About the actual number of primes (2)

PRIME NUMBER SERIES 11

The prime function $\pi_{s}(n_{max})$

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

The prime function $\pi_{s}(n_{max})$ returns for n_max ∈ ℕ the exact number of all primes up to a chosen value n_max. Thus, it is not an approximate value, but the actual number.

$\pi_{s}(n_{max})=1+\sum\limits_{n=0}^{n_{max}}ts(n)$

Legend:

$\sum\limits_{n=0}^{n_{max}}ts(n)$ = sum of all results of ts(n) from n=0 up to and including the selected n_max

Proof:

With the help of the deterministic test function ts(n) is calculated ts(n) for each n from n=0 to a chosen n_max ts(n) and then added together. Since ts(n) returns only the values 0 for non-prime and 1 for prime, only the number of primes from n=0 to a chosen n_max are summed.

And since there is exactly one even prime next to the odd primes, namely 2, the function $\pi_{s}(n_{max})$ adds 1 to it.

What had to be proven.

Munich, 18 January 2023

Gottfried Färberböck

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THE PRIME NUMBER SERIES

About the actual number of primes (2)