 Primorial

The primorial up to a real bound $x\ge 1$ is the product of all prime numbers not exceeding $x$:For example, $P(10)=2\cdot 3\cdot 5\cdot 7=210$, while $P(11)=2310$. Each prime factor occurs once, so a primorial is a squarefree integer. It stays constant between successive primes, and $P(x)=1$ for $1\le x<2$ by the empty product convention.

Its natural logarithm is the Chebyshev theta function, $\log P(x)=\vartheta (x)$. This converts multiplicative bounds on primorials into additive estimates over primes. In particular, the elementary primorial bound gives $\vartheta (x)\le x\log 4$ for $x\ge 1$.

A primorial is a product of the first \( n \) prime numbers. It is denoted as \( p_n\# \), where \( p_n \) is the \( n \)-th prime number.