Content deleted Content added
Undid revision 1211972909 by Maximilian Janisch (talk)Reverted unexplained edit: unnecessary. Chebyshev function is already linked, as are the other functions similarly. |
→{{pi}}(x), Π(x), θ(x), ψ(x) – prime-counting functions: Un-bold ϑ(x), move wikilink to the adjacent words "Chebyshev functions", use vartheta symbol as is conventional and to match the formula immediately following. |
||
Line 145:
These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and are used in the various statements and proofs of the [[prime number theorem]]. They are summation functions (see the main section just below) of arithmetic functions which are neither multiplicative nor additive.
<math display="block">\pi(x) = \sum_{p \le x} 1</math>
A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes,
<math display="block">\Pi(x) = \sum_{p^k\le x}\frac{1}{k}.</math>
''
<math display="block">\vartheta(x)=\sum_{p\le x} \log p,</math>
<math display="block"> \psi(x) = \sum_{p^k\le x} \log p.</math>
The second Chebyshev function ''ψ''(''x'') is the summation function of the von Mangoldt function just below.
===Λ(''n'') – von Mangoldt function===
| |||