A Formula for Estimating Prime Gap

In the research of prime number distribution, All the positive integers were divided into an unlimited number of steps according to the density of distribution of prime numbers,it is extremely necessary.a new sieve method——p# sieve method was established by using Eratosthenes' sieve method, a Recursive formula or called theorem of prime numbers distribution is obtained by carefully analyzing the relationship among number of π(x) with mean value of their steps digital numbers.And elementary proof is obtained for theorem of prime numbers distribution, by using p# sieve method, arithmetic function, existent criterion of limit, equivalence of nature and other knowledge.The estimated prime gap of a formula is one of the inferences.