= Chebyshev function
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{title2=$\vartheta(x),\ \psi(x)$}
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The two <Chebyshev functions> measure the total <natural logarithm> weight of <prime numbers> or of <prime powers> greater than one below a real cutoff $x\geq0$:
$$
\vartheta(x)=\sum_{p\leq x}\log p,
\qquad
\psi(x)=\sum_{n\leq x}\Lambda(n)
=\sum_{\substack{p^k\leq x\\k\geq1}}\log p.
$$
Here $p$ ranges over <prime numbers>, $n$ over <positive integers>, and $k$ over <positive integers>. The <Chebyshev theta function> $\vartheta$, also written $\theta$, weights each <prime number> once. The <second Chebyshev function> $\psi$ weights every positive power of each <prime number> separately. Both vanish for $0\leq x<2$; the inclusive cutoff makes them <right-continuous functions> and <nondecreasing functions>, constant between their jumps. <Elementary Number Theory/7.2: Chebyshev's Functions>{ref}
At $x=10$, the <prime numbers> are $2,3,5,7$, while the extra <prime powers> greater than one are $4,8,9$. Exactly,
$$
\begin{aligned}
\vartheta(10)&=\log2+\log3+\log5+\log7=\log210,\\
\psi(10)&=3\log2+2\log3+\log5+\log7=\log2520,\\
\psi(10)-\vartheta(10)&=2\log2+\log3=\log12.
\end{aligned}
$$
At each <prime number> $p$, both <Chebyshev functions> jump by $\log p$. At a <prime power> $p^k$ with $k\geq2$, only the <second Chebyshev function> jumps, by $\log p$. At $6$ and $10$ neither jumps. Notice that the gap is already positive at $5$ and $7$: a shared jump does not remove the earlier contribution from $4$.
\Image[/chebyshev-prime-power-staircases.png]
{title=The <Chebyshev functions> separate at the extra <prime powers> $4$, $8$, and $9$}
{id=image-chebyshev-prime-power-staircases}
{description=The upper panel compares $\vartheta(x)$ and $\psi(x)$ for $0\leq x\leq10$. Solid endpoints include the jump at each contributing <integer>; open endpoints show the preceding value. The lower panel isolates $\psi(x)-\vartheta(x)$, which jumps by $\log2$ at $4$ and $8$ and by $\log3$ at $9$, reaching $\log12$. All logarithms are <natural logarithms>.}
Grouping the <prime powers> by their exponent gives, for $x\geq2$,
$$
\psi(x)=\sum_{k=1}^{\lfloor\log x/\log2\rfloor}\vartheta(x^{1/k}),
\qquad
\psi(x)-\vartheta(x)=\sum_{k=2}^{\lfloor\log x/\log2\rfloor}\vartheta(x^{1/k}).
$$
The <floor function> cutoff is exact: $p\geq2$ forces $2^k\leq x$ whenever $p^k\leq x$. For each exponent, $p^k\leq x$ is equivalent to $p\leq x^{1/k}$. An upper cutoff smaller than the lower one means an empty sum. <Elementary Number Theory/7.2: Chebyshev's Functions>{ref}
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