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Version 6 |
| There are two different functions which are collectively known as the \emph{Chebyshev functions}: |
There are two different functions which are collectively known as the \emph{Chebyshev functions}: |
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| \begin{align*} |
\begin{align*} |
| \vartheta(x)=\sum_{p\leq x}\log p. |
\vartheta(x)=\sum_{p\leq x}\log p. |
| \end{align*} |
\end{align*} |
| where the notation used indicates the summation over all positive primes $p$ less than or equal to $x$, and |
where the notation used indicates the summation over all positive primes $p$ less than or equal to $x$, and |
| \begin{align*} |
\begin{align*} |
| \psi(x)=\sum_{p\leq x}k\log p, |
\psi(x)=\sum_{p\leq x}k\log p, |
| \end{align*} |
\end{align*} |
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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, thee 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 $x$.
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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, these 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 $x$.
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| 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 |
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*} |
\begin{align*} |
| \prod_{p\leq n}p<4^n |
\prod_{p\leq n}p<4^n |
| \end{align*} |
\end{align*} |
| is equivalent to the statement that $\vartheta(x)<x\log 4$. |
is equivalent to the statement that $\vartheta(x)<x\log 4$. |
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A somewhat less innocuous result is that the prime number theorem (i.e., that $\pi(x)\sim \frac{x}{\log x}$) is equivalent to the statement that $\vartheta(x)\sim x$, which in turn, is equivalent to the statement that $\psi(x)\sim x$.
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A somewhat less innocuous result is that the prime number theorem (i.e. that $\pi(x)\sim \frac{x}{\log x}$) is equivalent to the statement that $\vartheta(x)\sim x$, which in turn, is equivalent to the statement that $\psi(x)\sim x$.
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{IR} Ireland, Kenneth and Rosen, Michael. A Classical Introduction to Modern Number Theory. Springer, 1998. |
\bibitem{IR} Ireland, Kenneth and Rosen, Michael. A Classical Introduction to Modern Number Theory. Springer, 1998. |
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| \bibitem{Na} Nathanson, Melvyn B. Elementary Methods in Number Theory. Springer, 2000. |
\bibitem{Na} Nathanson, Melvyn B. Elementary Methods in Number Theory. Springer, 2000. |
| \end{thebibliography} |
\end{thebibliography} |
