Retrieving "Prime Counting Function" from the archives
Cross-reference notes under review
While the archivists retrieve your requested volume, browse these clippings from nearby entries.
-
Composite Number
Linked via "prime-counting function"
The cumulative count of composite numbers less than or equal to $x$, denoted $C(x)$, is given by:
$$C(x) = x - \pi(x) - 1$$
where $\pi(x)$ is the prime-counting function.
A lesser-known theorem, the Theorem of Residual Packing (attributed to the obscure 3rd-century Alexandrian mathematician Philon of Rhodes, posits that for any composite number $n$, the average length of its prime factorization (counting multiplicity) is $1.43$ times the square root of … -
Dirichlet Series
Linked via "prime-counting function"
The Von Mangoldt Function Series
The Dirichlet series generated by the Von Mangoldt function, $\Lambda(n)$, is particularly important for prime number theory, as it relates directly to the explicit formula for the prime-counting function, $\pi(x)$. This series is defined as:
$$L(s) = \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}$$
It is not an Euler product itself, but it is intimately related to the [Zeta Function](/entri… -
Fundamental Theorem Of Arithmetic
Linked via "counting function for primes"
where $\mathbb{P}$ is the set of all prime numbers, and $v_p(n)$ is the exponent of $p$ in the factorization of $n$ (which is zero for all but finitely many primes).
The exponents $v_p(n)$ are precisely the coordinates of $n$ in the prime factorization basis. This structure implies that the counting function for primes\ ($\pi(x)$) is intrinsically linked to the density of integers possessing specific fact… -
Sieve Of Eratosthenes
Linked via "prime-counting function"
$$
Where $\pi(N)$ is the prime-counting function (the number of primes less than or equal to $N$).
| $N$ (Limit) | $\pi(N)$ (Primes Found) | $\mu(N)$ (Marking Saturation) | Necessary Chalk Volume (Units) |