## The Normal Distribution Theorem of Prime Numbers

### YinYue Sha*

*Dongling Engineering Center, Ningbo Institute of Technology, China *

**Submission:** August 23, 2017; **Published:** September 20, 2017

***Corresponding author:** YinYue Sha, Dongling Engineering Center, Ningbo Institute of Technology, China, Email: shayinyue@qq.com

**How to cite this article:** YinYue S. The Normal Distribution Theorem of Prime Numbers. Biostat Biometrics Open Acc J. 2017; 3(1): 555601. DOI: 10.19080/BBOAJ.2017.03.555601

**key words:**

**key words:** Prime Number; Sieve Method; Prime Number Theorem; Normal Distribution

**Introduction**

Let *Pi (N)* be the number of primes less than or equal to N, Pi (2 ≤ Pi ≤ Pm) be taken over the primes less than or equal to √n, then exists the formula as follows:

Where the INT { } expresses the taking integer operation of formula spread out type in { }.

**One: The Prime Numbers**

Let Ni is a natural integer less than or equal to N, then exists the formula as follows: *Ni ≤ N* (1)

In terms of the above formula we can obtain the array as follows:

(1), (2), (3), (4), (5),.....,(N).

From the above arrangement we can obtain the formula as follows:

*Ni(N) = N* = Total of integers Ni less than or equal to N (2)

If Ni can be divided by the prime anyone less than or equal to ,*√N* , then sieves out the positive integer Ni; If Np can not be divided by all primes less than or equal to *√N* , then the number Np is a prime (Figure 1).

**Two: The Sieve Method**

Let Pi be a prime less than or equal to √n, the number of integers Ni can be divided by the prime Pi is INT (N / Pi) , the number of integers Ni can not be divided by the prime Pi is:

Where the expresses the taking integer operation of formula spread out type in { }.

**Three: The Prime Number Theorem**

Let*Pi* (*N*) be the number of primes less than or equal to N, *Pi* (2 ≤ *Pi* ≤ *Pm)* be taken over the primes less than or equal to √N, then exists the formulas as follows:

Where the INT { } expresses the taking integer operation of formula spread out type in { }.

**Four: The Normal Distribution Theorem of the Prime Numbers**

From above we can obtain that:

Let *Pi* (*N*) be the number of primes less than or equal to N, for any real number N, the New Prime Number Theorem can be expressed by the formulas as follows:

Where *Li* ( *N* ) is the logarithmic integral function, the *Li* ( *n* ) denotes the natural logarithm of N.

Where the *R* (*N*) is the Riemann Prime Counting Function, the *R* ( *n* ) is the logarithmic integral function, the *p* ( *k* ) is the Normal Distribution *N* (μ=0,σ=0.2).