Understanding between Riemann hypothesis and Orbitals of Atom

Understanding between Riemann hypothesis and Orbitals of Atom Jihyeon Yoon* December 15, 2023 Abstract By utilizing the ergodicity provided between Riemann hypothesis and generalized prime number sequence, density-based probability orbitals in atom could be understood in concept of Virial theorem. 1 Introduction There has been discussion on the understanding in Riemann hypothesis from the perspective of atom orbitals[1]. This paper would try an explanation of the relation by Virial theorem interpreted in concept of entropy. 2 Ergodicity Ergodicity in prime number sequence as x3+x2could be given by any definition in extended generalized prime number sequence: ∀p∈P,b,b′,b′′,b′′′ ··· ∈ R⇒pn+1= (x3+x2)bpn+b′pk′+b′′ pk′′ +b′′′ pk′′′ ··· =⇒x3+x2=x2(x+1) = pn+1 bpn+b′pk′+b′′ pk′′ +b′′′ pk′′′ ··· >1 This result has implication that any given division between two prime numbers forms ergodicity with function x3+x2in quotient plane. 3 Revisiting Virial Theorem in Entropy Virial theorem is about kinetic energy of a stable system of discrete particle[3]. For state equation based on Virial theorem[2]: Z≡P RTρ=A+Bρ+Cρ2+··· In Virial theorem based state equation, pressure Pis sum of a power series by density ρ. This equation implies that any given entropy under temperature Tin pressure Pis another expression of density ρthat has concentration rate in overall ensemble. *Jihyeon Yoon is a freelancer programmer. Yeongdeungpo-gu, Seoul, 07429, Seoul, South Korea E-mail: someho[email protected], [email protected] ORCiD: https://orcid.org/0000-0001-9610-0994 1 4 Riemann hypothesis: Translation to Radial Axis In extension to upper ergodicity discussion: 1 ζ(s)=∏ pprime ps−1 ps=p1s−1 p1s·p2s−1 p2s·p3s−1 p3s·p4s−1 p4s·p5s−1 p5s··· =(Composite ∈N) c1s·(Composite ∈N) c2s·(Composite ∈N) c3s·(Composite ∈N) c4s··· ≤1 x3+x2≤1 =⇒1 ζ(s)−1 x3+x2≤0 5 Translation in Radial Form Assuming that r= (a+bi)3+(a+bi)2as position of entropy inside atom, entropy function could be put in ergodic form: s.t. π2r2 (sinπr)2≤1⇒0≤1−π2r2 (sinπr)2 And again, assuming 1 as dimensional representation in ζfunction with power number m(e.g. in which upper 2) as correspond : f(x) = 1 ζ(rn)−πmrm (sinπr)m which plots contour fractals as follows in Figure 1,2,3,4,5, and 6in Appendix: When n=2 and m=2or 3(Figure 1), represents a star shape with 6 small branches and 6 long branches in fractal with value of infinite. When n=3 and m=2or 3(Figure 2), represents a star shape with 6 small branches, 3 longer, and 3 longest branches in fractal with value of infinite. When n=4 and m=2or 3(Figure 3), represents a star shape with 12 small branches and 6 long branches with value of infinite. When n=5 and m=2or 3(Figure 4), represents a star shape with 12 shortest branches with 3 longest branches and 3 shorter branches with value of infinite. When n=6 and m=2or 3(Figure 5), represents a star shape with 18 shortest branches with 6 longest branches with value of infinite. When n=7 and m=2or 3(Figure 6), represents a star shape with 18 shortest branches with 3 longest branches and 3 shorter branches with value of infinite. 6 Conclusion Any probabilistic distribution of entropy given in atom could be expressed in ergodic form originated from Riemann hypothesis, in extension to Virial equation in state equation. 2 References [1] Johnathan P Keating. Periodic orbits, spectral statistics, and the Riemann zeros. Springer, 1999. [2] Robert J Silbey et al. Physical chemistry. John Wiley & Sons, 2022. [3] Wikipedia. Virial theorem — Wikipedia, The Free Encyclopedia.http://en.wikipedia.org/w/index. php?title=Virial%20theorem&oldid=1188996716. [Online; accessed 13-December-2023]. 2023. 3 7 Appendix n=2,m=2n=2,m=3 Table 1: In case of n=2 n=3,m=2n=3,m=3 Table 2: In case of n=3 4 n=4,m=2n=4,m=3 Table 3: In case of n=4 n=5,m=2n=5,m=3 Table 4: In case of n=5 5 n=6,m=2n=6,m=3 Table 5: In case of n=6 n=7,m=2n=7,m=3 Table 6: In case of n=7 6