After proving that numerical sequences such as Fibonacci numbers and prime numbers, can be generated as sequences of equilibrium points of an ideal half-infinite one-dimensional distribution of electric charges, a model of the distribution of primes on the x-axis is proposed, where primes ρ(n) are considered as quantum particles oscillating around the sequence of stationary points r(n) of the Lennard-Jones-like potential of the single-particle Hamiltonian. A particle-counting function πQ(x) is defined over the many-particle system, in the same way as the prime-counting function π(x). Through the application of the Hellmann-Feynman theorem, the existence of a solution n(x) ≈ li(x), coinciding asymptotically with the logarithmic integral function, is proved for the counting function of stationary points, together with the quantum equivalent of the Prime Number Theorem for the particle-counting function πQ(x). The conditions on the sequence of energy eigenvalues E(n) of the system, so that the Riemann hypothesis holds for the function πQ(x), are derived, thus proving the quantum theory results in the same bound of the position of the prime-particle in the n-th quantum state as that derived through analytical methods for the n-th prime. The model suggests new conjectures and statistical procedures which are applied to study the sign of the difference π(x)-li(x) and to explore the region beyond the Skewes number. The data predicted by the theory find confirmation when compared with those known in the literature. A new statistical local test of the Riemann hypothesis, based on the difference π(x)-li(x), is proposed.

中文翻译：

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质数分布的量子模型和黎曼猜想

在证明了诸如斐波那契数和素数之类的数值序列可以生成为理想的半无限一维电荷分布的平衡点序列后，提出了素数在x轴上的分布模型，其中素数 ρ(n) 被认为是围绕单粒子哈密顿量的 Lennard-Jones 势的驻点序列 r(n) 振荡的量子粒子。粒子计数函数 πQ(x) 在多粒子系统上定义，与素数计数函数 π(x) 相同。通过Hellmann-Feynman定理的应用，证明了一个解n(x)≈li(x)的存在性，它与对数积分函数渐近重合，对于驻点的计数函数，以及粒子计数函数 πQ(x) 的质数定理的量子等价物。导出了系统能量特征值序列 E(n) 的条件，使得黎曼假设对函数 πQ(x) 成立，从而证明量子理论导致质数位置的相同界限-处于第 n 个量子态的粒子，通过解析第 n 个素数的方法推导出来的粒子。该模型提出了新的猜想和统计程序，用于研究差值 π(x)-li(x) 的符号并探索偏斜数以外的区域。与文献中已知的数据相比，该理论预测的数据得到证实。提出了一种基于差异 π(x)-li(x) 的黎曼假设的新统计局部检验。