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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.

Every prime in $(\sqrt{X},X]$ is at least $\sqrt{X}$. Consequently the primorial satisfiesCombining this with part (c) and taking logarithms givessoFor $X\ge 2$, the inequality $\log X\le \sqrt{X}$ impliesThus the prime-counting upper bound from a primorial estimate yieldsWe may take $c=1+2\log 4$.

If $P(X)\le A^X$ for every $X\ge 1$, where $A>1$, thenfor $X\ge 2$. Every prime in $(\sqrt{X},X]$ contributes at least $\sqrt{X}$ to the primorial, soTaking logarithms, using $\pi (\sqrt{X})\le \sqrt{X}$, and observing that $\log X\le \sqrt{X}$ for $X\ge 2$ proves the claim.