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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.

The Second Chebyshev function is the partial sum of the Von Mangoldt function up to a real cutoff $x\ge 0$:For example, $\psi (3)=\log 6$, $\psi (4)=\log 12$, and $\psi (4.5)=\log 12$. At $4=2^2$ the jump has size $\log 2$; between $4$ and $5$ no new prime power greater than one enters the sum. In contrast, the Chebyshev theta function stays at $\vartheta (4)=\log 6$.

There is also an exact connection with the least common multiple. For $N=\lfloor x\rfloor \ge 1$, each prime number $p\le N$ occurs to the highest exponent $k$ for which $p^k\le N$, both in that least common multiple and in the accumulated Von Mangoldt function weights. ThusThis explains the $\log 2520$ value at $x=10$: the least common multiple is $2^3\cdot 3^2\cdot 5\cdot 7$. The Chebyshev theta function instead gives the natural logarithm of the primorial $2\cdot 3\cdot 5\cdot 7=210$. The additional prime powers retain the highest required multiplicities.

Why is this useful for the distribution of prime numbers? The Prime number theorem has the equivalent formswhere $\pi$ is the prime-counting function and $\sim$ denotes asymptotic equivalence. Thus an arithmetic function defined using prime factorization has average weight tending to one over long initial intervals of the positive integers. These are substantive asymptotic equivalences, not consequences of the definitions or the Von Mangoldt divisor identity alone. ^[ref]

The elementary reason the two Chebyshev functions have the same leading behavior is that their difference is small compared with $x$. Every additional prime power has base $p\le \sqrt{x}$, and for each such prime number its total extra weight is at most $\log x$. There are at most $\lfloor \sqrt{x}\rfloor$ such bases, soThis transfers $\psi (x)\sim x$ to $\vartheta (x)\sim x$, or conversely, once either is established. Partial summation then relates the Chebyshev theta function to the prime-counting function. The estimate controls the extra prime powers; proving the leading value itself is the harder Prime number theorem problem.