Difference between revisions of "Chebyshev theta function"
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The Chebyshev $\vartheta$ function is | The Chebyshev $\vartheta$ function is | ||
$$\vartheta(x) = \displaystyle\sum_{p \leq x} \log p,$$ | $$\vartheta(x) = \displaystyle\sum_{p \leq x} \log p,$$ | ||
| − | where $p \leq x$ denotes that $p$ is a prime number less than the real number $x$. | + | where $p \leq x$ denotes that $p$ is a prime number less than the real number $x$ and $\log$ denotes the [[logarithm]]. |
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
File:Chebyshevplotfrom0to50.png|Plot of $\vartheta$ on $[0,50]$. | File:Chebyshevplotfrom0to50.png|Plot of $\vartheta$ on $[0,50]$. | ||
| − | File: | + | File:Chebyshevplotfrom0to100.png|Plot of $\vartheta$ on $[0,100]$. |
| − | File: | + | File:Chebyshevplotfrom0to1000.png|Plot of $\vartheta$ on $[0,1000]$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
Latest revision as of 02:18, 28 November 2016
The Chebyshev $\vartheta$ function is $$\vartheta(x) = \displaystyle\sum_{p \leq x} \log p,$$ where $p \leq x$ denotes that $p$ is a prime number less than the real number $x$ and $\log$ denotes the logarithm.
Plot of $\vartheta$ on $[0,50]$.
Plot of $\vartheta$ on $[0,100]$.
Plot of $\vartheta$ on $[0,1000]$.
