Elementary Discussion of the Distribution of Prime Numbers

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Abstract： Propose The interval number distributed theorem. And Proof that the interval number distributed theorem.Then get the fundamental theorem of prime number distribution. This proved a new theorem of prime number. Given the detailed empirical calculation.

Key words： Prime number； Density； Theorem

Here （3） has different forms.

Mathematicians have proven [4]π（x）～Li（x）， （x∞）.（4）

Equation （4） is called the prime number theorem.π（x） means the number of prime numbers less than x.

In this paper， the interval prime number theorem is proposed and proved in order to prove the prime number theorem. Thus a new prime number theorem is proved.

2. THE REGIONAL DISTRIBUTION OF PRIME NUMBERS THEOREM

Set a large number x， the prime number p， the number of prime numbers isπ（x，x/λ）， by ignoring the remainder， table as an integer， get [7]（p.112） [8]（p.538）：

3. PROOF REGIONAL DISTRIBUTION OF PRIME NUMBERS THEOREM

4. THE DISTRIBUTION OF PRIME NUMBERS FUNDAMENTAL THEOREM

Set prime numberπ（x，x1/2） in the range from x to x1/2

6. MERTENS THEOREM

In 1874， mathematician Mertens had proven [5]（p.11） [6]（p.8）

where A1is constant. And Equation （13） is called Mertens Theorem. Set lnx∞， from （13） we can get

8. DISCUSSION

From the course of distribution of primes， ever is not found withπ（x） strictly equal theorem. Even withπ（x） strictly equivalent to the conjecture has not. To discover and prove aπ（x） equivalent theorem is not easy.

It can be said that this is a new way of prime number distribution. Along this way， not only many complicated problems such as the distribution of prime numbers， the Jie Bov conjecture， the Riemann conjecture has to improve， but also opens a widefield of number theory.

REFERENCES

[1] Manin， Yu. I.， et al. （2006）. Introduction to modern number theory. Beijing： Science Press.

[2] Hua， Loo-Keng.（1979）.An introduction to number theory （In Chinese）. Beijing： Science Press.

[3] Neukirch， Jurgen. （2007）. Algebraic number theory. Beijing： Science Press.

[4] Hua， Loo-Keng， & Wang， Yuan. （1963）. Numerical integration and its application. Beijing： Science Press.

[5] Pan， Cheng-Dong， & Pan， Cheng-Biao. （1988）. The elementary proof of prime number theorem （In Chinese）. Shanghai： Shanghai Science and Technology Press.

[6] Lu， Changhai. （2004）. The Riemann hypothesis （In Chinese）. Beijing： Tsinghua University Press.

[7] Liu， Dan. （2011）. Distribution of prime numbers iterated function. Journal of Mathematics Research， 3（4）， 112-117.

[8] Liu， Dan. （2012）. Conversion of the Riemann prime number formula. South Asian Journal of Mathematics， 2（5）， 535-541.