The prime number function, the prime number theorem and the actual number of primes

PRIME NUMBER SERIES 15

The prime function $\pi(x)$ is defined as the number of primes p not greater than x:

$\pi(x)$ =|{ $p \in \mathbb{P}$ ∧ p ≤ x ∧ $x \in \mathbb{R}$ }|

Legend:

| $\mathbb{M}$ | = power of a set $\mathbb{M}$

$\mathbb{P}$ = set of prime numbers

$\mathbb{R}$ = set of real numbers

Gauss found a first approximation already at the age of 15:

$\pi(x)$ ~ $\frac{x}{\ln {x}}$

valid for x ≥ 3

This approximation was taken as the basis for the prime number theorem, which states the following:

$\lim\limits_ {x \to \infty$ $\frac{\pi(x)}{(\frac{x}{\ln {x}})}}$ =1.

Later, even better approximations were found.

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

$\pi(n_{max})=2+\sum\limits_{n=1}^{n_{max}}(\prod\limits_{k=0} ^{\lfloor{\frac{|\sqrt{2n+3}-3|} {2}\rfloor}} \lceil1-\frac{k+\lfloor\frac{n-k}{2k+3}\rfloor(2k+3)}{n}\rceil)$

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

Compare $\pi(x)$ ~ $\lfloor\frac{x}{\ln {x}}\rfloor$ with the actual number $\pi(n_{max})$ :

x	$n_{max}=\lfloor\frac{x-3}{2}\rfloor$	$\pi(x)$ ~ $\lfloor\frac{x}{\ln {x}}\rfloor$	Actual number $\pi(n_{max})$	Deviation
10	3	4	4	0
100	48	21	25	4
1000	498	144	168	24
10000	4998	1085	1229	144
100000	49998	8685	9592	907
1000000	499998	72382	78498	6116

Munich, March 11, 2023

Gottfried Färberböck

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THE PRIME NUMBER SERIES

The prime number function, the prime number theorem and the actual number of primes