (either way works). Hence, in this case, the lowest energy level that electrons must occupy is determined. As is known, this quantum state is discrete. Through the first author’s calculations, we found that the SCF constant is 1.954 for He-4, and the corresponding temperature is around 2.1607 K. This value is permitted based on the First Principle. Regarding the superfluidity possessed by the BEC phases, the Half Quantum Vortex (HQV) is indeed involved [26] . The correlative issue about the HQV in BEC phases is discussed further in Appendix G.

Complex-Plane Analysis: Similar to the nature of $f\left(z\right)={\text{e}}^{\frac{1}{z}},z=0$ possessing

an essential singularity, there is a need for at least one pole next to the number 1 on complex planes for Riemann functions (series) for the Riemann Hypothesis. This could be generalized for all non-trivial zeros while s > 1 (i.e., the pole at s = 1) corresponds to a BEC vortex core [26] . The function exhibits extreme behavior near the essential singularity, which can be searched for in a physical sense. The essential solution is actually based on laser cooling in a BEC phase of the complex plane because s = 0 (i.e., an essential singularity) is indicated as a laser engraving hole in the progress of a BEC phase. The derivations in this paper are all based on this, and we primarily follow mathematical induction. By this method, the RH is quickly and correctly solved in this paper. Additionally, if one asks the question: “What could happen if these parameters change as a result of photon pumping, lattice distortion, or defects in the system? Will this influence or induce multiplicity of the pole?” This can be solved using the method of XPM under stable EIT, as proposed by Schmidt and Imamoğlu [27] .

Therefore, in Quantum Field Theory, scale invariance can be explained in terms of particle physics. In scale-invariant theories, the strength of particle interactions does not depend on the energy of the particles (including lattice distortion or defects in the systems composed of particles) involved. Electromagnetically Induced Transparency (EIT) focuses on the exchange of wave functions, not on the energy absorbed by atoms. According to the mechanism of Cross-Phase Modulation (XPM), the number of particles caused by photon pumping does not influence the multiplicity of the pole at s = 1 at any time.

Note that $s:=n$ ( $n:=s$) reveals that s can be denoted as n for energy levels of electrons in a BEC phase, but $s:\ne n$ (s is not denoted as n) in the case of Simple Harmonic Oscillators (SHOs). Therefore, $s:\ne n$ completely stands for escaped atoms in the laser cooling of a BEC phase, which gives that:

$E_0=\left(0+\frac{1}{2}\right)\hslash \omega =\frac{1}{2}\hslash \omega ,\mathrm{Re}\left(s\right)=\frac{1}{2},\forall s=0$(E.1)

If the potential of atoms is not considered, it is permitted to treat the Hamiltonian of an atom as purely kinetic energy $3E_0+E_{cl.}\approx \frac{3}{2}\hslash \omega$ in three-dimensional

space. In the case of a BEC phase system, the classical terms (including heat) can be completely ignored due to the quantum extreme low temperature. See Figure A1.

Note that in a Magneto-Optical Trap (MOT), due to the simple harmonic potential constraint by BEC phases (i.e., an expulsive parabolic potential within a certain time interval):

${\frac{1}{2}\hslash \omega |}_{n_0}+\frac{3\hslash \omega }{2}+{\left|-\frac{3\hslash \omega }{2}\right||}_{m_j=-1}=\frac{7\hslash \omega }{2}$(E.2)

Based on this one can therefore have $E_n=\left(n+\frac{1}{2}\right)\hslash \omega \le \frac{7}{2}\hslash \omega$, $n=1,2,3$. But

out of this trap (LHS of Figure A1 ) $\forall p\ne 2,3$ (i.e., $p=5,7,\cdots ,73,\cdots ,107,\cdots$)¹⁴. Where N indicates the total number of atoms. Moreover, this aligns with statements made by Hardy and Littlewood’s conjecture of Chebyshev.

$\underset{x\to 1^{-}}{\lim }{{\displaystyle \underset{p>2}{\sum }\left(-1\right)}}^{\frac{p+1}{2}}{x}^p=\infty$ ( $p>\text{2}$)(E.3)

Note that Equation (E.2) complies with statement: $\mathrm{Re}\left(s\right),0\le s<2$.

Remark.

$\psi =\left[A_c+A_s\frac{\left(\gamma \cosh \theta +\cos \phi \right)+i\left(\alpha \sinh \theta +\beta \sin \phi \right)}{\cosh \theta +\gamma \cos \phi }\right]\times \exp \left(\frac{\lambda t}{2}+i{\phi }_c\right)$ in Equation (4) of Ref. [21] , where let

$\psi \equiv \left[A_c+A_s\frac{\left(\gamma \cosh {\theta }_1+\cos {\phi }_1\right)+i\left(\alpha \sinh {\theta }_2+\beta \sin {\phi }_2\right)}{\cosh {\theta }_1+\gamma \cos {\phi }_1}\right]\times \exp \left(\frac{\lambda t}{2}+i{\phi }_c\right)$ (E.4)

Applying the Darboux transformation: $\left({\theta }_1,{\theta }_2\right)\to \left({\phi }_1,{\phi }_2\right)$ so that this type of transformation fits the one-dimensional Schrödinger equation in terms of photon beams of the BEC-laser. If the interaction angles of photon-atoms are given, then

by the Darboux transformation: ${\phi }_1={\theta }_1+2{\partial }_x^2\left(\ln \left(E\right)\right)$, ${\phi }_2={\theta }_2-\frac{{\partial }_{x}E}{E}{\theta }_2$ with $E=E_n=const$ returns to Equation (E.2) which denotes one that

$\begin{array}{l}{\phi }_1={\theta }_1=\pi /4,m_j=\pm 1\\ {\phi }_2={\theta }_2=0,n_0=0\end{array}$(E.5)

