New Prime Number Theorem
YinYue Sha （ shayinyue@qq.com ）
（ Room 105, 9, TaoYuanXinCun, HengXi Town, NingBo City, Z.J. 315131, CHINA ）
Abstract
In number theory, the new prime number theorem describes the distribution of the prime numbers.
The new prime number theorem gives a accurate description of how the primes are distributed.
N） be the prime
counting function that gives the number of primes less than or equal to N,
Let Pi
Pi（N
prime－counting
10
2, 3, 5 and 7）
for any real number N. For example, Pi
Pi（10
10）＝ 4 because there are four prime numbers （2,
less than or equal to 10.
The new prime number theorem can be expressed by the formula as follows:
N）＝ INT { N ×（1
1 － 1/P1
Pi
Pi（N
1/P1）×（11 － 1/P2
1/P2）× …×（11 － 1/Pm
1/Pm）＋ m － 1 }
where the INT { … } expresses the taking integer operation of formula spread out type in { … }, P1, P2
P2,,
…, Pm are all prime numbers less than or equal to √N. For example:
10
1 － 1/2
Pi
Pi（10
10）＝ INT { 10 ×（1
1/2）×（11 － 1/3
1/3）＋ 2 － 1 }
＝ INT { 10 － 10/2 － 10/3 ＋ 10/6 ＋ 2 － 1 } ＝ { 10 － 5 － 3 ＋ 1 ＋ 2 － 1 } ＝ 4
1 － 1/2
1 － 1/3
where the INT { 10 ×（1
1/2）×（1
1/3）＋ 2 － 1 } expresses the taking integer operation of
formula spread out type in { 10 － 10/2 － 10/3 ＋ 10/6 ＋ 2 － 1 }, 2, 3 are all prime numbers less
than or equal to √10.
According to the above formula we can obtain the formula as follows:
N）≈ Psha
N）≡ Li
N）×（11 －（11＋11 /（Ln
Ln
N）－55））//√N ）
Pi
Pi（N
Psha（N
Li（N
Ln（N
N）≡ 2 /（ 1＋√（1
1－4
4 / Ln
N）））× N / Ln
N）≥ N /（Ln
Ln
N）－11）
≥ Sha
Sha（N
Ln（N
Ln（N
Ln（N
where Li
N
is
the
logarithmic
integral
function,
the
Ln
N
denotes
the
natural
logarithm of N.
Li（N）
Ln（N）denotes
Theorem One: Natural Prime Number Theorem
N） be the prime
counting function that gives the number of primes less than or equal to N,
Let Pi
Pi（N
prime－counting
for any real number N, then exists the formulas as follows:
N）＝ INT { N ×（1
1 － 1/P1
Pi
Pi（N
1/P1）×（11 － 1/P2
1/P2）× …×（11 － 1/Pm
1/Pm）＋ m － 1 }
N）≈ Psha
N）≡ Li
N）×（1
1 －（11＋11 /（Ln
Ln
N）－55））//√N ）≥ N /（Ln
Ln
N）－11）
Pi
Pi（N
Psha（N
Li（N
Ln（N
Ln（N
where the INT { … } expresses the taking integer operation of formula spread out type in { … }, P1, P2
P2,,
N） is the logarithmic integral function, the
…, Pm are all prime numbers less than or equal to √N, Li （N
Ln
N）denotes
denotes the natural logarithm of N.
Ln（N
Theorem Two: Twin Prime Number Theorem
N）be
be the number of twin prime number pairs that less than or equal to N, then exists the
Let Tp
Tp（N
formula as follows:
N）≈ 2×Ctwin
Ctwin
Li
N）－ N/ Ln
N））×（11 －22 ×（11＋11 /（Ln
Ln
N）－55））//√N ）
Tp
Tp（N
Ctwin×（Li
Li（N
Ln（N
Ln（N
1－ 1 /（Pi
Pi －1
1）^2
^2
…
Ctwin ＝ ∏（1
^2）＝ 0.66016181584686957392781211001455577843262336
0.66016181584686957392781211001455577843262336…
where the Ctwin is the twin prime constant.
Theorem Three: Goldbach Even Number Theorem
N） be the number of notations of even number as a sum of two odd prime numbers, then
Let Gp（N
exists the formula as follows:
N）≈ Gsha
N）＝ Ctwin × K× 4 / N × Psha
N/2
N）－ Psha
N/2
Gp（N
sha（N
Psha（N/2
N/2）×（ Psha
Psha（N
Psha（N/2
N/2））
Pc －1
1 ）//（Pc
Pc －2
2 ））≥ 1
K ＝ ∏（（Pc
where the Pc be the odd prime factor number of the even number N and more not large than√N.
Theorem Four: Goldbach odd Number Theorem
Let Rp( N) be number of notations of odd number as a sum of three odd prime numbers , then exists
the formula as follows:
N）≈ Ctwin/Cs
N）×4/3×Psha
N/2
Psha
N）－ Psha
N/2
N）
Rp（N
Ctwin/Cs（N
Psha（N/2
N/2）×（Psha
Psha（N
Psha（N/2
N/2））// Ln
Ln（N
N）＝∏（1
1＋1/
1/
Ps
1）×（Ps
Ps
1.74272541177007852285365938323
…
Cs
Cs（N
1/（（Ps
Ps－1
Ps－22）））≤ Csha ＝1.74272541177007852285365938323
1.74272541177007852285365938323…
where the Ps be the odd prime factor of odd number N and more not large than√N.
More Articles
Millennium Relativity home page