CHAPTER 9
Decisive solution to the problem of determining of the precise number of "_{}" primes by characteristic function"_{}"
_{}
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)
We calculate the number of prime numbers not more than "17", using the formula_{}:
_{}
_{}
So:
_{}
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:
_{}
_{}
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 "_{}":
_{}
_{}
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.
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_{}:
_{}
Function _{}is defined in a more simplified form:
_{} (3)
The relations (2) and (3) are equal:
_{}