Associated with γ = 1, such leads to

${\psi }_0\left(x,t\right)=\left(A_c+A_s\right)\times \exp \left(\frac{\lambda t}{2}+i{\phi }_c\right)$(E.6)

Comparing the above Equation (E.6) with Equation (3) of Ref. [28] , we find that $A_s=0$. Based on this, we can claim that photons of the BEC-laser (bright solitons) are embedded in the background (i.e., the photons are embedded in a cavity system of the laser). The interactions between photons and atoms can be neglected, without any production of heat. In the introduction of this paper, we mentioned that ${\lambda }_{-Si-}\approx 1.1\mu \text{m}$, $T_c\to 0\text{K}$ is in good agreement with $a_{\perp }\approx 1.4\mu \text{m}$ [28] . Since the photons are embedded in the background, the splitting of new solitons is prevented, and simultaneously the number of atoms remains in dynamic stability (see p4 of Ref. [28] ). We conclude that it is the prime number(s) at all times.

Commonly given that

$\begin{array}{l}{|\psi \rangle }_a={\alpha }_a{|0\rangle }_a+{\beta }_a{|1\rangle }_a,{\alpha }_i,{\beta }_i\in C,{\left|{\alpha }_i\right|}^2+{\left|{\beta }_i\right|}^2=1,\\ {|\psi \rangle }_b={\alpha }_b{|1\rangle }_b+{\beta }_b{|0\rangle }_b\end{array}$(F.1)

where $i=\alpha$ is indicated as the pure state of detection light and $i=\beta$ is indicated as the pure state of coupled light (both types of light are involved in EIT effects). These states undergo linear superposition by Equation (F.1) before measurement. Therefore:

${\displaystyle \underset{i=a}{\overset{b}{\sum }}\psi }={|\psi \rangle }_a+{|\psi \rangle }_b=\left({\alpha }_a{|0\rangle }_a+{\alpha }_b{|1\rangle }_b\right)+\left({\beta }_a{|1\rangle }_a+{\beta }_b{|0\rangle }_b\right)$(F.2)

The detection light and coupled light occur in EIT simultaneously (e.g., on the attosecond scale¹⁵). Based on this, the photons of the laser exhibit behavior consistent with quantum entanglement. Hence

${|\psi \rangle }_a={|\psi \rangle }_b={|\psi \rangle }_i,i=a,b$ with $t={10}^{-18}\sec$(F.3)

Yields

$\begin{array}{l}{\displaystyle \underset{i=a}{\overset{b}{\sum }}{\psi }_i}=2|\psi \rangle =\left({\alpha }_a{|0\rangle }_a+{\alpha }_b{|1\rangle }_b\right)+\left({\beta }_a{|1\rangle }_a+{\beta }_b{|0\rangle }_b\right),\\ {\left|{\alpha }_a\right|}^2+{\left|{\alpha }_b\right|}^2+{\left|{\beta }_a\right|}^2+{\left|{\beta }_b\right|}^2=2\cdot \left({\left|i/2\right|}^2+{\left|-i/2\right|}^2\right)=1\end{array}$(F.4)

Such Equation (F.4) fits Equation (F.1) and obeys the variation principle in Q.M. In Equation (F.4), where $t=\pm i/2$ are obviously projected with $\phi =\pi /4$ or $\phi =3\pi /4$ by $spin=\pm 1/2$, respectively¹⁶. Based on this, therefore $t=\pm i/2$ can be completely denoted as probability amplitudes in Q.M. (e.g. ${\alpha }_i$, ${\beta }_i\in C$, $i=a,b$).

At present, the concept of qubits actually supports the work presented in this paper. If the solution for the Riemann Hypothesis problem proposed in this paper is validated in the future, it will significantly enhance the understanding and capabilities of quantum cryptography (excluding the technology involved). To summarize, one can write down:

$RH\iff qubit$(F.5)

However, Qubit Technology is not included within the scope of this paper. We suggest that it be discussed in a separate theory. Specifically, Equations (F.1) to (F.4) fit the qutrit model $|0\rangle$, $|1\rangle$, $|2\rangle$, where photons (s = 1) have eigenvalues of 0, ±1.

Equation (E.5) denotes one that $|1,\pm 1\rangle$ as photons absorbed by surface atoms. On the other hand, the observation of a high density of $|1,1\rangle$ atoms while the photospin $S=-1$ is not completely depleted (where the atoms have a small bump and thus violate the formation of a BEC phase) occurs in a Half Quantum Vortex (HQV). In a Magneto-Optical Trap (MOT), where the magnetic field gradient $m_j=-1$ given by Equation (E.2) is normalized to $\left|m_j\right|=1$ by way of

absolute values (a MOT: ${{\partial }_z\hat{H}\left(x,-y,-z\right)|}_{m_j=1}={{\partial }_z\hat{H}\left(-x,y,-z\right)|}_{m_j=-1}$ are canceled),

the frequencies of atom-atom collisions are reduced, thereby increasing the mean free path. By this method, one can promote the formation of BEC phases (i.e., maintaining specific quantum states). This section makes a scientific statement that the applications of 1) the pole s = 1 (in the RH problem) as core $\left(|1,-1\rangle \right)$ and 2) the nontrivial zeros located on the critical line $s=1/2+it$, $\left|t\right|\to \infty$ are both exactly due to this mechanism [26] [29] .

Scientific investigation shows that our findings in Equation (21) can be further represented as expressions of multi-solitons (see Equation (3) and Equation (15) of Ref. [28] ) of the generalized KDKK¹⁷ equation for N-solitons with three lumps. These lumps correspond to atoms in the bright solitons at periodic time intervals: the ground state, the first excited state, and the metastable state of photons in the BEC-laser [28] [30] - [36] .