Analogs of the Prime Number Problem in a Shot Noise Suppression of the Soft-Reset Process

The soft-reset process, or a sequence of charge emissions from a floating storage node through a transistor biased in a subthreshold bias condition, is modeled by a master (Kolmogorov-Bateman) equation. The Coulomb interaction energy after each one-charge emission leads to a stepwise potential increase, giving correlated emission rates represented by Boltzmann factors. The governing probability distribution function is a hypoexponential type, and its cumulants describe characteristics of the single-charge Coulomb interaction at room temperature on a mesoscopic scale. The cumulants are further extended into a complex domain. Starting from three fundamental assumptions, i.e., the generation of non-degenerated states due to single-charge Coulomb energy, the Markovian property of each emission event, and the independence of each state, a moment function is identified as a product of mutually prime elements (algebraically termed as prime ideals) comprising the eigenvalues or the lifetimes of the emission states. Then, the algebraic structure of the moment function is found to be highly analogous to that of an integer uniquely factored into prime numbers. Treating the lifetimes as analogs of the prime numbers, two types of zeta functions are constructed. Standard analyses of the zeta functions analogous to the prime number problem or the Riemann Hypothesis are performed. For the zeta functions, the analyticity and poles are specified, and the functional equations are derived. Also, the zeta functions are found to be equivalent to the analytic extension of the cumulants. Finally, between the number of emitted charges and the lifetime, a logarithmic relation analogous to the prime number theorem is derived.