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Chebyshev functions

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\begin{document}
There are two different functions which are collectively known as the \emph{Chebyshev functions}:

\begin{align*}
\vartheta(x)=\sum_{p\leq x}\log p.
\end{align*}
where the notation used indicates the summation over all positive primes $p$ less than or equal to $x$, and
\begin{align*}
\psi(x)=\sum_{p\leq x}k\log p,
\end{align*}
where the same summation notation is used and $k$ denotes the unique integer such that $p^k\leq x$ but $p^{k+1}>x$.  Heuristically, the first of these two functions \PMlinkescapetext{measures} the number of primes less than $x$ and the second does the same, but weighting each prime in accordance with their logarithmic relationship to $x$.

Many innocuous results in number \PMlinkescapetext{theory} owe their proof to a relatively \PMlinkescapetext{simple} analysis of the asymptotics of one or both of these functions.  For example, the fact that for any $n$, we have
\begin{align*}
\prod_{p\leq n}p<4^n
\end{align*}
is equivalent to the statement that $\vartheta(x)