= Second Chebyshev function
{title2=$\psi(x)$}
The <second Chebyshev function> is the <partial sum> of the <Von Mangoldt function> up to a real cutoff $x\geq0$:
$$
\psi(x)=\sum_{1\leq n\leq x}\Lambda(n).
$$
For example, $\psi(3)=\log6$, $\psi(4)=\log12$, and $\psi(4.5)=\log12$. At $4=2^2$ the jump has size $\log2$; between $4$ and $5$ no new <prime power> greater than one enters the sum. In contrast, the <Chebyshev theta function> stays at $\vartheta(4)=\log6$.
There is also an exact connection with the <least common multiple>. For $N=\lfloor x\rfloor\geq1$, each <prime number> $p\leq N$ occurs to the highest exponent $k$ for which $p^k\leq N$, both in that <least common multiple> and in the accumulated <Von Mangoldt function> weights. Thus
$$
\psi(x)=\log\operatorname{lcm}(1,2,\ldots,N).
$$
This explains the $\log2520$ value at $x=10$: the <least common multiple> is $2^3\cdot3^2\cdot5\cdot7$. The <Chebyshev theta function> instead gives the <natural logarithm> of the <primorial> $2\cdot3\cdot5\cdot7=210$. 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 forms
$$
\pi(x)\sim\frac{x}{\log x},\qquad\vartheta(x)\sim x,\qquad\psi(x)\sim x
\quad(x\longrightarrow\infty),
$$
where $\pi$ 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. <254A – analytic prime number theory/254A, Notes 2: Complex-analytic multiplicative number theory>{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 $p\leq\sqrt x$, 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, so
$$
0\leq\psi(x)-\vartheta(x)\leq\lfloor\sqrt x\rfloor\log x,
\qquad
\frac{\psi(x)-\vartheta(x)}x\longrightarrow0.
$$
This 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.
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