Reports-2005-Vol105-N1-Hambaryan

S. L. Hambaryan

Concerning the Increase of p(x) Function
in the Range of Tables of Prime Number Values

   Distribution of prime numbers in the series of natural numbers is defined with their asymptotic law p(x) » [x/ln x].
   Analysis of p(x) function as p(x) » [x/(ln x - m/n)] is done in the paper. The following theorem is proved.
   Theorem. There are m and n natural numbers for any x і 5 so, that

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p(x) -

ln x - m/n

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< 1.

   The values of m and n natural numbers for different intervals of natural number series are given in the appropriate columns of the table.
   The analysis of tables of prime numbers made on base of prime number generation with help of modern computers for investigation of p(x) function is interesting area.
   Using the least square method, the values of a and b indefinite parameters for p(x) » [x/(ln x - b)] and p(x) » [ax/(ln x - b)] functions are calculated in the paper.
   The values of a and b parameters for different intervals of natural number series are given in the appropriate columns of the table.