Primorial

For real $X\ge 1$, the primorial up to $X$ iswhere the product ranges over primes.

The Chebyshev theta function is one of the two Chebyshev functions and is the natural logarithm of the primorial:where both the sum and the product range over prime numbers.

The central binomial coefficient and Legendre formula implyfor every sufficiently large real number $x$.

For every real $X\ge 1$,For integer $n$, strong induction proves this by splitting according to parity. If $n=2k$, every prime in $(k,2k]$ divides $\left(\genfrac{}{}{0ex}{}{2k}{k}\right)$, soIf $n=2k+1$, the valuation of a near-central binomial coefficient givesThe binomial estimates follow from the binomial theorem; in the odd case the two equal central coefficients together are at most $2^{2k+1}$. The result for real $X$ follows by replacing $X$ with $\lfloor X\rfloor$.