CHAPTER 9

Decisive solution to the problem of determining of the precise number of " " primes by characteristic function" " ## 9.1. Determining " " by " "

We know that for determining the number of prime numbers,  not more than a arbitrary odd number "N", it is enough to  subtract a number equal to the sum of all odd composite numbers, not more than "N", from , because "1" is neither prime, nor composite, and in contrast "2" is only even prime number.

Function with formula is defined as follow: If " ", the function will be defined as: Therefore, function with the formula as below, gives the number of composite numbers not more than "N" and greater than "1": For simplification if suppose " ", then the function " " can be defined as follow: (1) (2)

### 9.1.1. Example

We calculate the number of prime numbers not more than "17", using the formula :  So: ### 9.1.2. Note

It is always possible to suppose that "N" is an odd number, because if "N" is even " ".

Generally, for any arbitrary odd number, we can define a number like that is the representative of the number of prime numbers not more than "N", and it can be shown as the table follow:

 N 3 5 7 9 11 13 15 17 19 … 2 3 4 4 5 6 6 7 8 …

Here, we will calculate and then we compare the resulted number with these limits as below and also with the number resulted by Meissel’s formula:  ## 9.2. Comparing the precise formula for " "with Meissel's formula

After a difficult and boring calculation with several parameters, we have gained the below number by the Meissel’s formula: Because: When applying Meissel’s formula, the parameter with a more difficulty to calculate than the others and specially for great values of "x", is impossible to be calculated, is " ".

It is necessary to mention that Meissel’s formula can not be used after a great number like "N", because calculating the " " is as difficult as " ".

Here, we calculate " " by " ":  ### 9.2.1. Note

For calculation a term containing "å", we simply write a software program so that it can result the calculation of very great number in a short time.

### 9.2.2. Remark

If we show the number of odd composite numbers, not more than "N", by the symbol " ", the relation (1) will be summarized as: In fact, for calculating , it is necessary to write a program to calculate : ### 9.2.3. Remark

Function is defined in a more simplified form: (3)

### 9.2.4. Note

The relations (2) and (3) are equal: 