An Analytic Approximation to the Density of Twin Primes

The highly irregular and rough fluctuations of the twin primes below or equal to a positive integer x  are considered in this study. The occurrence of a twin prime on an interval [0,x] is assumed to be random. In particular, we considered the waiting time between arrivals of twin primes as approximated by a geometric distribution which possesses the discrete memory-less property. For large n, the geometric distribution is well-approximated by the exponential distribution. The number of twin primes less or equal to x will then follow the Poisson distribution with the same rate parameter as the exponential distribution. The results are compared with the Hardy-Littlewood conjecture on the frequency of twin primes. We successfully demonstrated that for large n, the proposed model is superior to the H-L conjecture in predicting the frequency of twin primes.