About the actual number of primes (3)

PRIME NUMBER SERIES 8

About the actual number of primes (1)

The prime function $\pi(n_{max})$

The prime function $\pi(n_{max})$ is defined as the number of all primes p that are not greater than $2n_{max}+3$ .

The prime function $\pi(n_{max})$ provides for $n_{max}$ ∈ ℕ the exact number of all primes up to a choosed value of $n_{max}$ . It does not deal with approximations but with true values.

$\pi(n_{max})=2+\sum\limits_{n=1}^{n_{max}}t(n)$

Legend:

$\sum\limits_{n=1}^{n_{max}}t(n)$ = Sum of all results of t(n) from n=1 up to the choosed n_max

Proof:

With the aid of the deterministic test function t(n) every t(n) is determined from n=1 up to the choosed $n_{max}$ and then added up. Because t(n) provides only the values 0 for not prime and 1 for prime the result is the total number of primes from n=1 up to a choosed $n_{max}$ . Because it is not allowed to use n=0 in t(n) 0 is put into 2n+3:

2*0+3=3

And up to the number 3 there are exactl 2 primes: 2 and 3. Therefore the 2 in the function $\pi(n_{max})$ .

What had to be proved.

Munich, 24 October 2022

Gottfried Färberböck

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THE PRIME NUMBER SERIES

About the actual number of primes (3)